Quantum black holes in the horizon quantum mechanics model at the Large Hadron Collider

Quantum black hole production at the Large Hadron Collider is investigated using the horizon quantum mechanics model. This model has novel implications for how black holes might be observed in collider experiments. Black hole production is predicted to be possible below the Planck scale, thus leading to the intriguing possibility that black holes could be produced even if the Planck scale is slightly above the collider centre of mass energy. In addition, the usual anticipated resonance in the black hole mass distribution is significantly widened in this model. For values of the Planck scale above the current lower limits, the shape of the black hole mass distribution is almost independent of the Planck scale and depends more on the number of extra dimensions. These model features suggest the need for alternative search strategies in collider experiments.


Introduction
Low-scale gravity provides an interesting possibility for gaining insight into the hierarchy problem. A wide variety of models based on different paradigms [1,2,3] have been proposed. A speculative, but intriguing, possibility of most models is the production of quantum black holes in hadron colliders [4,5].
The cross section for black hole production is typically chosen to be the classical geometric formσ ≈ πr 2 g , where r g is the gravitational radius which is a function of the black hole mass M and depends on the fundamental parameters of the model. In the large extra dimensions paradigm proposed in Ref. [1,2], the model parameters are the higher-dimensional Planck scale M D and total number of space-time dimensions D. We will consider the case of a tensionless non-rotating spherically symmetric solution for the gravitational radius [6].
In proton-proton collisions, only a fraction of the total centre of mass energy √ s is available in the hard-scatter process. We define sx a x b ≡ sτ ≡ŝ, where x a and x b are the fractional energies of the two colliding partons (assumed massless) relative to the proton energies. The full particle-level cross section σ is obtained from the parton-level cross sectionσ by using [7] σ pp→BH+X (s) = where a and b are the parton types in the two protons, and f a and f b are parton distribution functions (PDFs) for the proton. The sum is over all possible quark and gluon pairings. While several pre-factors to the cross section have been suggested (see Ref. [7] for a summary) they are not important for this study and will not be considered. The usual ansatz is that black holes can not be produced with M below some minimum mass threshold M th . This is emphasized by the use of the Heaviside step function Θ in Eq. (1). The value of M th is typically taken to be M D for quantum black holes or a few times M D for classical black holes. Unfortunately, results depend on the subjective choice of the M th cutoff.
A modification to the typical model of black hole formation in hadron colliders is made by the horizon quantum mechanics (HQM) model [8,9]. The HQM model makes a modification to black hole production by treating the source of the black hole and its horizon as individual quantum objects with their own wave functions. This serves to make the location of both the source and horizon fuzzy. The system exhibits properties of a black hole when the source is located within the quantized horizon, with the probability of the system being a black hole given by where P H (r H ) is the probability density of horizon radii r H and P S (r < r H ) gives the probability that the source is within r H . Explicit expressions of these probabilities are giving in Ref. [8,9]. Qualitatively, the use of the HQM probability in the calculation of the proton-proton cross section is akin to replacing the step function located at M th with a sigmoid-like function that varies with M/M D and depends on D.
The purpose of the work presented here is to evaluate the impact of the HQM model on the production of quantum black holes with emphasis on the signatures for experiments at the Large Hadron Collider (LHC). We begin with a brief description of Monte Carlo (MC) event generation in the HQM model, with more details of the implementation described in Appendix A. Possible reasons for differences between our results and a previously published result [10] can be found in Appendix B. We discuss the effects of HQM on the total proton-proton cross section and the differential proton-proton cross section as a function M . The possibility of quantum black hole detection in the HQM model in LHC experiments is discussed.
We make use of the following conventions. When comparing models, the QBH model refers to the quantum black hole model with Heaviside step function turn-on typically used by ATLAS and CMS searches at √ s of 7 TeV [11,12,13],8 TeV [14,15,16,17,18,19], and 13 TeV [20,21,22,23,24,25,26,27] that does not include any HQM effects. The HQM model will be the model with horizon quantum mechanics effects included. The only difference between these two models is their production turn-on behaviour in M/M D for different D. The total number of space-time dimensions D = n + 4, where n is the number of extra dimensions, unlike in Ref. [10], where D = n + 3 only represents the total number of spacial dimensions.

Black hole production probability
For the purpose of cross section calculations along with event generation, the Qbh 3.00 MC quantum black hole event generator 1 is used [28]. In this model [29,30,31], we consider tensionless non-rotating black holes. HQM effects are added to the proton-proton cross section by including the factor P BH of Eq.
where P BH requires another numerical integration. The cross section formula is now independent of M th and the model has one less free parameter. In order to visualise how the HQM probability varies with M, M D , and D, we have computed the integral in Eq. (2) explicitly, as shown in Fig. 1.
The probability curves suggest some interesting phenomena that are not seen in the QBH model. First, instead of a step function at M = M D , the new curves are smooth. The most notable consequence is that there is a finite probability that a black hole can be formed with M < M D . Second, we see that the probability for a black hole to be produced near M D is suppressed for high D. In other words, one generally expects more black holes to be produced for low D. This is at odds with the usual effect of dimensionality in the QBH model, where greater D corresponds to a greater geometric cross section. A third observation is that most of the curve is significantly above the value of M/M D = 1. And lastly, the slope in the curves at P BH = 0.5 are not particularly steep.
We can roughly quantify the extent to which the P BH curves create a threshold in the M distribution by considering the midpoint of each curve as the point where P BH = 0.5. These values are shown in Table 1. For D = 6, the black hole mass threshold rises to slightly above the usual M D threshold in the QBH model. For D = 10, the threshold is more than twice M D . This means that more dimensions will cause heavy suppression of black hole production in the HQM model, unlike the QBH model in which more black holes will be produced at higher D. 3 Proton-proton total cross section We start by analyzing how the inclusion of HQM impacts the proton-proton total cross section as a function of M D and D. There are two competing factors at play. On one hand, we are multiplying the parton-level cross section by a factor between 0 and 1, which in general decreases the cross section. On the other hand, we are considering a wider range of possible M than in the QBH model. In addition, while it is unreasonable to think of producing events in the QBH model if M D > √ s, the smooth cutoff imposed by HQM allows for black holes when M D is above the collider energy. The phenomena are shown in Fig. 2.
The inclusion of HQM suppresses the total cross section for low M D but predicts a higher cross section than the QBH model at high M D . It is also interesting to note how the role of dimensionality is reversed in the two models. For a given M D , higher cross sections occur at lower D in the HQM model, except for a small region below about 2 TeV. Also, in the HQM model the cross section at a given M D is significantly different for different D as M D increase. Thus over most of the M D range, dimensionality is significantly more important in the HQM model. It is also useful to determine the M D value at which the HQM model cross section crosses over the QBH model cross section, and thus where the HQM model might become more significant. For D = 6, D = 8, and D = 10, the crossovers in M D occur at approximately 5.4 TeV, 8.2 TeV, and 9.7 TeV, respectively. To understand which region of M D is interesting, we consider the current lower-limits, at the 95% confidence level, on M D of 9.9 TeV, 6.3 TeV, and 5.3 TeV for D = 6, D = 8, and D = 10, respectively, set by the CMS [32] and ATLAS [33] experiments. At these M D limits, black hole production in the HQM model is still well below the QBH model except for D = 6 where the HQM model predicts a cross section of about three orders of magnitude higher than the QBH model.
The lower limits on M D are based on graviton searches in the same large extra dimensions paradigm [1,2] as used for black hole models, and we thus take them to be applicable to both the QBH and HQM models considered here. Of particular importance for observing quantum black holes in experiments is the number of black hole events we are able to produce. Typically, a minimum of ten signal events is sought to form a reasonable claim of discovery 2 In Fig. 3, we plot the luminosity required to produce ten events in proton-proton collisions at √ s = 13 TeV. Analysis performed by ATLAS and CMS using the full run-2 dataset typically quote a luminosity of about 139 fb −1 . Using this luminosity, more than ten events can be produced in the QBH model for M D less than about 8.7 TeV, 9.2 TeV, and 9.5 TeV for D = 6, D = 8, and D = 10, respectively. The lower limits on M D would exclude D = 6 black holes in the QBH model. The current best lower limit from a direct QBH search is M th = M D > 9.4 TeV for D = 10 [27]. Even with a luminosity of 1 ab at √ s = 13 TeV, the limit on M th is unlikely to go above about 10.5 TeV. Thus, the QBH model is being significantly restricted even at current luminosities.
The LHC is able to produce black holes at much higher values of M D in the HQM model for most D. At a current luminosity of 139 fb −1 , values of M D in the HQM model are not constrained by the lower limits on M D , and quantum black holes could exist in the LHC experiment's current datasets. However, as we will see next it will be non-trivial to detect HQM black holes in current ATLAS and CMS datasets even if produced.

Proton-proton differential cross section
The inclusion of HQM in quantum black hole production has notable implications on the M distribution of black holes. Since the cross sections of QBH and HQM models typically differ For a small M D , the HQM model gives the peak structure of the QBH model, but this changes for higher M D , and M is distributed over a wide range: 2 M 10 TeV. This difference in shape is a direct consequence of the shapes of the PDFs and the P BH curve from HQM. The PDFs fall rapidly as parton energies approach √ s/2. For M D = 12 TeV in the QBH model, a very small cross section is expected since M is limited to the range 12 < M < 13 TeV. In the M D = 12 TeV HQM model, the lower mass for black holes is dictated by the P BH curve. Black hole masses below 2 TeV are suppressed since P BH ≈ 0, and likewise black holes with mass above about 10 TeV are suppressed by the PDFs. This interplay in the HQM model between the convolution of PDFs and P BH gives rise to the shape of the M distributions.
The peak in the QBH M distribution moves up with increasing M D since the model's definition of M th is a strict cutoff in M . In contrast, the HQM model M distribution does not appear to shift up much above M D 7 TeV. This phenomena is explored further in Fig. 5. While the QBH model M distribution moves up with increasing M D acting as a minimum mass threshold, the HQM model M distributions are much more spread out and the shape of the distributions do not change significantly once M D exceeds a few TeV. We also observe that in the HQM model it is very difficult to produce black hole masses above ∼ 11 TeV, even though M D is not limited. TeV. The value of the mean M to which the trend converges is dependent on D. The reason for this is an interplay between the P BH curves which approach zero as M approaches zero and the PDFs which approach zero as M approaches √ s. The consequence is a "pinching off" that serves to create a mass distribution that does not change shape significantly between the two mass regions where the production of black holes is vanishingly small. The mean M increases with D due to the P BH curve being shifting higher in M/M D with increasing D, as previously shown in Fig. 1.
Finally, the shape of the HQM model M distribution has implications on how black holes in this model may be detected in the ATLAS and CMS experiments. In the QBH model, black holes are expected to predominantly decay into two-body final states. The majority of these decay products would be quarks and gluons that would hadronize to produce jets. For  Quantum black hole events are simulated using the same selection criteria, at the particle level, as in the ATLAS analysis. We understand that particle-level selection will only roughly emulate the geometrical acceptance of events in the the ATLAS detector, but the signal yields should be indicative of a full experimental analysis. 4 Figure 7a) shows an example QBH resonance for M D = 9.5 TeV and D = 10. This resonance is beyond the highest dijet mass event obtained by ATLAS. In addition, the decisive lack of such a resonance structure in the dijet mass spectrum has allowed ATLAS to limit black holes in the QBH model to M th > 9.4 TeV for D = 10 at the 95% confidence level [27]. Thus, the QBH model in its simplest form is close to being ruled out.
For the HQM model, dijet distributions are shown in Fig. 7b), Fig. 7c

Discovery potential in the dijet mass distribution
In order to predict the discovery potential for observing quantum black holes, we take into consideration both the number of events above background and the significance of the signal. For the significance, we use the gaussian approximation without uncertainty: where "signal" is the number of signal events above background and "background" is the number of background events excluding signal events. We understand that this formula will break down with small number of events, and that we should really include background uncertainties. However, such an analysis is beyond the scope of this work, and is unlikely to change the qualitative findings. We consider a significant observation to be greater than 5σ. Using a cut-and-count method, significance is calculated by counting events above M th . While this is natural for the QBH model, it is perhaps not so meaningful for the HQM model since many of the events have M < M D . For the sake of comparison, we consider two approaches to calculating the significance for the HQM model. The first is the usual definition, where we consider M D as a cutoff. In this method M D values beyond √ s can not be probed. In the second method, we consider all black hole events and count the background from the least massive signal event. We understand that the latter method would be extremely difficult, and probably not even desirable, to realize in an experiment's analysis, but it might be more indicative of a shape-fit procedure that might likely be used.
The event count and significance are presented in Fig. 8 and Fig. 9, respectively. While counting HQM model events over the entire mass range gives the greater number of events, the method of counting HQM model events only above M D give better significances. This could have been anticipated given the large number of background events at low dijet masses. Using either approach to calculating the significance, the discovery potential at allowed values M D is less for the HQM model than the QBH model. Since the ATLAS background that we are using does not extend beyond 8.1 TeV, and because of simple significance formula Eq. (4), the significance curves in Fig. 9 end at M D = 8 TeV.
Using the M > M D counting method and by noting the minimum M D value given by the ten event and 5σ criteria, we assess the possibility of detecting HQM black holes in ATLAS and CMS. For D = 10, the number of signal events is greater than ten for M D 7.5 TeV. The corresponding significance is greater than 5σ for M D 7.4 TeV, and this sets the upper limit on M D to observe black holes in the HQM model. For the D = 6 case, greater than ten events occurs when M D 8.0 TeV and the significance is greater than 5σ at M D 8.0 TeV. However, with only one background event, the significance as defined in Eq. (4) does not have have meaning. In any case, the lower limit on M D from the CMS experiment [32] for D = 6 is 9.9 TeV at the 95% confidence level, thus eliminating the HQM model for D = 6.
Given the increase in luminosity and √ s in subsequent LHC runs, these discovery potentials stand to increase somewhat. With this thought in mind, we make some predictions at √ s = 13 TeV on the luminosity required at a given M D for a meaningful discovery. We assume that the number of background events, based on the data from Ref. [27], scales linearly with luminosity. When calculating the significance using M > M D as a cutoff in the cut-and-count method, we have made the additional assumption that event-count is the limiting factor for M D > 8 TeV as this is the highest dijet mass at which the ATLAS background estimate is given. The results are shown in Fig. 10 where we only consider luminosities above 139 fb −1 . It is seen that the increase in probing M D with a reasonable increase in luminosity is not very significant, indicating that we are close to exhausting the search for black holes in both QBH and HQM models using the dijet mass distribution at √ s = 13 TeV. Although we have used a very simplistic approach to estimating the discovery potential, this conclusion is unlikely to change with a more robust estimate.

Angular search
Given the small potential for observation of HQM black holes in the dijet mass distribution, a discovery in the invariant mass variable is unlikely. Alternatively, an angular search may be   performed to distinguish an enhancement of events due to black hole production above QCD background [11,12,20,22,23]. The HQM model does not yet predict any modification to the usual decays in the QBH model; there is no difference between the two models in terms of the shape of angular distributions.
An example angular search could be in the variable χ defined as where y 1 and y 2 are the rapidities of the two jets. QCD t-channel scattering constituting the background is approximately constant in χ, while s-channel resonances tend to be enhanced at low χ. Because of this, an angular search could help uncover the wide s-channel mass enhancement that is predicted by HQM. Since the predictions of an angular search are highly dependent on the analysis and detector details, we leave it to the ATLAS and CMS collaborations to perform such a search.

Conclusions
Microscopic black hole formation as predicted by HQM was implemented in the Qbh MC event generator to investigate the impact on possible black hole production at the LHC. The inclusion of the HQM model serves to decrease the total black hole cross section for small M D , but the new model is not restricted by a threshold mass requirement. Therefore, HQM predicts black holes may be produced at M D ∼ √ s. The HQM model is also highly dependent on dimensionality and predicts that a greater number of events may be produced with a smaller D. The M distribution is also greatly affected by HQM with a much wider spread of black hole masses. In other words, there is no resonance structure in the HQM model. This wide M distribution converges to a constant shape for large M D , which can be considered to be one of the defining features of the HQM model.
The predicted signal in the dijet mass distribution along with ATLAS run-2 data were used to estimate the number of signal events and significance. Observations of quantum black holes governed by HQM were predicted to be limited to M D 8.0 TeV for D = 6 and M D 7.2 TeV for D = 10. Most likely the mass spread of HQM black holes is too large to allow observation using a resonance search alone, and an angular search is an exciting possibility. Some of the above results were first mentioned in Ref. [10]. Unfortunately, that paper could only make use of ATLAS and CMS results at a centre of mass energy of 8 TeV. We view our analysis as more comprehensive, close to experiment, and up to date.
Lastly, although the HQM model has been used, we do not believe the results presented here depend specifically on the formula presented in Ref. [9]; similar results would be obtained for any non-step-like threshold mass production of black holes.

Acknowledgments
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

A Monte Carlo event generation
In order to visualise how the HQM probability P BH varies with D, M D , and M , we computed the integral in Eq. (2) explicitly using numerical integration. As shown in Fig. 1, good accuracy was achieved with the use of Simpson's method and an adequate large number of subdivisions.
A more elegant means of producing the appropriate P BH factor can be performed by MC integration. As a check, we have also produced the curves in Fig. 1 using MC sampling. By integrating and inverting the P H distribution, random values of the horizon radius r H can be sampled using a uniform distribution of random numbers. Since P H is a probability density function, using random r H values to calculate P S for a large number of samples effectively computes the expected value for P S (or equivalently, P BH ). For completeness, we present this calculation.
We begin from Eq 3.7 in Ref. [9]: where in this appendix we use the notion of Ref. [9] except we take the total number of spatial dimensions to be d. We have define a = (d − 2)/(2m) and s = d/[2(d − 2)], and used ∆ = m as in Ref [9]; Γ(s, x) = ∞ x t s−1 e −t dt is the upper incomplete Gamma function. In addition, we are using Planck units since we are only interested in lengths and masses relative to M D .
By taking the Heaviside step function to be one, the indefinite integral can be computed: Substituting a lower limit of R d = [2m/(d − 2)] 1/(d−2) and upper limit of r H , to allow calculation of the cumulative distribution function, gives If we generate a uniform random real number u in the interval (0, 1) [or 1 − u in the interval (1, 0)] and set it equal to Eq. (8), we can solve for r H by inverting the incomplete Gamma function with respect to its second parameter: Note that Q −1 (s, Q(s, x)) = x, where Q −1 is the inverse of the regularized upper incomplete Gamma function Q(s, x) = Γ(s, x)/Γ(s). There are numerical methods to optimise this inversion. Upon randomly sampling the horizon radii from Eq. (9), we return values of P S (r < r H ) as given by Eq. 3.5 in Ref. [9]: where γ(s, x) = x 0 t s−1 e −t dt is the lower incomplete Gamma function. The above random horizon generation can simply be looped over with an average of all P S values giving an approximate value for P BH . We easily recreate the same probability curves as in Fig. 1 which used Simpson's method.
Both the MC method and Simpson's method for calculating P BH have been implemented in Qbh. Despite both methods producing the same results, there are technical pros and cons of each method. The MC HQM calculation just presented is the default method.
One additional technicality should be mentioned. Since black hole production in the HQM model allows for M less than M D there is no lower-mass cutoff in the generator. Instead, the P BH curve imposes its own smooth limit as it becomes arbitrarily small. To sample M via a power transformation of the cross section used to increase efficiency, we choose an arbitrary minimum of 100 GeV since in practise it is exceedingly rare to generate an event with M this low. For example, selecting a 200 GeV minimum has a negligible impact on the results.
We point out that our curves of P BH are identical to the corresponding figure in Ref. [10] within our ability to read values from their figure. Equation (7) in Ref. [10] disagrees with Eq. (3.8) Ref. [9], although the later cites the former. We believe Eq. (7) in Ref. [10] has the inverse power of (m d /m) and a normalization difference of (D − 2) 2 . If the formula in the paper was actually use to generate the plot, the curves continue to increase above one with increasing mass and do not represent probability distributions.

B Comparison of Qbh and BlackMax event generators
A first HQM study was made in Ref. [10] and the discrepancy between their results and those presented here should be addressed. In Ref. [10], the black hole event generator BlackMax [34] was used. It is designed using the same cross section formula as Qbh, with some additional prefactors, so we expect approximate agreement between the two results. However, it is consistently found that cross sections reported in Ref. [10] are approximately an order of magnitude larger than we obtain with Qbh.
We present two plausible reasons for the discrepancy. The first is the event generator parameters. Results were produced using BlackMax 2.02.0 in the "standard configuration". We interpret this to mean the set of default parameters included in the BlackMax 2.02.0 tarball, changing only the options absolutely required for the simulation at hand (i.e. √ s, M D , and D).
Additionally, no PDF set was specified in Ref. [10]. While we used the CTEQ6L1 [35] PDF set in this analysis, a different PDF set could give different results. An alternative explanation is that some parameters are misreported. BlackMax accepts a parameter for the number of extra dimensions n, which is equivalent to D = 4 + n in our convention. Erroneously setting n = D (or n = d) as input to BlackMax reproduces the results reported in Ref. [9] closer, which suggest the difference may be superficial.
As a sanity check, we embarked on a more complete comparison between BlackMax and Qbh to ensure our results are consistent both with and without the HQM modifications. In summary, we can get good agreement between the two generators after accounting for small differences between models. This suggest the results in Ref. [10] are misreported, or possibly even erroneous.
First, the CTEQ6L1 PDF set was chosen for both Qbh and BlackMax. Both generators can be linked to LHAPDF 6.2.1 [36] as version 6 contains support for the legacy interfaces used by BlackMax. We do not build BlackMax with Pythia as it is not needed for our comparisons. BlackMax can be built using a modern version of gcc (we used 5.4) provided the necessary Fortran libraries are included in the system. Some depreciated compiler flags may need to be removed depending on the particular compiler version. The parameters that need to be changed (in the parameter.txt file of the BlackMax distribution) in order to bring Qbh and BlackMax into good agreement are as follows: 1. The Choose_a_case option should be set to 1 to consider tensionless non-rotating black holes.
2. The choose_a_pdf_file option should be set to 10042 to use the CTEQ6L1 PDF set.
3. The other_definition_of_cross_section option should be set to 2. By default Black-Max uses an angular momentum form factor which Qbh does not.
4. The Mass_loss_factor, momentum_loss_factor, and Angular_momentum_loss_factor should be set to 0.0 in BlackMax since these options are not considered in Qbh.

The
Qbh member function qbh->setQscale(false) should be called in the user-defined main.cc file. This uses the black hole mass as the QCD scale. This scale is hardcoded in BlackMax.
6. Comment out line 456 in the default BlackMax.c source file. This line modifies the differential cross section by accounting for the position of the black hole with respect to the extra dimensions. Qbh does not include this feature.
7. To include HQM modifications in BlackMax, we read in the P BH values from a file. As BlackMax is written in C, we do not have access to the Boost gamma function library that we use in Qbh for calculating P BH . Since we know that both presented methods of including P BH in Qbh are identical, we simply read the nearest-mass value from a pre-calculated table into BlackMax in order to modify the parton-level cross section.
Reading in values is done at the beginning of the main function into a global array. The cross section is modified within the "cross section by Monte Carlo integrals" section of BlackMax.c. The variables of interest are: Mb the randomly generated black hole mass; Mpl the Planck scale; and fact the parton-level cross section factor. The modification is made after the fact=-2*u*Mb*A1*Lx0 line in the BlackMax source code.
With these modifications there is good agreement between the generators. The only major discrepancy is between the non-HQM BlackMax and Qbh cross sections when M D 11 TeV.
Here, the Qbh cross section is about 2% higher than the BlackMax value.
It is important to note that this difference is not seen in the HQM calculation, suggesting that it is likely a result of the rising minimum mass value. The precise reason for this remains unknown, but we will outline what has been tried. First, Qbh and BlackMax use different power transforms when sampling masses. Manipulating the exact transform does not produce a large change in the calculation. Second, BlackMax and Qbh include the PDF samples in the proton-level cross section in slightly different ways. The consequence of manipulating the methodology was also found to be negligible. The only facet of the calculation which was not subject to intense scrutiny was the process of sampling of the PDFs. While both generators use the same CTEQ6L1 PDFs with LHAPDF 6.2.1, Qbh uses the LHAPDF 6 interface while BlackMax uses the legacy Fortran interface. Furthermore, the control flow for evaulation of the PDFs is significantly different. This suggests that the remaining discrepancy may be a consequence of this. If further tweaking of the generators is required, we suggest the investigation resumes here.
In summary, aside from a difference in cross sections at higher M D in the non-HQM case, there is good agreement between the generators. Importantly, this difference is only a matter of a small percent uncertainty as opposed to an order of magnitude disagreement seen in Ref. [10]. This gives us confidence in the implementation of our modified Qbh generator.