Top Quark Production from Black Holes at the CERN LHC

LHC is expected to be a top quark factory. If the fundamental Planck scale is near a TeV, then we also expect the top quarks to be produced from black holes via Hawking radiation. In this paper we calculate the cross sections for top quark production from black holes at the LHC and compare it with the direct top quark cross section via parton fusion processes at next-to-next-to-leading order (NNLO). We find that the top quark production from black holes can be larger or smaller than the pQCD predictions at NNLO depending upon the Planck mass and black hole mass. Hence the observation of very high rates for massive particle production (top quarks, higgs or supersymmetry) at the LHC may be an useful signature for black hole production.


INTRODUCTION
Andrew Chamblin was a very good friend and a much valued collaborator-we greatly miss him. This paper was initiated by Andrew.
It is now generally accepted that the scale of quantum gravity could be as low as one TeV [1] and hence there can be graviton, radion and black hole production at LHC [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. If such processes occur then LHC collider experiments [19,20] can probe TeV scale quantum gravity. One of the most exciting aspects of this will be the production of black holes in particle accelerators. These 'brane-world' black holes will be our first window into the extra dimensions of space predicted by string theory, and required by the several brane-world scenarios that provide for a low energy Planck scale [21]. As the black hole masses at the LHC are relatively small (3)(4)(5)(6)(7) and the temperatures of the black holes are very high (∼ 1 TeV) the black holes can be a source for top quark production via Hawking radiation. In fact there can be an enormous amount of heavy (supersymmetry and higgs) particle production from black holes [22,23], much more than expected from normal pQCD processes. This comes about from two competing effects as the Planck scale increases: 1) top quark production from black holes increases because the temperature of the black holes increases as the Planck scale increases for fixed black hole masses (see below) and 2) the cross section for black hole production decreases [24,25,26,27]. Reviews of this exciting field are given in [28].
In this paper we calculate top quark production cross sections from TeV scale black hole at the LHC via Hawking radiation and compare them with the direct pQCD parton fusion processes at next-to-next-to-leading order (NNLO). We find that the top quark production cross sections from black holes at the LHC can be larger or smaller than those from pQCD processes at NNLO depending on the value of the TeV scale Planck mass and the black hole masses. We find that as long as the temperature of the black holes is of the order of TeV, the top quark production cross section from the black holes does not depend very much on the top quark mass M t . On the other hand the direct pQCD production cross section at NNLO is sensitive to M t . This provides us with an important conclusion: if TeV scale black holes are indeed formed at the LHC, then one signature of this will be an unusually copious production of massive (Top quarks, Higgs and SUSY) particles, which is not possible via pQCD processes. Hence if we observe very high rates for massive particle production at the LHC, this might provide indirect evidence that TeV scale black holes are being produced.
The paper is organized as follows. In Sec. II we present the computation for the rate of top quark production from black holes via Hawking radiation at the LHC. In Sec. III we sketch the pQCD techniques at NNLO for top quark production at the LHC. In Sec. IV we present and discuss our results.

II. TOP QUARK PRODUCTION FROM BLACK HOLES AT THE LHC
If black holes are formed at the LHC then they will quickly evaporate by emitting thermal Hawking radiation. The emission rate per unit time for top quark with momentum p = | p| and energy Q = p 2 + M 2 t can be written [14] as where σ s is the grey body factor and T BH is the black hole temperature, which depends on the number of extra dimensions and on the TeV scale Planck mass. c s is the multiplicity factor. The temperature of the black hole is given in [3], namely where R S is the Schwarzschild radius of the black hole, M P is the TeV scale Planck mass, M BH is the mass of the black hole and d is the number of extra dimensions. The grey body factor in the geometrical approximation is given by [24,25,26] where we take Γ s = 2 3 for spin half particles. The total number of top quarks emitted from the black holes is thus given by: where t f is the total time taken by the black hole to completely evaporate, which takes the form [4]: C depends on the extra dimensions and on the polarization degrees of freedom, etc. However, the complete determination of t f depends on the energy density present outside the black hole which is computed in [27] where the absorption of the quark-gluon plasma [29] by a TeV scale black hole at the LHC is considered (this time is typically about 10 −27 sec). The value we use throughout this paper is t f = 10 −3 fm which is the inverse of the TeV scale energy.
This result in Eq. (4) is for top quark emission from black holes of temperature T BH . To obtain the top quark production cross section from all black holes produced in proton-proton collisions at the LHC we need to multiply the black hole production cross section with the number of top quarks produced from a single black hole. The black hole production cross section σ BH in high energy hadronic collisions at zero impact parameter is given in [3,15], In this expression x a (x b ) is the longitudinal momentum fraction of the parton inside the hadron A(B) and We use µ = M BH as the scale at which the parton distribution functions are measured. ab represents the sum over all partonic contributions. The black hole production cross section in a binary partonic collision is given by [3]σ where d denotes the number of extra spatial dimensions. The total cross section for top quark production at LHC is then given by We will compare this cross section for top quark production via black hole resonances with the top quark cross section produced via pQCD processes at NNLO, as will be explained in the next section.

III. TOP QUARK PRODUCTION VIA PQCD PROCESSES AT THE LHC
The top quarks at LHC are mainly produced in tt pairs. At the LHC proton-proton collider, the QCD production process involves quark-antiquark and gluon-gluon fusion mecha-nism. The gluon-gluon fusion processes give the dominant cross section (about 90 percent).
This subprocess at high energy is the main reason for larger rate of the cross section compared to Tevatron at Fermilab. The single top quark production occurs via electroweak process. The single top quark production cross section (∼ 300 pb ) is smaller compared to tt total cross section (∼ 970 pb) at LHC at √ s =14 TeV pp collisions. Hence we will not consider the single top quark production cross section [30] in this paper. We will consider tt pair production using parton fusion processes at LHC and will compare them with the top quark production cross section from black holes.
At the next-to-next-to-leading order (NNLO) one needs to compute the following partonic subprocesses. On the leading-order (LO) level we have q +q → tt, g + g → tt.
In NLO we have in addition to the one-loop virtual corrections to the above reaction the following two-to-three body processes At NNLO level we receive the two-loop virtual corrections to the LO processes in eq. (9) and one-loop virtual corrections to NLO reactions in eq. (10). To these contribution one has to add the results obtained from the following two-to-four body reactions g + g → tt + g + g, g + g → tt + q +q, g + q(q) → tt + q(q) + g, q +q → tt + g + g, q +q → tt + q +q, q + q → tt + q + q,q +q → tt +q +q, After the phase space integrals has been done the partonic cross sectionσ is rendered finite by coupling constant renormalization, operator renormalization and the removal of collinear divergences. The renormalization scale µ R is set equal to the mass factorization scale µ F .
The cross section for top quark production in proton-proton collisions at the LHC is given by σ = a,b=q,q,g whereσ ab is the partonic level cross section for top quark production. For the details, see [31,32]. Reviews of present status of top quark physics at LHC can be found in [33].

IV. RESULTS AND DISCUSSIONS
In this section we will compute the top quark production cross section from black hole at √ s = 14 TeV in pp collisions and will compare them with the top quark production via parton fusion processes at NNLO. The top quark production from black holes is described in section II. For the black hole production we choose the factorization and normalization scale to be the mass of the black hole. As the temperature of the black hole at the LHC is ∼1 TeV there is not much difference in the top quark production cross section from black holes if the top quark mass M t is increased from 165 to 180 GeV. For black hole mass M BH much closer to the Planck mass M P the string corrections are important. In this situation string ball production becomes important [15]. For this reason we will choose black hole mass M BH to be larger than the Planck mass M P [15,22,23,27] in our computations below. In Fig. 1 we present the black hole production cross section at the LHC. The y-axis is the black hole production cross section σ BH in pb and the x-axis is the black hole mass M BH in TeV. The solid, dashed, dot-dashed and dotted curves are for Planck masses of 1, 2, 3 and 5 TeV respectively. The number of extra dimensions d = 4. As can be seen from the figure the cross sections decrease rapidly when both the Planck and black hole masses increase.
These black hole production cross sections will be multiplied with the number of top quarks produced from a single black hole to obtain the top quark production cross section from a black hole at the LHC. TeV in each case. The lower two lines are for black hole masses equal to 6 and 10 TeV respectively with the Planck mass equal to 2 TeV in each case.
In Fig. 2 we present results for the average number of top quarks produced from a single black hole as a function of top quark mass. The y-axis is the average number of top quark production from a single black hole and the x-axis is the mass of the top quark in GeV.
The upper two lines are for black hole masses equal to 3 and 5 TeV respectively with the Planck mass equal to 1 TeV in each case. The lower two lines are for black hole masses equal to 6 and 10 TeV respectively with the Planck mass equal to 2 TeV in each case.It is clear that the average number of top quark produced from a single black hole is much larger for smaller black hole mass. This is because as the mass of the black hole becomes smaller the temperature becomes larger (∼ TeV) and the thermal radiation of top quarks are enhanced. This is the case from a single black hole emission. The black hole production cross section itself decreases at LHC as the mass of the black hole increases. Hence the total cross section of top quark production from black holes at LHC is a competitive effect from the above two factors (see eq. (8)). In Fig.3 we present the total top quark production cross section from black hole production and compare them with the pQCD predictions at NNLO. The former is given for Planck mass M P = 1 TeV and for two different choices of the black hole mass, namely M BH = 3, 5 TeV respectively. We plot for comparison the NNLO top quark cross section from [32] with TeV. We also find that, unlike standard model predictions, the top quark production from black hole is not sensitive to the increase in top quark mass. Hence we suggest that the measurement of an increase in cross section for heavy particle (top quark or Higgs [23] or SUSY [22]) production at the LHC can be a useful signature for black hole production.
We make a brief comment about the grey body factor used in this paper. The grey body factor which is used in Eq. (3) is only valid in the regime of massless quanta and when the energy of the emitted particle is small compared to the black hole mass. Therefore, the computed cross section eq. (8) gives only an approximation to the actual cross section. Since this approximation improves for more massive black holes, our conclusions should remain valid. Finally, we warn the reader that the Planck mass MP = 1 TeV used in this paper is somewhat at odds with the constraints posed by LEP data on contact interactions [34] which suggests the Planck mass is greater than 2.2 TeV. If these constraints are true, then black hole production at the LHC, albeit exciting, may not lead to measurable contributions even if the large extra dimension scenario is realized.