Charged Lepton Spectra from Hot-Spot Evaporation

Julian Moosmann and Ralf Hofmann, Charged Lepton Spectra from Hot-Spot Evaporation, ISRN High Energy Physics 2012, article ID 509793, 2012. Reproduced under the Creative Commons Attribution licence stated in the article, which permits unrestricted use, distribution and reproduction in any medium provided the original work is properly cited; the published version is at doi.org/10.5402/2012/509793. Equations are described in words and identified by the number they carry in the source; the reference-number markers have been removed and the figures are described rather than shown.

(On this site, the companion papers in Ralf Hofmann’s SU(2) programme are the thermal ground state and its nonthermal probes at /library/stm-9a9190b65f, the stability analysis of centre-vortex loops with one self-intersection at /library/stm-e1f7b72444, and the extension of the same construction to galaxies and the dark universe at /library/stm-b06591e288.)

Abstract

Spectra for the emission of charged leptons from evaporating hot-spots of preconfining phase in SU(2) Yang-Mills thermodynamics are computed. Specifically, we consider charged single and dileptons with their spectra being functions of energy and invariant mass, respectively. In the former case, our results relate to narrow and correlated electron and positron peaks measured in supercritical heavy-ion collisions performed at GSI in the 1980s. In the latter case, we point out how strongly the spectra depend on typical kinematic cuts (CDF analysis of Tevatron Run II data). We also propose a scenario on how muon events of anomalously high multiplicity and large impact-parameter modulus arise in the Tevatron data.

1. Introduction

Pure SU(2) Yang-Mills thermodynamics possesses a narrow preconfining phase characterized by condensed monopole-antimonopole pairs. The latter originate as isolated defects through caloron and anticaloron dissociation in the deconfining phase at higher temperatures. Shortly above the critical temperature for the Hagedorn transition towards the confining phase, the preconfining thermodynamics of the spatially infinitely extended gauge-theory system is dominated by this thermal ground state. The only propagating massive, dual gauge mode decouples, that is, acquires a very large Meissner mass. This domination leads to negative pressure of order minus the fourth power of the Hagedorn temperature. Once temperature falls below that value, the infinitely extended monopole-antimonopole condensate decays nonthermally into single and self-intersecting center-vortex loops which are interpreted as spin one-half fermions. Only onefold self-intersecting center-vortex loops are absolutely stable in an SU(2) Yang-Mills theory. Thus it is these fermions that are associated with charged leptons and their antiparticles.

As a consequence, there is one SU(2) Yang-Mills theory for each lepton family with the respective Yang-Mills scale matching the mass of the charged lepton: the electron scale of order the electron mass, the muon scale of order the muon mass, the tau scale of order the tau mass. This scenario, although at first sight ruled out by the apparent experimentally inferred pointlikeness of charged leptons, has predictive power under conditions where the creation of large-sized hot spots of preconfining ground state is feasible. These conditions include very high energies in particle collisions, high temperatures in extended systems, and local energy densities exceeding the fourth power of the electron mass, about ten to the eleventh in units of keV to the fourth.

Recall that, according to SU(2) Yang-Mills thermodynamics, the apparent pointlikeness of leptons when probed by high-momentum transfer standard-model processes is a consequence of the presence of a large number of unstable excitations in the interaction region. Their combined effect together with the free shiftability of vortex intersection points is efficiently and successfully described perturbatively by the standard model. In this context, a succession of vertices and off-shell propagators, that is Feynman diagrams, mediates the transformation of structureless initial into the according final states.

Recently, anomalous, high-multiplicity muon events of symmetrically, positive-negative, distributed and power-like decaying, as opposed to exponentially cut-off, rate as a function of impact factor were reported to occur in high-energy particle collisions. These data are generated by protons and antiprotons colliding head-on at a centre-of-mass energy of 1.96 TeV, Tevatron Run II. The invariant mass spectrum of dimuons created outside the vacuum of the beam pipe exhibits anomalous structure in the GeV region. Also, taking an additional muon into account, the according experimental rate is not explicable by any known standard model processes. We are aware of the fact that the experimental situation in the case of anomalous multimuon events produced at the Tevatron as of yet is not clearly cut and contested. Motivated by existing experiments, our theoretical dimuon spectra, obtained by investigating hot-spot evaporation in the muon theory under certain kinematic cuts, are intended to offer guidelines for future analysis of collider data in the TeV realm.

We also consider single-particle spectra for positron and electron emission from evaporating hot spots of the electron theory. Here the motivation arises due to the observation of universal narrow-peak structures in supercritical heavy-ion collisions at the Gesellschaft fur Schwerionenforschung Darmstadt more than two decades ago.

1.1. Experimental situation: multilepton events

1.1.1. Electron and positron emission in supercritical heavy-ion collisions (GSI)

In supercritical heavy-ion collision systems such as uranium on thorium, thorium on thorium, thorium on curium, uranium on uranium and uranium on curium, of combined nuclear charge between 180 and 188 and thus well above criticality at 173, the emission of narrow positron peaks of several tens of keV width and centre at 250 keV, 330 keV, and 400 keV was detected. These narrow peaks sit on a broad background which is explained by conventional, parameter-free theory. Later the correlated emission of narrow peaks of electrons with similar characteristics was observed. Both signatures decisively were ruled out to be due to any nuclear reactions when bombarding energies near the Coulomb barrier are applied. The emergence of these narrow peaks was theoretically approached by postulating large delay times of the order 3 times ten to the minus twenty-first of a second which, however, are hard to reconcile with the smaller average delay time in deep inelastic reactions.

The width of the narrow electron and positron peaks in sum energy is narrower than that of the single positron lines. Also, from an analysis of Doppler broadening, one concludes that the emitting source moves very slowly, at most five hundredths of the speed of light. At the same time, compared to the situation of a weak constraint on opening angle, between 40 and 170 degrees, no sizable detector activity was seen in the search for correlated back-to-back emission of electrons and positrons. The combination of the latter two signatures excludes the pair, that is two-particle, decay of a neutral particle state in the mass range between 1.5 MeV and 2 MeV — which from an analysis of the invariant mass spectrum alone would not be excluded.

Furthermore, the prediction of a strong dependence of peak energies and intensities on the combined nuclear charge, derived from the hypothesis of a long-lived, giant dinuclear complex needed for spontaneous pair creation, does not match with the experimental data. Rather, the data point to an independence of the peak features on that charge. This universality is a puzzle in conventional strong-field QED which predicts strong power laws in it.

Considering all signatures mentioned above — large delay times, pointing to the formation of spatial regions with negative pressure and hence to the existence of droplets of preconfining ground state of an SU(2) Yang-Mills theory whose Hagedorn temperature is of order the electron mass, 511 keV; no pair decay but still a heavy neutral primary, pointing to droplets of energy content several times the electron mass evaporating isotropically; universality of peak energies and widths in single positron and pair spectra, pointing again to a new phase of matter, preconfining-phase droplets whose decay is insensitive to the physics of their creation — we tend to attribute them to the formation and subsequent evaporation of preconfining-phase hot spots of the electron theory, whose one-particle thermal emission spectrum is simply computed in section 2.

Let us now estimate the typical energy density associated with the Coulomb fields at closest approach of the two nuclei. Notice that the spontaneous creation of positrons in strong-field QED considered in explaining the anomalously high yield requires electronic binding energies of about minus twice the electron mass. Since the combined charge times the fine-structure constant is of order one, the associated Bohr radius is of order the inverse of that binding energy, and the maximal energy density in a supercritical heavy-ion collision is of order the inverse fourth power of that radius, about ten times the fourth power of the electron mass — which is larger, but not hierarchically larger, than what is minimally needed to traverse the Hagedorn transition in the electron theory. Thus preconfining-phase hot spots likely are generated in the form of small droplets, of spatial extent comparable with that Bohr radius.

In connection with the smallness of the droplets, an important question concerns the narrowness of the observed peaks. Notice that the observed width of the electron and positron peaks corresponds to a lifetime of about ten to the minus twentieth of a second. This, however, is the predicted lifetime for preconfining-phase hot-spot evaporation in the electron theory, with a slow dependence on the deposited energy. If the energy deposited during the formation of a hot spot is larger but comparable to the peak energy of the single-particle spectrum for thermal electron or positron emission, then the hot spot and its decay signatures must be interpreted quantum mechanically. The droplet’s thus computed lifetime, however, still should coincide with the inverse width seen in the experimental decay spectra. From the fact that the observed peaks are narrow in contrast to what is seen in the computed spectra, we would again conclude that the experimental hot spots are small droplets with a position uncertainty characterized by typical recoils; that is, their mass is only several times larger than the electron mass. This is compatible with the argument on available energy density for hot-spot creation given above.

For the case where the deposited energy in hot-spot creation would exceed the electron mass by many orders of magnitude, the observed peaks should broaden, and asymptotically the experimental one-particle spectra should match those computed in section 2.

1.1.2. Multimuon events at large impact parameter (Tevatron Run II)

For anomalous multimuon events ignited by proton-antiproton primary collisions at a centre-of-mass energy of 1.96 TeV, Tevatron Run II, total integrated luminosity 2100 inverse picobarns, and analyzed by the CDF collaboration — however, not verified by D0, which seems to apply a much tighter constraint on the range of investigated impact parameter — we are interested in the large number of dimuon events originating outside the beam pipe. Namely, an anomalous rate of dimuons, nearly independent of the sign of the impact parameter, was detected for an impact-parameter modulus greater than the beam-pipe radius of 1.5 centimetres. Notice that known QCD processes account for the dimuons created within the vacuum of the beam pipe, while the counts for dimuons originating from vertices outside it are in excess of standard model predictions.

The motivation and precise implementation philosophy of that analysis in imposing certain kinematic cuts during data analysis remains largely obscure to the present authors. In part this is explicable by a profound lack of experience in addressing the complex data situation posed by TeV range primary collisions. Nevertheless, we would like to point out what our evaporating hot-spot model predicts about the shape of the dimuon invariant mass spectrum at large impact-parameter modulus when applying typical kinematic constraints chosen by the experimentalists.

In that analysis a subset of the Run II data, subject to certain kinematic constraints in transverse momentum and opening angle, with integrated luminosity now only 742 inverse picobarns, was analyzed, and so-called ghost events were isolated. These events share the following features: neither the approximate symmetry under reversal of the sign of the impact parameter and the decay of the rate distribution of multimuon events with increasing impact-parameter modulus, nor the large magnitude of the rate of events containing two or more muons outside the beam pipe, nor the distribution of rate for a given charge composition of dimuon events in invariant mass, is explained by known standard model processes. Probably depending on the imposed kinematic cuts, a transverse momentum of at least 2 GeV, the measured invariant mass spectra of dimuons at large impact-parameter modulus typically have large weight in the GeV region. To link this to our muon-theory approach, we will in section 3 present a number of theoretical results on dimuon spectra based on a single evaporating hot spot when varying kinematic cuts.

Since such a calculation unrealistically assumes an impact-parameter independence of the process creating the hot spot, and since it ignores the physics of prerequisite processes, neither the impact-parameter dependence nor the overall normalization of the dimuon spectra is predicted here. For a prediction of these two features, additional deliberations and assumptions must be made. It is clear, however, that a preconfining hot spot of the muon theory with a total mass of about 1 TeV is almost at rest and thus emits muons isotropically, explaining the observed approximate symmetry under sign reversal of the impact parameter in the distribution of the experimental yields: emission towards the beam pipe is as likely as away from it at a given distance.

We tend to believe that the reason why QCD predictions and experimental data agree well for events originating inside the beam pipe is the absence of heavy nuclei whose strong electric fields trigger hot-spot creation assisted by the rare TeV photons of electromagnetic proton-antiproton annihilation. Outside the beam pipe, the detector material provides for these nuclei. Assuming the probability for the conversion of a TeV photon into a muon-theory hot spot inside the Coulomb potential of a heavy nucleus to be sufficiently small, the almost unadulterated total photon flux decreases geometrically in a power-like way as a function of the impact parameter while the evaporation physics is independent of it. This would explain the slow, power-like decay of multimuon activity at large impact-parameter modulus.

This paper is organized as follows. In the next section we compute the spectrum for emission of single charged leptons, arising from evaporation of static SU(2) hot spots, as a function of lepton energy. Specifically for the electron theory, we also determine hot-spot lifetimes in dependence of the deposited energy and observe agreement with the inverse width of the narrow peaks detected at GSI. In section 3 we compute the spectra for the emission of charged dileptons from the evaporation of static SU(2) hot spots as a function of the pair’s invariant mass and under various kinematic cuts. As a result, these spectra sensitively depend on the kinematic cuts applied. Section 4 discusses our results in view of present and future analysis of TeV-range collider data.

2. Single-lepton spectrum from hot-spot evaporation

For temperatures just above the Hagedorn temperature, the SU(2) Yang-Mills system is ground-state dominated. Once the creation of a bubble of some initial radius containing this new phase of matter has taken place, its evaporation is simply determined by the content of stable, final particle species — the number of self-intersections in the center-vortex loop, spin, and charge — by energy conservation, and by the value of the Hagedorn temperature.

Since we are interested in leptons far away from the emitting hot-spot surface, we can address the problem of calculating their spectra thermodynamically, setting the temperature equal to the Hagedorn temperature, which is 11.24 divided by two pi, times an order-unity factor y expressing a theoretical uncertainty, times the mass of the charged lepton state.

Leaving the angular integration explicit and taking the singly self-intersecting state, equation 2.1 in the source gives the number of charged leptons emitted per unit time and surface as the integral over momentum and over the two angles of the following quantity: a degeneracy factor of four — twofold for spin and twofold for charge — divided by eight pi cubed times the on-shell energy, multiplied by the cubed momentum divided by one plus the exponential of the on-shell energy over the Hagedorn temperature. Here the momentum is the modulus of the spatial on-shell momentum of a single charged lepton far away from the emitting surface, and the lepton mass can be set equal to 511 keV for the electron theory and 105.6 MeV for the muon theory.

Equation 2.2 in the source gives the differential yield per unit on-shell momentum, time and surface as two over pi squared, times the cubed momentum divided by the on-shell energy, divided by one plus the exponential of the on-shell energy over the Hagedorn temperature. Equation 2.3 in the source gives it instead per unit on-shell energy: two over pi squared, times the squared energy minus the squared lepton mass, divided by one plus the exponential of the energy over the Hagedorn temperature.

To obtain the differential number of emitted charged leptons per bin of energy, equation 2.4 in the source multiplies that differential yield by the actual surface of the spherical hot spot, four pi times the squared radius, and integrates over time from zero to the evaporation time.

Equation 2.5 in the source gives the shrinking radius: the initial radius minus the emission current at the Hagedorn temperature divided by the energy density of that phase, times the elapsed time. Equation 2.6 in the source gives the evaporation time as that energy density divided by the emission current, times the initial radius.

Equation 2.7 in the source collects the ingredients: the energy density of the Hagedorn phase is 22.48 times the fourth power of y times the fourth power of the lepton mass; the initial radius is the cube root of three over four pi, times the fraction of centre-of-mass energy going into hot-spot creation, divided by that energy density; and the total current is the sum of the massless and massive contributions. Equation 2.8 in the source gives each contribution as the degeneracy factor times the integral over momentum of the cubed momentum divided by two pi squared, divided by one plus the exponential of the on-shell energy over the Hagedorn temperature. For the massless species the mass is zero and the degeneracy is two, since neutrinos are Majorana. The parameter y is the ratio of the Yang-Mills scale to the charged lepton mass.

The typical inverse width of single electron or positron peaks seen in supercritical heavy-ion collisions, equation 2.9 in the source, is about one over seventy keV, that is 9.4 times ten to the minus twenty-first of a second. It compares well with the lifetime of hot spots belonging to the electron theory calculated by assuming recoil-free, that is thermal, emission. Recall that the evaporation time varies rather weakly with the centre-of-mass energy. Setting both order-unity parameters to one, equation 2.10 in the source tabulates it: 10 MeV gives 1.0 times ten to the minus twenty-first of a second; 100 MeV gives 2.15 times ten to the minus twenty-first; 1 GeV gives 4.63 times ten to the minus twenty-first; 10 GeV gives 1.0 times ten to the minus twentieth; 100 GeV gives 2.15 times ten to the minus twentieth; and 1 TeV gives 4.63 times ten to the minus twentieth.

Notice that the deposited-energy parameters enter only into the normalization of the spectrum and not into its shape. For reasons of better interpretability, it is advantageous to factor out a dimensionful quantity — a pure number divided by the lepton mass — obtained by rescaling the energy and the Hagedorn temperature by the lepton mass in the first factor. Equation 2.11 in the source defines that number as pi squared times the deposited-energy fraction, times the centre-of-mass energy over the squared lepton mass, divided by an integral over a dimensionless variable of its cube times the sum of the massless and massive Fermi factors. Equation 2.12 in the source evaluates it for the electron theory at a centre-of-mass energy of 10 MeV: 0.569661 and 1.13932 for y equal to one at the two values of the deposited-energy fraction, and 9.88092 and 19.7618 for y equal to one half.

The remaining factor is then a dimensionless spectral shape function of the dimensionless energy, the energy divided by the lepton mass, and of y, as set out in equation 2.13 in the source. Figure 1 in the source plots that shape function against dimensionless energy from zero to twenty for y equal to one half, solid, and y equal to one, dashed; the curves rise to a maximum of about 0.3 and fall away smoothly.

Notice the broad spectral shape. As mentioned in section 1.1, this is a consequence of our assumption that the hot spot can be treated as a classical, recoil-free object during the entire history of its thermal evaporation. Obviously, this assumption breaks down when the mass of the hot spot becomes comparable to the lepton mass. The emission of charged leptons then is subject to a quantum mechanical decay, with the width of the latter still being described by the lifetime computed from recoil-free, thermal emission.

For the example of the electron theory, at a centre-of-mass energy of 10 MeV and with the deposited-energy fraction set to one, equation 2.14 in the source gives the number of single positrons and electrons per energy bin of one electron mass emitted at the spectral maximum: 1.14 times 0.3, that is 0.342, for y equal to one; and 19.8 times 0.064, that is 1.27, for y equal to one half. Appealing to the number of counts per energy bin in the experimental data, these numbers can be used to obtain an estimate for the total number of hot spots created in supercritical heavy-ion collisions at given integrated luminosity.

3. Dilepton spectrum from hot-spot evaporation

Let us now derive predictions for the shape of the invariant mass spectrum of dileptons emitted from the surface of an evaporating hot spot. Under certain kinematic constraints, this is the situation relevant to the CDF analysis. Considering a charged lepton far away from the hot-spot surface and thus in thermal equilibrium, it is reasonable to assume that it is not temporally and spatially correlated to another charged lepton emerging from the same hot spot. Moreover, it is experimentally impossible to resolve the hot spot spatially or temporally at presently available energies. Thus, the entire hot-spot evaporation is seen as an instant of extremely high-multiplicity emission of charged leptons emerging from a point.

Having said the above, it is clear that the number of charged lepton pairs with their members in given momentum and angular bins, equation 3.1 in the source, is the squared surface-time factor from the single-lepton calculation, multiplied by the product of two independent Fermi factors, one for each lepton, divided by four pi to the sixth.

Equation 3.2 in the source gives the squared invariant mass of a charged lepton pair as twice the quantity: the squared lepton mass, plus the product of the two on-shell energies, minus the product of the two momenta times the cosine of the angle between them. Equation 3.3 in the source solves that for the first momentum in terms of the invariant mass, the second momentum and the opening angle, keeping the only acceptable root. Equation 3.4 in the source expresses the cosine of the opening angle in terms of the two polar and two azimuthal angles.

Equation 3.5 in the source then gives the number of lepton pairs per invariant mass bin as the squared surface-time factor times the integral, over the second momentum and the four angles, of the Jacobian factor and the two Fermi factors evaluated at the solved momentum and opening angle — with the integration subject to kinematic constraints. For example, towards the infrared one may cut off the modulus of each lepton’s momentum component in the plane transverse to the beam direction. Moreover, the opening angle of the lepton pair may be constrained by requiring its cosine to lie between a fixed number and one. A constraint on that cosine implies constraints on the polar and azimuthal angles.

As in section 2 it is advantageous to rescale the momentum, the invariant mass and the Hagedorn temperature by the lepton mass, and to factor the result as in equation 3.6 in the source: a normalisation constant, equal to the square of the single-lepton constant, divided by the lepton mass, times a dimensionless spectral shape function of the dimensionless invariant mass and of y. Constraints on the transverse momentum are then expressed in terms of the dimensionless transverse momenta. The integral associated with the shape function is performed using Monte Carlo methods.

Figure 2 in the source presents four plots of that dimensionless dilepton shape function against dimensionless invariant mass, for y equal to one half and one, at two values of the opening-angle cut — minus one, that is no cut, and 0.8, the cut chosen by CDF — and two values of the transverse-momentum cut — zero, and 9.47, corresponding to a representative 1 GeV infrared cut on both transverse momenta in the muon pair analyzed by CDF. Panel (a), no cuts at all, peaks near 0.4 in shape units; panel (b), the opening-angle cut alone, peaks near 0.06; panel (c), the transverse-momentum cut alone, peaks near 0.0008; and panel (d), both cuts, peaks near 0.0004.

Notice how severely the kinematic cuts on transverse momentum and opening angle influence the spectral shape, in view of the localization of maxima and their widths. Notice also that for the muon theory the unconstrained case corresponds to a spectral maximum of about 0.6 GeV while, with the transverse-momentum cut applied and no opening-angle cut, the maximum shifts to about 2.3 GeV.

Setting the centre-of-mass energy to 1 TeV — a TeV photon generated at the Tevatron and depositing its energy into a muon-theory hot spot by virtue of the CDF detector material — and the lepton mass to the muon mass of 105.6 MeV, equation 3.7 in the source gives the normalisation constant as 7.6 times ten to the fourth and 3.04 times ten to the fifth for y equal to one at the two values of the deposited-energy fraction, and 2.29 times ten to the seventh and 9.14 times ten to the seventh for y equal to one half.

For example, considering the muon theory at a centre-of-mass energy of 1 TeV with the deposited-energy fraction set to one, equation 3.8 in the source gives the number of muon pairs, regardless of the charge of their participants, emitted at the spectral maximum per invariant-mass bin of one muon mass: 3.04 times ten to the fifth times 0.44, that is 1.34 times ten to the fifth, with no cuts; times 0.07, that is 2.13 times ten to the fourth, with the opening-angle cut; and times four times ten to the minus fourth, that is 122, with both cuts applied. A comparison of these numbers with the experimental counts of dimuon events per bin of invariant mass can be used to obtain an estimate for the total number of hot spots created within a given value of total integrated luminosity in TeV-range collider experiments.

4. Summary and conclusions

Concerning the data analysis in TeV-range collider experiments, the lesson of the present paper is that certain prejudices — kinematic cuts — applied to experimental data in extrapolating known physics, the standard model, into the unknown necessarily hide potentially important signatures. The possibility that metastable hot spots of a new phase of matter are not only created in ultrarelativistic heavy-ion collisions, where due to quantum chromodynamics they are expected, but also in isolated TeV-range proton-antiproton or proton-proton collisions, suggests a scenario completely different from the perturbative standard model approach for the deposition of centre-of-mass energy into collision products. Rather than creating secondaries of energy comparable to the centre-of-mass energy, energy would then be redistributed on a large multiplicity of charged, low-energy leptons stemming from hot-spot evaporation. Because of the missing confinement mechanism in products of SU(2) Yang-Mills theories with equal electric-magnetic parity, it is clear that only the leptonic sector exhibits these large multiplicities. In view of the large data stream that will be generated by the LHC, we hope that the present work will contribute to the process of overcoming these prejudices.

An exception to this rule is the deposition of centre-of-mass energy into the mass of intermediate vector bosons propagating in the preconfining phase of SU(2) Yang-Mills theory. A genuine prediction here is that two more but much heavier copies of vector-boson triplets must eventually be excited. For the muon theory the mass of the neutral boson is hierarchically larger than the muon mass, since this vector mode decouples at the Hagedorn temperature. In this context, it is worth pointing out the potential resonance peak at an invariant mass of about 240 GeV — a hierarchy of about two thousand relative to the muon mass — seen in the electron-positron invariant mass spectrum generated at Tevatron Run II and reported by both the CDF and D0 collaborations. Notice that resonances in this channel are practically free of hadronic contaminations. If true, then this resonance is a plausible candidate of the decoupling dual gauge-field mode of the muon theory.

Concerning the interpretation of narrow-width and correlated single electron or positron peaks in supercritical heavy-ion collisions, we believe that the present paper has established a plausible connection to the creation of preconfining hot spots of the electron theory and their subsequent evaporation.

Endnotes

1. Strong-field QED, local decay of the QED vacuum: occupation of the 1s state of binding energy — in natural units where the speed of light, the reduced Planck constant and Boltzmann’s constant are one — of about minus twice the electron mass, by a state of equal energy transferred from the Dirac sea.

2. By thermal we mean that the hot spot is considered sufficiently large not to wobble about indeterministically; that is, one can assume a smoothly shrinking spherical surface due to recoil-free evaporation.

3. In principle, this is calculable under specific assumptions. For example, one could presume that, inside the Coulomb potential of the nucleus, the incoming photon perturbatively creates a single pair of TeV dimuons which in turn originates a perturbative cascade producing an ever increasing number of low-energy muons until their number density reaches criticality for the Hagedorn transition.

4. Microscopically, Lambert’s law states that at a given point of the emitting surface only the momentum component perpendicular to this surface gets Fermi-distributed at the Hagedorn temperature far away from the hot-spot system and thus is measurable. Thus the factor of one quarter in the earlier work actually should be replaced by unity.

5. Because of the large reservoir of to-be-emitted charged leptons in a hot spot and the assumed absence of external fields, the evaporation physics is charge and spin-orientation blind. In reality, this cannot entirely be true because of a possible fluctuating, CP violating axion field, whose homogeneous incarnation is of cosmological relevance, that slightly prefers the emission of negative over positive charge. However, for the total yields, this effect is negligible.

6. For electrons and positrons at a centre-of-mass energy of about 1 TeV, the evaporation time is about ten to the minus nineteenth of a second and the initial radius ten to the minus tenth of a metre. For muons and antimuons at comparable energies, these time and distance scales are even smaller. Since the CDF II single-hit resolution is one micrometre, an evaporating hot spot acts like a point-like vertex experimentally.