- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

ISRN High Energy Physics

Volume 2012 (2012), Article ID 509793, 12 pages

http://dx.doi.org/10.5402/2012/509793

## Charged Lepton Spectra from Hot-Spot Evaporation

^{1}Laboratorium für Applikationen der Synchrotronstrahlung, Universität Karlsruhe (TH), P.O. Box 6980, 76128 Karlsruhe, Germany^{2}Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

Received 7 December 2011; Accepted 25 December 2011

Academic Editors: E. Anassontzis, H. Hayashii, and S. Paganis

Copyright © 2012 Julian Moosmann and Ralf Hofmann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 (anti)caloron dissociation in the deconfining phase at higher temperatures [1–3]. 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 . Once temperature falls below , the infinitely extended monopole-antimonopole condensate decays nonthermally into single and self-intersecting center-vortex loops (CVLs) which are interpreted as spin-1/2 fermions [1, 2, 4–6]. Only onefold self-intersecting CVLs are absolutely stable in an SU(2) Yang-Mills theory; see [6–8]. 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: , ,. 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 [5], high temperatures in extended systems [4], and/or local energy densities exceeding keV^{4}. Recall that, according to SU(2) Yang-Mills thermodynamics, the apparent pointlikeness of leptons when probed by high-momentum transfer standard-model (SM) processes is a consequence of the presence of a large number of instable 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 SM. In this context, a succession of vertices and off-shell propagators (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 [9]. These data are generated by protons and antiprotons colliding head-on at 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 SM 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 [9–11]. Motivated by existing experiments [9, 10], our theoretical dimuon spectra, obtained by investigating hot-spot evaporation under certain kinematic cuts, are intended to offer guidelines for future analysis of collider date in the TeV realm.

We also consider single-particle spectra for , emission from evaporating hot spots. Here the motivation arises due to the observation of universal narrow-peak structures in supercritical heavy-ion collisions at the Gesellschaft für Schwerionenforschung Darmstadt (GSI) 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 , , , , of combined nuclear charge between 180 and 188 and thus well above criticality at , the emission of narrow positron peaks of several tens of keV width and center at 250 keV, 330 keV, and 400 keV was detected [12–14]. 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 [15]. 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 s which, however, are hard to reconcile with the smaller average delay time in deep inelastic reactions; see [12] and references therein.

The width of the narrow 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 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* (two-particle) decay of a neutral particle state in the mass range between 1.5 MeV to 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,^{1} does not match with the experimental data. Rather, the data point to an independence of the peak features on . This universality is a puzzle in conventional strong-field QED which predicts strong power laws in .

Considering all signatures mentioned above (large delay times formation of spatial regions with negative pressure pointing to the existence of droplets of preconfining ground state of an SU(2) Yang-Mills theory of scale keV; no pair decay but still a heavy neutral primary droplets of energy content several times evaporating isotropically; universality of peak energies and widths in single positron and pair spectra again, 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 whose one-particle thermal emission spectrum is simply computed; see 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 [16] requires electronic binding energies . Since the associated Bohr radius is , and the maximal energy density in supercritical heavy-ion collision is which is larger (but not hierarchically larger!) than what is minimally needed to traverse the Hagedorn transition in . Thus preconfining-phase hot spots likely are generated in the form of *small* droplets (spatial extent comparable with ).

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/or positron peaks corresponds to a lifetime of about s. This, however, is the predicted lifetime for preconfining-phase hot-spot evaporation (with a slow dependence on the deposited energy); see (2.10) below. 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,^{2} then the hot spot and its decay signatures must be interpreted quantum mechanically. The droplet’s thus *computed* lifetime [5], 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 . 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 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 TeV (Tevatron Run II, total integrated luminosity 2100 ) and analyzed by the CDF collaboration (however, not verified by D0 [10] which seems to apply a much tighter constraint on the range of investigated impact parameter [11]), 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 . 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 with are in excess to SM predictions.

The motivation and precise implementation philosophy of [9] 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; see below, when applying typical kinematic constraints chosen by the experimentalists.

In [9] a subset of the Run II data, subject to certain kinematic constraints in transverse momentum and opening angle (integrated luminosity now only 742 ) was analyzed, and so-called ghost events were isolated. These events share the following features: neither the approximate symmetry under and the decay of the rate distribution of multimuon events with increasing , the large magnitude of the rate of events containing two or more muons for , nor the distribution of rate for a given charge composition of dimuon events in invariant mass is explained by known SM processes. Probably depending on the imposed kinematic cuts ( GeV), the measured invariant mass spectra of dimuons at large typically have large weight in the GeV region. To link this to our 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 total mass *~*1 TeV is almost at rest and thus emits muons *isotropically* explaining the observed approximate symmetry under in the impact-parameter distribution of the experimental yields (Emission towards the beam pipe is as likely as away from it at given distance.).

We tend to believe that the reason why QCD predictions and experimental data agree well for events originating at is the absence of heavy nuclei whose strong electric fields trigger hot-spot creation assisted by the rare TeV photons of electromagnetic annihilation. Outside the beam pipe, the detector material provides for these nuclei. Assuming the probability for the conversion of a TeV photon into an hot spot inside the Coulomb potential of a heavy nucleus to be sufficiently small,^{3} the almost unadulterated total photon flux decreases (geometrically) in a power-like way as a function of while the evaporation physics is independent of . This would explain the slow, power-like decay of multimuon activity at large .

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 , we also determine hot spot life times 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 SU(2) Yang-Mills system is ground-state dominated. Once the creation of a bubble of radius containing this new phase of matter has taken place, its evaporation is simply determined by the content of stable, final particle species (number of self-intersections in center-vortex loop, spin, and charge), energy conservation, and 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 ( is the charged lepton state of the SM) where is of order unity expressing a theoretical uncertainty. In the following we strongly appeal to the results of [5].

Leaving the angular integration in (7) of [5] explicit^{4} and setting , the number of charged leptons emitted per unit time and surface reads
where (2-fold spin degeneracy, two-fold charge degeneracy^{5}) is the modulus of the spatial on-shell momentum of a single charged lepton far away from the emitting surface, and can be set equal to keV for and MeV for . From (2.1) we have for the differential yield per unit on-shell momentum, time, and surface
or alternatively, the differential yield per unit on-shell energy , time, and surface
To obtain the differential number of emitted charged leptons per bin of energy, we multiply by the actual surface of the spherical hot spot and subsequently integrate over time from zero to [5]:
According to (5) of [5], one has
and according to (6) of [5]
where
In (2.8) one has and (neutrinos are Majorana), in (2.7) represents the fraction of c.m. energy going into hot-spot creation ( of order unity), and where is the Yang-Mills scale of the SU(2) gauge theory. This implies ; see for example [17].

The typical inverse width of single electron or positron peaks seen in supercritical heavy-ion collisions [12–15] compares well with the life time of hot spots belonging to calculated by assuming recoil-free, that is, thermal emission. Recall that by (2.6) and (2.7), varies rather weakly with . Setting , we have Notice that, according to (2.4), the quantities and enter only into the normalization of the spectrum and not into its shape. For reasons of better interpretability, it is advantageous to factor out the dimensionful quantity where denotes the number obtained by rescaling and by in the first factor of (2.4). Explicitly, we have where . For and MeV, this yields The remaining factor in (2.4) then is a dimensionless function of its dimensionless arguments and : In Figure 1 the function is plotted in dependence of for and . Notice the broad spectral shape of . 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 emission of charged leptons then is subject to a quantum mechanical decay with the width of the latter still being described by the life time computed from recoil-free, thermal emission.

For the example of , at MeV and at , (2.12) and Figure 1 tell us that the number of single positrons *and* electrons per energy bin of keV and emitted at the spectral maximum is
Appealing to the number of counts per energy bin in the experimental data [13–15], the numbers in (2.14) 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 analysis presented in [9]. 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.^{6} 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 momentum bins and and angular bins , , , and reads For the invariant mass of a charged lepton pair, we have where . Solving (3.2) for , the only acceptable solution is Moreover, we have Appealing to (3.1), (3.3), and (3.4), we have for the number of lepton pairs per invariant mass bin where the integration over , , and is 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 where is a fixed number between 1 and −1. According to (3.4) the kinematic constraint on implies constraints on and on .

As in Section 2 it is advantageous to rescale , , and by in (3.5) and to factor the result in analogy to (2.13) into the following form: where and . Constraints on the transverse momentum are then expressed in terms of the dimensionless version . The integral associated with is performed using Monte Carlo methods.

In Figure 2 plots of are presented as a function of at and , for (latter cut chosen by CDF [9]), and (latter cut corresponding to a representative 1 GeV infrared cut on both transverse momenta in the muon pair analyzed by CDF). Notice how severely the kinematic cuts on and influence the spectral shape in view of the localization of maxima and their widths. Notice also that for the unconstrained case ( and ) corresponds to a spectral maximum of about 0.6 GeV while, for and , the maximum shifts to about 2.3 GeV.

Setting TeV (a TeV photon generated at the Tevatron and depositing its energy into an hot spot by virtue of the CDF detector material) and MeV, one has For example, considering , TeV, (3.7), and Figure 2 imply that the number of muon pairs (regardless of the charge of their participants) emitted at the spectral maximum per invariant-mass bin of is A comparison of (3.8) with the experimental counts of dimuon events per bin invariant mass [9] 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 SM) 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 (RHIC), 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 SM approach for the deposition of center-of-mass energy into collision products. Rather than creating secondaries of energy comparable to , 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 [1, 2], 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 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 mass of the neutral boson is hierarchically larger than (since this vector mode decouples at ). In this context, it is worth pointing out the potential resonance peak at GeV (hierarchy ) seen in the invariant mass spectrum generated at Tevatron Run II and reported by both collaborations CDF [18] and D0 [19]. 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 [1, 2].

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 and their subsequent evaporation.

#### Endnotes

- Strong-field QED, local decay of the QED vacuum: occupation of the state of binding energy (use of natural units ) by a state of equal energy transferred from the Dirac sea.
- By thermal we mean that the hot spot is considered sufficiently large to
*not*wobble about indeterministically; that is, one can assume a smoothly shrinking spherical surface due to recoil-free evaporation. - 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.
- 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 far away from the hot-spot system and thus is measurable. Thus the factor 1/4 in the first half of (7) of [5] actually should be replaced by unity.
- 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 [17]) that slightly prefers the emission of negative over positive charge. However, for the total yields, this effect is negligible.
- For electrons/positrons and TeV, the evaporation time is s and the initial radius m [5]. For muons/antimuons and comparable values of , these time and distance scales are even smaller. Since the CDF II single-hit resolution is m, an evaporating hot spot acts like a point-like vertex experimentally.

#### References

- R. Hofmann, “Nonperturbative approach to Yang-Mills thermodynamics,”
*International Journal of Modern Physics A*, vol. 20, no. 18, pp. 4123–4216, 2005. View at Publisher · View at Google Scholar - R. Hofmann, “Erratum: Nonperturbative approach to Yang-Mills thermodynamics (International Journal of Modern Physics A (2005) 20, 18, (4123-4216)),”
*International Journal of Modern Physics A*, vol. 21, no. 31, pp. 6515–6523, 2006. View at Publisher · View at Google Scholar · View at Scopus - R. Hofmann, “Yang-Mills thermodynamics,” arXiv:0710.0962 [hep-th].
- F. Giacosa, R. Hofmann, and M. Schwarz, “Explosive Z pinch,”
*Modern Physics Letters A*, vol. 21, no. 36, pp. 2709–2715, 2006. View at Publisher · View at Google Scholar · View at Scopus - F. Giacosa and R. Hofmann, “Bursts of low-energy electron–positron pairs in tev-range collider physics,”
*Modern Physics Letters A*, vol. 24, no. 24, pp. 1937–1942, 2009. - R. Hofmann, “Yang-Mills thermodynamics at low temperature,”
*Modern Physics Letters A*, vol. 22, no. 35, pp. 2657–2668, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Moosmann and R. Hofmann, “Evolving center-vortex loops,” arXiv:0804.3527 [hep-th].
- J. Moosmann and R. Hofmann, “Center-vortex loops with one selfintersection,” arXiv:0807.3266 [hep-th].
- T. Aaltonen, et al., “Study of multi-muon events produced in p-pbar collisions at sqrt(s)=1.96 TeV,” arXiv:0810.5357 [hep-ex].
- D0 collaboration, http://www-d0.fnal.gov/Run2Physics/WWW/results/prelim/B/B57/.
- T. Dorigo, http://dorigo.wordpress.com/tag/anomalous-muons/.
- W. Greiner and J. Reinhardt, “Quo vadis electron-positron lines?”
*Physica Scripta*, vol. T56, article 203, 1995. - H. Backe, L. Handschug, F. Hessberger et al., “Observation of positron creation in superheavy ion-atom collision systems,”
*Physical Review Letters*, vol. 40, no. 22, pp. 1443–1446, 1978. View at Publisher · View at Google Scholar · View at Scopus - T. Cowan, H. Backe, M. Begemann et al., “Anomalous positron peaks from supercritical sollision systems,”
*Physical Review Letters*, vol. 54, no. 16, pp. 1761–1764, 1985. View at Publisher · View at Google Scholar · View at Scopus - T. Cowan, H. Backe, K. Bethge et al., “Observation of correlated narrow-peak structures in positron and electron spectra from superheavy collision systems,”
*Physical Review Letters*, vol. 56, no. 5, pp. 444–447, 1986. View at Publisher · View at Google Scholar · View at Scopus - J. Reinhardt, U. Müller, B. Müller, and W. Greiner, “The decay of the vacuum in the field of superheavy nuclear systems,”
*Zeitschrift für Physik A*, vol. 303, no. 3, pp. 173–188, 1981. View at Publisher · View at Google Scholar · View at Scopus - F. Giacosa and R. Hofmann, “A Planck-scale axion and SU(2) Yang-Mills dynamics: present acceleration and the fate of the photon,”
*European Physical Journal C*, vol. 50, no. 3, pp. 635–646, 2007. View at Publisher · View at Google Scholar · View at Scopus - “CDF collaboration,” CDF/PUB/EXOTIC/PUBLIC/9160.
- “D0 collaboration,” D0 Note 5923-CONF.