Black solitons as the origin of the hadronic temperature in Salam-Strathdee strong gravity: soliton existence, ab initio Hawking emission, and transverse-energy phenomenology

In 1977 Abdus Salam and J. Strathdee proposed that the universal hadronic temperature - the roughly 100 MeV thermal scale seen in the transverse-energy spectra of high-energy collisions - is the Hawking temperature of 'black soliton' solutions of an Einstein-like, strongly coupled, massive spin-2 field (the 'f-field' of strong gravity), with transplanted Kerr-Newman-type horizon relations predicting solitons of mass about 1.5 GeV and decay widths of order tens of MeV. The quantitative validation program this proposal requires has never been completed. First, no horizon-containing (including rotating or charged, Kerr-Newman-type) soliton solutions of the massive strong-gravity equations have been constructed or excluded in the hadronic regime where mediator mass, inverse temperature, and soliton radius are all of comparable magnitude. Second, no ab initio derivation of the thermal emission of such solitons - mode-dependent absorption cross-sections, greybody factors, and power - exists for the strong-coupling, massive-mediator regime; only weak-gravity Hawking and Page formulae transplanted outside their domain of validity are available. Third, the conversion of the total-energy thermal emission into measured transverse-energy spectra, and hence the confrontation of the numerical predictions with hadronic data, has never been carried out. Later literature settled only fragments: the spherical, massless-mediator sector (the authors' own sequel; the Isham-Storey exact solutions, whose would-be horizon was shown in 2011 to be a singular surface, while more general interaction potentials do admit regular bigravity black holes - static, spherical solutions of one specific two-parameter interaction potential rather than the $f$-field mass term, lying outside the $m_F R \sim 1$ regime and the rotating or charged cases targeted here); the energetics of strong-gravity-restricted evaporation of ordinary black holes (Qadir and Sayed 1979); and modern explanations of hadronic thermality now use QCD mechanisms instead. The problem - completing or decisively refuting the black-soliton account of the hadronic temperature within the strong-gravity framework - remains open.

Background

Two seemingly unrelated thermal facts were established by the mid-1970s.

On the hadron side, soft particle production in high-energy collisions is approximately thermal: the transverse-energy ($E_T$) spectra - energy summed over particles emitted transverse to the beam direction - exhibit a universal temperature of order 100 MeV that depends only weakly on collision energy, the empirical descendant of the limiting ('Hagedorn') temperature implied by the exponentially growing number of hadronic states. On the gravitational side, Hawking (1975) proved that a black hole - a classical, event-horizon-containing soliton of Einstein's equations - radiates every particle species thermally, at a temperature fixed by the surface gravity (the gravitational acceleration at the horizon); for a spinless, spherical hole this is $kT = 1/(4\pi R)$ with $R$ the horizon radius. Page (1976) computed the detailed emission: the flux in each particle-spin channel is a Planck spectrum multiplied by greybody factors - spin- and frequency-dependent absorption cross-sections $\sigma_s(\omega)$ encoding the scattering of emitted modes off the curved exterior geometry.

In between stood 'strong gravity'. Isham, Salam and Strathdee (1971) had proposed that alongside ordinary gravity there acts a second, strongly coupled spin-2 field $f_{\mu\nu}$ (the f-field), a flavour-singlet (quark-flavour-blind) mediator of hadronic forces with coupling $G_F$ of order GeV$^{-2}$ (roughly $10^{38}$ times Newton's constant) and mass $m_F$ of order the GeV scale, with $m_F^2 \approx 4.5\ \mathrm{GeV}^2$ estimated from the Pomeron trajectory - the leading vacuum Regge exchange of soft (low-momentum-transfer) high-energy scattering, whose linear relation between a state's spin and squared mass fixes a spin-2 state's mass. A massive spin-2 propagator differs from the massless graviton propagator in its trace terms, so the f-field's Einstein-like equations are not equivalent to Einstein gravity with a rescaled coupling.

Salam and Strathdee (1977) connected the two facts. Assuming an Einstein-like, strongly interacting, massive spin-2 field, they proposed to identify temperature in hadronic physics with strong surface gravity: 'black soliton' solutions of the f-field equations would Hawking-radiate thermally, explaining the observed thermal $E_T$ spectra. Transplanting Kerr-Newman-type horizon relations (Kerr-Newman being the rotating, charged black-hole solution of Einstein's theory) and the Hawking/Page emission formulae into the strong-gravity theory, the source summarizes: soliton radius $R = 1/(4\pi kT)$ from the empirical temperature (about 120 MeV), mass $M = R/(2G_F) \approx 1.5$ GeV, decay width of order tens of MeV - numbers of the right hadronic size. The paper acknowledged that these were transplants: the derivations used hold for massless mediators and weak surface gravity, whereas here the soliton mass, the inverse temperature, and the mediator mass are all of comparable magnitude. The open questions it left are: do the assumed horizon-containing solitons exist in the massive theory (including the rotating or charged, Kerr-Newman-type cases)? Can the thermal emission be derived ab initio - greybody factors and mode-dependent cross-sections in the strong, massive regime? How does total-energy thermal emission convert into measured transverse-energy spectra? How do the order-of-magnitude uncertainties in $G_F$ and $m_F$ propagate to the predictions? And can colour-carrying solitons be long-lived, given that ordinary charged black holes rapidly neutralize by discharge and pair emission (Zaumen 1974; Gibbons 1975)?

The modern mainstream explains the universal hadronic temperature through QCD: statistical hadronization (production of hadrons from a thermalized fireball, with the final particle composition frozen in at the 'freeze-out' stage near 160 MeV, where inelastic particle production ceases), and horizon-analogy pictures in which the colour-confinement vacuum plays the role of the horizon. The strong-gravity program was largely abandoned after the mid-1980s, but its field equations survive as ancestors of today's bimetric and massive-gravity theories (theories with two metric tensors or a graviton mass, constructed ghost-free in the 2010s). The question the source left open - a completed, quantitative black-soliton account of the hadronic temperature within strong gravity, or a decisive exclusion of it - has not been answered.

Problem Statement

Within the framework proposed by Salam and Strathdee (1977) - an Einstein-like, strongly coupled, massive spin-2 'f-field' with coupling $G_F$ of order GeV$^{-2}$ and mass $m_F$ of order the GeV scale - determine whether the black-soliton explanation of the hadronic temperature can be validated or falsified. The single objective is a quantitative account of the observed thermal transverse-energy spectra as Hawking-type radiation of strong-gravity solitons, with three jointly necessary components:

(1) Existence. Construct or rule out classical horizon-containing (including Kerr-Newman-type, i.e. rotating or charged) soliton solutions of the massive strong-gravity field equations in the physically relevant regime where the mediator mass, the inverse temperature, and the soliton radius are all of comparable hadronic magnitude ($m_F R \sim 1$), taking into account that the massive spin-2 propagator's trace structure differs from the massless graviton's.

(2) Emission. Derive ab initio the thermal emission of such solitons - Hawking-type temperature, mode-dependent absorption cross-sections $\sigma_s(\omega)$, greybody factors, and total power - valid in this strong-coupling, massive-mediator regime, where the weak-gravity Hawking and Page derivations lie outside their formal domain of applicability.

(3) Phenomenology. Convert the total-energy thermal emission into the experimentally measured transverse-energy ($E_T$) distributions, propagate the order-of-magnitude uncertainty of the phenomenologically fixed $G_F$ and $m_F$ through the predicted soliton mass, radius, width, and temperature, and address whether colour-carrying solitons could be sufficiently long-lived to be observed, so that the hypothesis is actually confronted with hadronic data.

The falsification branch is a demonstration, within the stated framework, that no such solitons exist, or an exhibited pathology of the framework that makes any consistent thermal-emission computation impossible.

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