ORI-C
LEVEL-1 HADRONIC STABILISATION
An operationalised relational architecture
Structural accessibility, effective formation and survival
Didier Daloze
Observation • Regulation • Integration • Coherence
ORI-C framework | ori-c.be | 2026
Abstract
We propose a phenomenological model of hadronic stabilisation in three stages: structural accessibility, effective formation within an evolving medium, and then persistence of the state or population formed. The minimal version uses a single QCD combination built from independent confinement and chirality indicators, explicitly relates the production source to the formation rate, and distinguishes the survival of a stable hadron from the evolution of a population of resonances subject to decay, scattering and regeneration. The framework replaces neither QCD nor microscopic models of hadronisation. It supplies a computable and falsifiable synthesis architecture for comparing the conditions of emergence and persistence of hadronic channels.
Purpose of the document
This note presents a model of hadronic stabilisation founded on three distinct operations: the physical accessibility of a state, its formation within an evolving medium, and its capacity to retain its identity after formation.
The architecture distinguishes these stages. The first measures structural accessibility. The second describes effective formation against dissociation and expansion. The third follows either the survival of an identified state when its intrinsic width is negligible, or the observable evolution of a population of resonances when decay and regeneration cannot be separated.
Central principle A level-1 composite stabilisation requires that a state be physically accessible, that it form within the time available, and that it survive long enough against the processes able to dissolve it.
Architecture of the model
1. Structural accessibility Composition, correlations and the QCD regime make the hadronic state possible.
2. Effective formation The formation rate dominates dissociation and the dilution of the medium sufficiently.
3. Observable persistence A stable hadron retains its identity; a population of resonances evolves up to a fixed observation window, with losses and regeneration.
1. Structural accessibility
Qₕ(T, μB) = σ[Aₕ(T, μB) + RQCD(T, μB) - κₕ]
RQCD(T, μB) = wC Cconf(T, μB) + (1 - wC) Bχ(T, μB)
Qh lies between 0 and 1. The minimal version does not carry five free coefficients per channel. It combines the availability of the constituents with a global indicator of the QCD regime. The confinement and chirality components remain computed separately from independent data, and are then combined by a global weight fixed before calibration.
Choosing a sum in the latent argument of the sigmoid treats Aₕ and RQCD as two distinct and partly compensatory contributions to accessibility. A product would impose a strict conjunctive gate: if one of the two terms became zero, accessibility would be strongly suppressed even where the other remains high. The multiplicative variant remains testable in the sensitivity analysis, but it is not retained in the minimal model so as not to introduce from the outset a stronger structural suppression and an additional coupling.
Structural criterion
Qₕ ≥ ΘQ
Normalisation and QCD observables
All the factors of the structural part are dimensionless and bounded. This constraint makes the score comparable between several hadronic states and prevents a variable from dominating solely because of its unit.
Confinement and chirality indicators
Cconf(T, μB) = 1 - L̂R(T, μB) ; Bχ(T, μB) = Δ⟨q̄q⟩(T, μB) / Δ⟨q̄q⟩(0, 0)
Cconf can be built from a renormalised and then normalised Polyakov loop, with L̂R close to 0 in the confined regime and rising toward the deconfined regime. One then uses Cconf = 1 − L̂R. The loop remains an indicator of the change of regime, not an exact order parameter in QCD with dynamical quarks of physical masses.
Bχ must be built from a subtracted or renormalised chiral condensate, then normalised. The direct ratio of the bare condensate to its vacuum value is not sufficient, since the bare condensate depends on the regularisation and contains divergent contributions.
Separation between composition and correlations
Ah represents the relative availability of the flavours compatible with the state h. Gh describes additional correlation information. In the minimal model, Gh = 1 and introduces no free coefficient. It is reintroduced only if a channel observable brings information distinct from that contained in Ah.
Cross term
The product Cconf × Bχ no longer belongs to the minimal model. The two indicators are first interpolated separately from lattice observables, then combined into a global indicator RQCD with a weight common to all species. A non-linear function remains available as an extension, but its parameter must be justified by an improvement on independent data.
Possible non-linear extension
K(Cconf, Bχ; λ) = [Cconf Bχ + λ Cconf² Bχ²] / (1 + λ)
An extension may replace the simple product by this normalised coupling function. The parameter λ tunes the non-linearity without modifying the bounds of the variable when λ ≥ 0. This writing constitutes a phenomenological freedom inspired by models in which the Polyakov sector and chiral dynamics are coupled within a common potential. It is neither an identity of QCD nor a term to be introduced before the data justify an additional parameter.
Structural variables
Symbol ORI-C dimension Interpretation
Aₕ Composition Relative availability of the compatible flavours, without a mechanical penalty tied to the number of valence quarks alone.
Gₕ Configuration Optional correlation observable; Gₕ = 1 in the minimal model.
Cconf QCD regime Normalised indicator of confinement, interpolated independently.
Bχ QCD regime Normalised indicator of chiral breaking, interpolated independently.
RQCD Combined QCD regime Global combination of Cconf and Bχ with a weight fixed before calibration.
κₕ Anchoring Value determined by the condition Qₕ(T*) = 0.5; it is not freely adjusted per channel.
Formation in an expanding medium
The formation of a hadron modifies an abundance within a medium that is diluting and cooling. The volume source Rform,h and the rate Γform,h are related by the density of precursor configurations npre,h. The formation score measures a conditional kinetic efficiency; it does not by itself determine the yield, which also depends on npre,h.
dnₕ/dτ + θ nₕ = Rform,h - Rdiss,h - Rdecay,h
Symbol ORI-C dimension Interpretation
nₕ Density Density of the hadronic type h.
npre,h Precursors Density of the configurations able to feed channel h.
Rform,h Source Volume production, related by Rform,h = Γform,h npre,h.
Γform,h Formation Rate of conversion of a precursor configuration into channel h.
Γdiss,h Dissociation Rate of dissociation, absorption or transformation in the medium.
Γdecay,h Decay Rate of intrinsic disappearance of the identified state.
θ Expansion Local rate of volume expansion of the medium.
Formation score
Fₕ = Qₕ Γform,h / (Γform,h + Γdiss,h + θ)
All rates are expressed in the same unit, for example fm⁻¹. Γform,h is a rate per precursor configuration, while Rform,h is a source per unit volume and time. This closure relates the score to the abundance equation without defining Γform,h as Rform,h/nₕ, which would make it singular when nₕ is small.
Cosmic expansion and collisions
θ = 3H(t) (homogeneous Universe) ; θ = ∇μuμ (expanding medium)
In the homogeneous Universe, θ corresponds to the rate of volume expansion. In a heavy-ion collision, it
must be replaced by the local rate of expansion of the fluid.
Formation criterion
The formation threshold must be exceeded for a minimum duration. An instantaneous crossing does not suffice to establish that a hadronic unit has genuinely individuated.
Δtₕ = ∫ 1[Fₕ(τ) ≥ ΘF] dτ ; Δtₕ ≥ τmin,h
Survival of a stable state and observable persistence of resonances
The survival of an identified hadron is a source-free propagator: it answers the question whether an already formed unit retains its identity up to a fixed observation time. This quantity suits stable or sufficiently long-lived hadrons. It does not by itself describe the final yield of a resonance continually destroyed and regenerated in the hadronic phase.
Sₕtag(tobs) = exp{-∫[τform,h→tobs] [Γdiss,h(τ) + Γdecay,h] dτ}
The upper bound of the integral is fixed by a physical window. For a stable hadron, one uses the end of the interacting medium. For a resonance, the tagged survival may be computed up to kinetic freeze-out, but the measured observable must come from the complete population equation, with regeneration sources and losses.
Operational definition of the formation time
The time τform,h is the first instant from which the formation score stays above its threshold for a continuous duration at least equal to τmin,h. The cumulative survival is then evaluated from that instant.
τform,h = inf{τ : Fₕ ≥ ΘF sur un intervalle continu de durée τmin,h}
The same exponential law describes the survival of a tagged unit, but its observational status differs. For a stable hadron, it directly constitutes a criterion of persistence. For a resonance, Γdecay,h remains finite and the probability decreases with the duration chosen; it must therefore be reported together with tobs and completed by the evolution of the population. No survival threshold proper to each resonance is fitted in the minimal model.
In the numerical prototype, the survival integral is computed up to an explicit observation time. Resonances are then confronted with a multichannel rate equation or with a transport calculation that includes regeneration.
Resonances: population, regeneration and observation window
Between chemical and kinetic freeze-out, resonance yields are modified by decay, by the scattering of decay products and by inverse reactions. The third stage therefore uses two diagnostics: the survival of a tagged unit and the population Nₕ(tobs) issuing from the complete equation. The second is the relevant diagnostic for ratios such as K*/K.
dnres,h/dτ + θ nres,h = Rform,h + Rregen,h - Rdiss,h - Γdecay,h nres,h
Minimal prescription for regeneration For each reversible reaction a + b to h, regeneration is computed from the densities of the products and the inverse rate. It does not constitute an independent parameter added to the population equation.
Rregen,h(τ) = Σa,b <σa+b→h vrel> na(τ) nb(τ)
At equilibrium, for each inverse pair: Rregen,h^eq = Γh→a+b^eq nh^eq
Detailed balance must be imposed channel by channel. Γdiss,h is therefore decomposed according to the reactions actually included, so as to avoid a global loss rate being artificially related to a single regeneration route. For K* to K + π and the inverse reaction, the same pair of reactions fixes loss and regeneration simultaneously, while the scattering of decay products intervenes separately in experimental reconstructibility.
Complete criterion of stabilisation
Qₕ ≥ ΘQ ; Δtₕ ≥ τmin,h ; stable hadron: Sₕtag(tobs) ≥ ΘS ; resonance: Nₕ(tobs) from
the population equation
The minimal model employs a global structural threshold and sets the formation threshold at 0.5 by operational definition. The persistence of a stable hadron is referred to a common time window. For resonances, no value Θsurv,h is fitted: the calculation must reproduce the observed population at tobs with the same loss and regeneration rates.
Integrated temporal criterion
Δtₕ = ∫[τi→τf] 1[Fₕ(τ) ≥ ΘF] dτ
This writing directly measures the cumulative duration during which the formation threshold is crossed. It is more legible than an integral weighted by the excess above the threshold. A weighted measure may be kept as a secondary diagnostic of the intensity of the crossing.
Calibration
Anchor point Choose a reference temperature T* and require the argument of the sigmoid to vanish at that temperature.
Cconf and Bχ Use the values actually obtained after normalisation. They are not necessarily equal to 0.5 at the same temperature.
Aₕ and Gₕ Combine flavour densities, coalescence kernels, susceptibilities or spectral weights according to the channel studied.
Rates Employ kinetic rates of the density × cross-section × relative-velocity type, with consistency through detailed balance where applicable.
Freeze-out Chemical freeze-out constrains the fixing of abundances. It does not by itself define the initial formation of hadrons.
Thresholds Use provisional values for sensitivity tests, then relate them to observable criteria proper to each species.
The chiral crossover provides a reference zone close to 156.5 MeV. The observables tied to confinement and chirality must nevertheless never be replaced by two identical functions. The minimal model uses two independent interpolations from the lattice. Their characteristic regions may overlap, but their shapes remain distinct. This precaution suffices to avoid an exact degeneracy of the coefficients.
Proposed operationalisation
The following expressions constitute a computable prototype. They serve to test the coherence of the framework, not to claim a derivation of complete hadronisation from real-time QCD.
Qₕ(T*) = 0,5 ⟺ κₕ = Aₕ(T*) + RQCD(T*)
The argument of the sigmoid is centred on a reference temperature T*. In the minimal version, κₕ is entirely determined by Ah and RQCD at that temperature. No independent coefficient cₕ, dₕ or eₕ is fitted per channel.
Control convention: Aₕ(T*) = RQCD(T*) = 0.5, hence κₕ = 1
This relation removes κₕ from the list of free parameters. The weight wC of the QCD combination is global and fixed at 0.5 in the first computation. Variants that separate confinement and chirality further are compared afterwards by independent validation, not added at initialisation.
The case Ah(T*) = RQCD(T*) = 0.5 gives κₕ = 1. It serves only to verify the implementation.
Initial thresholds
ΘQ = 0.5 Global structural threshold for start-up. Its variation belongs to the sensitivity analysis, not to a per-species fit.
ΘF = 0.5 Operational definition: Γform,h = Γdiss,h + θ for conditional kinetic efficiency.
Window tobs End of the interacting medium for stable hadrons; kinetic freeze-out or an explicitly stated experimental window for resonances.
τmin,h Establishment duration constrained by relaxation times or channel widths, without a survival threshold adjustable per resonance.
Caution on thresholds The values ΘQ = 0.5 and ΘF = 0.5 are start-up conventions. They must be subjected to a sensitivity analysis and cannot be fitted on the same data as those used to validate the model. For resonances, the population at tobs replaces a survival threshold proper to each species.
Composition and configuration
a_q = n_q(T, μ_q) / [n_q(T, μ_q) + n_q*] ; n_q* = n_q(T*, μ_q*)
The availability of each flavour is bounded by a smooth transformation, then combined by a weighted geometric mean. The exponents are normalised by the total number of valence constituents; a baryon is therefore not mechanically penalised by a cube against a square for a meson.
n_q(T, μ_q) = g_q/(2π²) ∫₀∞ p² dp / {exp[(√(p² + m_q²) - μ_q)/T] + 1}
Aₕ = ∏q a_q^(Nq,h / Nval,h)
The configuration can be related to a spectral weight integrated over a window Ωh around the channel considered:
The window, the weight wh and the renormalisation procedure must be fixed before fitting. Where the spectral function already contains the information about constituent availability, Ah and Gh may become partly redundant.
Parameterisation of the rates
Γform,h = ⟨σform,h vrel⟩ npre,h ; Γdiss,h = ⟨σdiss,h vrel⟩ nmed
Time evolution
In a collision, T(t), μB(t) and θ(t) come from a hydrodynamic calculation or a simplified expansion profile. In the primordial Universe, the relation whereby T varies as the inverse of the scale factor a is exact only when the entropic degrees of freedom remain constant. Around the QCD crossover, one uses rather:
T(t) a(t) g*s(T)^(1/3) ≈ constante
The trajectory μB(t) must be obtained with baryon-number conservation and an equation of state. The principal temporal criterion remains the effective duration above the formation threshold.
Numerical start-up parameterisations
The expressions below define a minimal computational scenario. They serve to check the behaviour of the framework, to measure its sensitivity and to prepare a calibration on data. They do not constitute a microscopic derivation of hadronisation.
Confinement and chirality
Cconf(T) = ½[1 - tanh((T - Tconf)/ΔTconf)] ; Bχ(T) = ½[1 - tanh((T - Tχ)/ΔTχ)]
Both functions are bounded between 0 and 1, but their parameters are fitted independently or replaced directly by interpolations of lattice data. Their centres may be close without their shapes being identical. Imposing Tc = Tχ and ΔTc = ΔTχ is forbidden in the starting set, since that would make the two contributions algebraically degenerate.
Identifiable reduction of the starting model
a_q(T, μ_q) = n_q(T, μ_q) / [n_q(T, μ_q) + n_q(T*, μ_q*)]
The first computation fixes Gh = 1, wC = 0.5 and employs only one RQCD combination common to all species. The composition and regime coefficients equal 1 and are not fitted per channel. κₕ is determined by the anchoring. Extensions reintroducing Gh, a non-linearity λ or separate weights are tested only after validation of the minimal model.
Availability of the constituents
The density nq is computed within the constituent model adopted, then transformed into aq = nq/(nq + nq*) with nq* = nq(T*). This definition equals 0.5 at the anchor, remains informative above T* and avoids the artificial plateau created by clip[nq/nq*, 0, 1]. The chemical potential remains zero in the first LHC test.
Starting prescription for npre,h
npre,h(T, μ) = Cpre,h ∏q nq(T, μq)^(Nq,h)
Cpre,h = {[Γdiss,h(Tch) + Γdecay,h] / Γform,h(Tch)} · nₕeq(Tch) / ∏q nq(Tch, μq)^(Nq,h)
This expression results from the stationary condition Γform,h npre,h = [Γdiss,h + Γdecay,h] nₕeq at Tch, with θ = 0. The simplified form Cpre,h = nₕeq/∏q nq^(Nq,h) is valid only when Γform,h = Γdiss,h + Γdecay,h in the static calibration scenario. Cpre,h carries the units needed for npre,h to remain a density.
In the first prototype, this closure supplies an explicit prescription for the density of precursor configurations. The normalisation is fixed once at Tch with the relation above, then kept along the trajectory. This approximation does not constitute a microscopic model of
coalescence. In a quantitative application it must be replaced by a Wigner kernel, a transport computation or a kinetic prescription proper to the channel.
Channel configuration
Gₕ(T, μB) = [∫Ωₕ wₕ(ω) ρₕ(ω; T, μB) dω] / [∫Ωₕ wₕ(ω) ρₕ(ω; T*, μB*) dω]
The first run sets Gₕ = 1 as the null hypothesis. It isolates the effect of composition and of the QCD regime without immediately introducing an additional spectral observable. A second run replaces Gₕ by a spectral weight or a normalised susceptibility in the channel studied. Comparing the two computations then measures the specific contribution of dynamic correlations and reveals any redundancy with Aₕ.
Starting spectral window
Ωh(Δh) = [mh - Δh , mh + Δh]
For a first computation, a symmetric window of half-width Δh = 100 MeV around the mass or the pole of the channel provides a concrete setting. The sensitivity analysis must then explore at least 50 to 200 MeV. This prescription is not universal. For a broad resonance, or one strongly modified by the medium, the window must follow the position of the pole and the spectral width, for example with Δh proportional to Γh(T, μB). The window Ωh, the spectral weight and the subtraction of the continuum must be fixed before calibration.
Phenomenological test rates
Γform,h(T) = αₕ T² exp(-mₕ/T) ; Γdiss,h(T) = βₕ T³
The coefficients αₕ and βₕ carry the dimensions needed to express the rates in fm⁻¹. These forms remain sensitivity tests. In the abundance equation, Γform,h multiplies npre,h; it must not be calibrated independently of a prescription for the precursors. A quantitative application replaces this pair by a coalescence kernel or by transport rates consistent with the inverse reactions.
Automatic calibration of the test rates
We define the conditional efficiency φₕ = Fₕ/Qₕ. Requiring φₕ(Tch) = ΘF fixes αₕ once βₕ and the expansion trajectory are chosen. The final yield remains controlled by npre,h and by the abundance equation.
φₕ(Tch) = ΘF ; Γform,h(Tch) = [ΘF/(1 - ΘF)] [Γdiss,h(Tch) + θ(Tch)]
αₕ = [ΘF/(1 - ΘF)] [βₕ Tch³ + θ(Tch)] / [Tch² exp(-mₕ/Tch)]
Conversion of the coefficients αₕ and βₕ
αₕ[GeV⁻¹] = 0,19732698 · αₕ[fm⁻¹·GeV⁻²] ; βₕ[GeV⁻²] = 0,19732698 · βₕ[fm⁻¹·GeV⁻³]
Thus βₕ = 1 fm⁻¹·GeV⁻³ corresponds to βₕ = 0.1973 GeV⁻² in natural units. Likewise αₕ = 1 fm⁻¹·GeV⁻² corresponds to αₕ = 0.1973 GeV⁻¹. The unit convention must be fixed before the computation and used without mixing in all rates.
For an accessibility Qh close to 1, the condition Fh/Qh = 0.5 gives Γform,h = Γdiss,h + θ. This equality defines a kinetic crossing, not detailed balance and not, by itself, chemical freeze-out.
Minimal numerical example
Illustrative example for the pion: Tch = 0.156 GeV, mπ = 0.140 GeV, ΘF = 0.5, βπ = 1 fm⁻¹·GeV⁻³ and τ = 7 fm/c, hence θ = 1/τ ≈ 0.143 fm⁻¹. The preceding formula gives απ ≈ 14.7 fm⁻¹·GeV⁻². The choice τ = 7 fm/c represents only a relatively long-lived Pb-Pb scenario. It serves to check the code and constitutes neither a physical calibration nor a universal duration; other values of τ, of θ and of expansion geometry must be explored in the sensitivity analysis.
Detailed balance and departure from equilibrium
At static chemical equilibrium: Rform,h^eq = Rdiss,h^eq + Rdecay,h^eq ; θ = 0
At chemical equilibrium in a static volume, the net collisional term vanishes: direct and inverse reactions are related by the equilibrium densities, the fugacities and the channels actually included. The expansion rate θ is not an inverse reaction and is not part of the microscopic detailed balance.
The condition Γform,h = Γdiss,h + θ obtained when ΘF,h = 0.5 therefore defines a kinetic crossing proper to the chosen score. It may be calibrated near chemical freeze-out, but it does not suffice to identify that freeze-
out. That identification requires solving the abundance equation and checking that inelastic reactions no longer restore equilibrium quickly enough against the expansion.
Simplified trajectory of a collision
T(τ) = T₀ (τ₀/τ)^(1/3) ; θ(τ) = 1/τ
The exponent equals 1/3 only in the conformal limit where cₛ² = 1/3. A realistic equation of state makes cₛ² temperature-dependent. The trajectory μB(τ) must be obtained from an isentropic evolution and must not be assumed to follow automatically the same power as T(τ).
Expansion rate used in the prototype
θ(τ) = ∇μuμ ; θBjorken = 1/τ ; θisotrope 3D = 3/τ
The standard Bjorken flow is longitudinal and boost-invariant; its expansion scalar equals 1/τ. The value 3/τ corresponds to an isotropic three-dimensional Hubble-type expansion. The code must therefore choose the geometry explicitly instead of using 3/τ as a generic Bjorken value.
Integration of the formation criterion
Δτₕ = ∫[τi→τf] 1[Fₕ(τ) ≥ ΘF] dτ ; Δτₕ ≥ τmin,h
Discretisation consists in summing the time steps for which the threshold is crossed. To define τform,h unambiguously, the principal criterion bears on a continuous interval of duration τmin,h. A sum of several separate episodes may be kept as a secondary diagnostic.
τmin,h(T) = ηₕ ℏc / T
This relation is written with T in energy units. When T is expressed in GeV and τ in fm/c, the factor ℏc ≃ 0.197 GeV·fm ensures dimensional consistency. ηh is a dimensionless coefficient to be explored in the sensitivity analysis. Values of order 1 to 2 give, around the crossover, durations close to 1 to 2.5 fm/c, without constituting universal formation times.
Interpretation of the minimum duration
τcorr,h ≈ τmin,h ; natural order of magnitude: ℏc/ΛQCD
τmin,h may be interpreted as an effective time for establishing the correlations of the channel: the time needed for the coloured degrees of freedom to produce a colour-singlet correlation robust enough to be treated as a hadronic unit. The scale 1/ΛQCD provides only a natural order of magnitude. It does not constitute a universal directly measurable time and must be compared with the parameterisation ηh ℏc/T as well as with the relaxation times extracted from the channel studied.
Constraining τmin,h by relaxation times The parameterisation ηh ℏc/T serves as a starting value. As soon as a relaxation time for the channel is available, ηh is fixed so as to reproduce that scale at the anchor point, then kept during validation.
ηh ≈ T* τrelax,h(T*) / (ℏc) ; τmin,h(T) ≈ max[ηhℏc/T, τrelax,h(T)]
For a broad resonance, the time needed to establish the channel must also remain compatible with its own lifetime.
τmin,h ≲ τlife,h = ℏc/Γh
This inequality is a criterion of consistency, not an identity between formation time and lifetime.
Checking the limiting cases
Very high temperature
Cconf and Bχ tend toward low values. The structural accessibility of hadrons remains low,
dissociation dominates and the formation score does not cross its threshold.
Low temperature
Cconf and Bχ become high. Hadronic states are structurally accessible and their survival can become very long. The rate of formation from free quarks no longer needs to remain high, since the quarks are already confined.
Rapidly expanding medium
A high structural accessibility does not guarantee formation if the medium evolves faster than the correlations are established. Separating the three stages prevents possibility, production and persistence from being confused.
Testing on known cases
Primordial Universe
The primordial case constitutes a distinct regime. Around the crossover, 3H is of the order of 10⁻¹⁹ fm⁻¹, far below the strong rates. The formation score there becomes practically independent of the expansion and tests above all the network of reactions. Baryonic evolution moreover requires an explicit baryon–antibaryon annihilation term; merging it into Γdiss,h would lose the distinction between the dissociation of a state and the annihilation of two populations.
Near the primordial crossover: 3H ≈ 3 × 10⁻¹⁹ fm⁻¹
Heavy-ion collisions
Heavy-ion collisions and small systems supply the principal kinetic lever. A continuous analysis in multiplicity, from pp to p-Pb and then Pb-Pb, varies the lifetime and the expansion rate far more gradually than a comparison with the primordial Universe. Stable yields constrain formation, while ratios of resonances to their stable states constrain persistence and regeneration.
Small systems: finite volume and expansion geometry In pp and p-Pb, a homogeneous Bjorken law does not suffice to represent the finite transverse size and the short lifetime of the system. The prototype may employ a finite-volume profile, solely as a sensitivity scenario, or receive θ(x,τ) directly from a transport model.
V(τ) = V0 [1 + (τ - τ0)/τR]^ν ; θ(τ) = d ln V/dτ = ν/(τ - τ0 + τR)
V0, τR and ν are fixed by the multiplicity, the initial size and the estimated lifetime. A quantitative application must replace this law by a hydrodynamic treatment suited to small systems or by a microscopic transport calculation, then compute the scores locally before the final integration.
Priority test: multiplicity dependence
The first falsifiable test compares the slopes of the h/π ratios as a function of charged multiplicity in pp, p-Pb and Pb-Pb. A rising hierarchy for K, Λ, Ξ and Ω is compatible with a formation progressively limited in small systems, but it is not unique to the ORI-C framework: statistical models with canonical suppression can also produce a dependence tied to strangeness. Discrimination therefore requires comparisons at similar mass or strangeness content, as well as the ratios K*/K, Λ(1520)/Λ and ϕ/K, sensitive to different relaxation times.
The specific prediction bears on the residuals after controlling for canonical volume, mass and strangeness: the channels whose Γform,h is lower must retain a steeper multiplicity slope. One and the same set of rates must describe several systems and centralities simultaneously.
Preferred test: continuous variation of θ between pp, p-Pb and Pb-Pb, at similar freeze-out
temperature
ALICE numerical benchmarks for calibration
For the 0–5% most central Pb-Pb collisions at √sNN = 2.76 TeV and at mid-rapidity, ALICE reports the integrated, charge-summed ratios K/π = 0.149 ± 0.010 and p/π = 0.046 ± 0.003. These values serve as control benchmarks for a first prototype, not as universal constraints: the calibration
must preserve the uncertainties, the exact definition of the ratios and the centrality class, and must then be validated on other centralities, multiplicities and systems.
Identifiability and validation
The analysis begins with the reduced model, without per-channel coefficients cₕ, dₕ and eₕ, with a global structural threshold and a formation threshold fixed at 0.5. Additional parameters are introduced only after an analysis of rank, correlation and likelihood profile. AIC, BIC or cross-validation can then determine whether the gain in fit justifies the added complexity.
Minimal numerical protocol
1. Minimal initialisation Interpolate Cconf(T) and Bχ(T) separately, fix RQCD with wC = 0.5, Gh = 1, ΘQ = ΘF = 0.5 and determine κₕ by the anchoring.
2. Kinetic closure Define npre,h and impose Rform,h = Γform,h npre,h. Then solve the abundance equation with the same rates as those of the score.
3. Per-species computation Evaluate Ah, Qh, Fh and τform,h. For stable states, compute the survival up to tobs; for resonances, solve the population with regeneration.
4. Calibration and identifiability Constrain the reduced model first. Check the rank and the correlations of the parameters before introducing Gh, λ or additional weights.
5. Multiscale validation Test the same set of rates on several multiplicities and systems, then on species or ratios not used in the fit.
Starting set for the code
The first computation uses T* = 156 MeV, two independent interpolations Cconf(T) and Bχ(T), Gh = 1, wC = 0.5, global composition and regime coefficients fixed at 1, ΘQ = 0.5 and ΘF = 0.5. κₕ is determined by the anchoring. No survival threshold proper to resonances is fitted. The rates are calibrated on part of the data and then tested on other multiplicities, systems and species.
Starting decisions for the implementation
Element Initial choice Status and sensitivity
npre,h Closure Cpre,h ∏q nq^(Nq,h), normalised at Tch Prototype only; to be replaced by coalescence or transport
βₕ Initial value fixed per channel before computing αₕ To be constrained by a microscopic prior or a sensitivity scan
ηₕ Initial interval 1 to 2 Check stability across species, systems and multiplicities
Tconf, ΔTconf, Tχ, ΔTχ Independent lattice interpolations Never impose two algebraically identical curves
T* 156 MeV Initial variation of ±10 MeV
ΘS 0.9 as a reference value for stable hadrons only
Explore ΘS within a predefined range, for example 0.80 to 0.95, and check that the conclusions do not depend on a fine tuning of the threshold. No survival threshold is fitted for resonances.
tobs End of the interacting medium or an explicitly stated experimental window
Must be common to comparisons within one and the same scenario
Geometry Bjorken, θ = 1/τ, for the first test Compare with 3/τ and with a local hydrodynamic treatment
Regeneration Rregen,h computed from the inverse reactions Detailed balance imposed channel by channel; no independent free normalisation.
Small systems Finite test volume or transport Compare the profile V(τ) with transport outputs; do not impose Bjorken by default.
Fluctuations Coupling disabled in the minimal model Introduce gh only after validating the version without critical slowing down.
Conditions of validation
The model must reproduce several observables with one and the same set of parameters and preserve its performance on data not used during the fit. An agreement obtained only after modifying the thresholds for each data set would signal a lack of identifiability rather than a predictive capacity.
The comparison may begin with integrated yields and the p/π and K/π ratios, then extend to resonances and centrality dependences. Momentum spectra and elliptic flow require a spatial extension of the model, since the three scalar scores do not describe the anisotropy of the medium.
Positioning relative to existing approaches
Coalescence models supply npre,h and Γform,h from quark distributions and Wigner kernels. The ORI-C framework organises these inputs by separating accessibility, formation efficiency and observable persistence. The closure Rform,h = Γform,h npre,h makes that relation explicit.
Freeze-out criteria compare reaction times with the rate of expansion. The present framework takes up that competition in Fh, but treats the population of resonances separately after chemical freeze-out. It constitutes a layer of synthesis only if the reduced model produces predictions common to several systems.
Limits of the model
Dimension Present limit
Spatial homogeneity The prototype is scalar and locally homogeneous. It describes neither spatial gradients, nor anisotropy, nor elliptic flow.
Chemical potential The dependence on μB remains simplified. An application to dense matter requires charge and strangeness constraints and a suitable equation of state.
Effective rates The effective rates and the precursor density jointly determine the yields. A calibration of the rate without a model of npre,h remains insufficient.
Channel observables Aₕ, Gₕ and Ωₕ depend on normalisation conventions, on the channel chosen, on the spectral window and on the treatment of the continuum.
Real-time dynamics Exponential survival describes a tagged unit, not the yield of a regenerated resonance. A window tobs and a population equation are necessary.
Identifiability Several sets of parameters may reproduce the same observable. A strict separation between calibration, sensitivity and independent validation remains indispensable.
Separation of regimes The primordial Universe and collisions do not sample the same competition with expansion; cosmology does not calibrate the θ term of collisions.
Parametric reduction The general version remains unidentifiable with few abundance ratios. The minimal model is compulsory before any per-channel extension.
Points of vigilance for a quantitative application
These points do not call the architecture of the model into question. They determine the level of confidence that can be granted to a numerical application and specify the developments needed to move from a coherence prototype to a quantitative tool.
Point of vigilance Consequence Route to improvement
Phenomenological rates
Progressively replace the effective laws by rates issuing from kinetic models, from coalescence kernels with Wigner functions, from transport cross-sections or from microscopic computations suited to the channel.
Predictive power depends strongly on the quality of the formation and dissociation rates supplied as input. A good calibration may reproduce data without guaranteeing that the microscopic dynamics is correctly described.
Point of vigilance Consequence Route to improvement
Thresholds and parametric complexity
Thresholds, rate coefficients, normalisations and minimum durations can create degeneracies: several sets of parameters may produce close results.
Strictly separate calibration and validation. Where the statistical framework permits, compare the variants by cross-validation and by criteria penalising complexity, such as AIC or BIC.
Spatial gradients and anisotropy
The scalar, homogeneous prototype describes neither local gradients of temperature and chemical potential, nor the anisotropy of the medium, nor elliptic flow v₂.
Couple the local scores to a 3D hydrodynamic treatment or to a transport code, then integrate the contributions spatially over the grid or over the relevant hypersurfaces.
Chemical potentials
Introduce simultaneously the baryon, electric and strangeness chemical potentials, the conservation constraints and a realistic equation of state.
The approximation μB ≃ 0 suits a first LHC prototype at mid-rapidity, but it becomes insufficient for the RHIC Beam Energy Scan programme, for low-energy collisions and for compact matter.
Redundancy between Aₕ and Gₕ
Systematically compare the model with Gₕ = 1 and then with a measured or computed Gₕ. A weak or unstable improvement would argue for merging the two variables or for an orthogonalised definition.
A spectral observable used for Gₕ may already contain part of the abundance information carried by Aₕ. Multiplying them then risks counting the same contribution twice.
Minimum duration τmin,h
The coefficient ηₕ in τmin,h = ηₕℏc/T remains free. Without an external constraint, it can absorb part of the discrepancies between model and data.
Constrain ηₕ by relaxation times, spectral widths, transport models or channel-specific estimates, then check that its value stays stable across several environments and data sets.
Confinement–chirality degeneracy
Use independent lattice interpolations and a single RQCD combination in the minimal model.
Two identical sigmoids make their coefficients indistinguishable, whatever the quantity of data.
Source and formation rate
Without Rform,h = Γform,h npre,h, the kinetic score does not constrain the abundance equation.
Define npre,h by coalescence, by transport or by a documented approximation, then use the same Γform,h in both equations.
Resonances and regeneration
A monotonic survival without a source fails to describe yields between chemical and kinetic freeze-out.
Fix tobs and solve a multichannel equation including inverse reactions and losses of decay products.
Two experimental regimes
The expansion term is negligible in cosmology but competitive in collisions.
Calibrate the dynamics with a multiplicity analysis across pp, p-Pb and Pb-Pb; use cosmology as a distinct test of the reaction network.
The methodological priority is to reduce the free parameters before increasing the complexity of the model. Each extension must bring a measurable improvement on independent data, without degrading the identifiability of the variables already present.
Lines of development
The following extensions are not necessary for the first prototype. They define the steps required to move from a homogeneous score to a quantitative model able to handle spatial observables, fluctuations and reaction networks.
Line Proposed extension Criterion of testing
Spatial extension
Check the conservation of charges and energy, the absence of double weighting of the rates, and then compare yields, spectra and v₂ with and without the ORI-C layer within the same host code.
Define a cell-by-cell interface: T, μB, uμ, θ, volume, constituent distributions, then compute Qₕ(x,p), Fₕ(x,p), Sₕtag(x,p) and the source SORI-C,h. Integrate over Σfo or sample the reactions within the transport.
Fluctuations and critical slowing down
Test trajectories with and without an increase of the relaxation time. Susceptibilities constrain the medium but do not directly determine a hadronic formation rate.
Relate relaxation times and rates to susceptibilities such as χ₄ᴮ/χ₂ᴮ. Near a critical region, dynamic slowing down may delay the adjustment of correlations and modify Fₕ.
Line Proposed extension Criterion of testing
Confinement–chirality coupling
Systematically explore the non-linear function K(Cconf,Bχ;λ) already introduced, instead of automatically retaining the simple product.
Compare λ = 0 with variants λ > 0 under independent validation. Adding λ is retained only if it improves the predictions beyond the gain expected from the increase in the number of parameters.
Microscopically better-founded rates
Replace the laws αₕT²exp(-mₕ/T) and βₕT³ by coalescence kernels, transport cross-sections or rates from effective models consistent with the channel.
Lattice QCD spectral functions can constrain structures and widths, but their continuation from Euclidean to real time does not automatically supply unique rates.
Identifiability before calibration
Establish which observables constrain each coefficient, threshold and normalisation before the global fit.
Combine Jacobian rank, likelihood profiles, Fisher information, VIF, sensitivity analysis and AIC/BIC criteria. Remove or merge parameters that remain persistently unidentifiable.
Local interface with a hydrodynamic or transport code
The coupling must be defined as a local numerical contract. At each time step and in each cell, the host code supplies T(x), μB(x), uμ(x), θ(x), the cell volume and, where the model permits, the constituent distributions. ORI-C then computes Qₕ(x,p), Fₕ(x,p), the rates and the formation source with the same inputs as those used in the abundance equation.
SORI-C,h(x,p) = Qₕ(x,p) Γform,h(x,p) npre,h(x,p)
Two interface modes are possible. In a hydrodynamic code such as MUSIC or iEBE-VISHNU, the local source is integrated over the particlisation hypersurface, and the particles produced can then be passed to an afterburner. In a transport code such as UrQMD, the same information enters as a local reaction probability during the step Δτ.
Pform,h(Δτ) = Qₕ [1 - exp(-Γform,h Δτ)]
The scores may serve either as diagnostics or as modulators of the microscopic sources and probabilities. They must not weight a second time a rate already calibrated with the same information. Yields, spectra and flow coefficients are obtained after spatial integration or particle sampling, not by a direct average of the scores.
E dNₕ/d³p = ∫Σfo pμ dσμ fₕ(x,p) Wₕ(x,p)
Σfo designates the particlisation hypersurface and Wₕ the conversion prescription adopted. In the minimal model, Wₕ must be defined once from Qₕ, Fₕ and the formation kernel, then kept identical in all comparisons.
Fluctuations and critical slowing down: test prescription Baryon susceptibilities characterise the fluctuations of the medium but do not directly determine a hadronic rate. A first extension may use them to modulate the relaxation time, then pass that slowing down on to the formation efficiency.
δχB = [(χ4B/χ2B)/(χ4B/χ2B)ref] - 1
τrelax,h = τrelax,h^0 [1 + gh max(0, δχB)]
Γform,h^eff = Γform,h^0 / [1 + θ τrelax,h]
gh is a coefficient that is global or shared by families of channels. This prescription remains phenomenological: it must be compared with the variant without coupling and is retained only if it improves independent data without creating a new degeneracy.
τmin,h = max[ηₕ ℏc/T, τrelax,h]
This test closure directly relates the slowing down of the medium to the minimum time for establishing the channel. It must be compared with the prescription without susceptibilities and becomes quantitative only after calibration of τrelax,h on an independent observable.
Open questions
Minimum time of broad resonances
The lifetime 1/Γₕ and the time for establishing correlations do not describe the same operation. The first measures the disappearance of the channel through decay; the second measures the time needed for its individuation. For a broad resonance to be treatable as a formed state, its correlation time must remain shorter than or comparable to its lifetime.
τlife,h = ℏc/Γₕ ; τcorr,h ≲ τlife,h
The comparison between ηₕℏc/T, the relaxation times of the channel and ℏc/Γₕ must therefore be carried out without positing
τmin,h = 1/Γₕ as an identity.
Individuated state or contribution to the continuum
For a broad resonance, the comparison τcorr,h ≲ τlife,h alone does not suffice to impose an ontological boundary. It supplies an operational criterion on the relevance of a quasi-particle description.
ξₕ = τcorr,h / τlife,h
When ξₕ ≪ 1, a population of formed states constitutes a coherent approximation. When ξₕ is of order 1, the result must be compared with an off-shell spectral description. When ξₕ > 1, the channel is treated primarily as a broad correlation or a contribution to the continuum, rather than as an on-shell species endowed with an independent formation time. These domains are modelling diagnostics, not universal physical thresholds.
Coupled channels
A hadron may be produced or destroyed through several routes. The ϕ, for example, may be related to s s̄ recombination and to reactions involving K⁺K⁻. The treatment of an isolated channel must then be replaced by a vector of abundances and a network of crossed sources and losses.
dnᵢ/dτ + θ nᵢ = Σj≠i [Rj→i(n,T,μ) - Ri→j(n,T,μ)] + Rext,i
This extension requires the consistent conservation of charges, detailed balance of the inverse reactions and a common definition of the thresholds throughout the network.
Detailed balance channel by channel
Regeneration and dissociation must be decomposed according to the microscopic routes c. For an inverse reaction a + b ⇌ h, each direct–inverse pair receives its own rate and satisfies detailed balance in the equilibrium state used for calibration.
Rregen,h(c) = ⟨σa+b→h(c) vrel⟩ na nb
Rdiss,h(c) = Γh→a+b(c) nₕ
Rregen,h(c),eq = Rdiss,h(c),eq
The total sources and losses are the sums over the channels: Rregen,h = Σc Rregen,h(c) and Γdiss,h = Σc Γdiss,h(c). For the ϕ, the constituent-recombination channel and the hadronic K⁺K⁻ channel are therefore parameterised separately, with their densities, kinematic thresholds and inverse reactions. A global regeneration constant must not compensate for several physically distinct routes.
Quantitative criterion of redundancy between Aₕ and Gₕ
Gₕ is reintroduced only after fitting the minimal model. Its redundancy is assessed from the sensitivity matrix or from a regression of Gₕ on Aₕ and the other predictors of the score.
VIF(Gₕ) = 1 / [1 - R²(Gₕ | Aₕ, RQCD, ...)]
A VIF above 5 constitutes a warning signal and a VIF above 10 a strong indication of collinearity, without any universal cut-off value. Gₕ is kept only if the rank of the Jacobian remains stable, if its confidence intervals are finite and if its addition improves the independent validation or the
AIC/BIC criteria. Failing that, it is fixed at 1, merged with Aₕ or replaced by an orthogonalised observable.
Sensitivity to the expansion geometry
The score Fₕ depends directly on θ. A Bjorken trajectory with θ = 1/τ, an isotropic expansion with θ = 3/τ and a realistic hydrodynamic treatment may therefore produce different formation windows. The geometry must be treated as an assumption of the model and subjected to a sensitivity analysis, not as a mere technical choice.
Sensitivity analysis of the trajectory
Sensitivity must bear on the parameters of the trajectory supplied to the model, not only on the hadronic coefficients. The minimal vector comprises the initial temperature T0, the initial time τ0, the lifetime of the medium, the trajectory μB(τ), the speed of sound or the equation of state, and the prescription for θ.
SO,pi = ∂ ln O / ∂ ln pi
O represents a yield, an abundance ratio, a formation time or a resonance modification. The local derivatives are completed by a global Morris- or Sobol-type analysis when the parameters interact. The variation amplitudes are taken within the uncertainties of the host code. The Bjorken, finite-volume and full-hydrodynamic geometries are compared as distinct scenarios. The model is considered robust only if the qualitative conclusions and the ordering of the channels remain stable across these variations.
Origin of the thresholds and link with observation
The threshold Θform,h = 0.5 has an operational interpretation within the bounded score: formation then equals the sum of dissociation and dilution. The structural and survival thresholds remain more conventional. They may be related to a requirement of robustness or to an experimental precision, but physical survival must not be confused with detection efficiency.
Stable hadron: Pobs,h = εdet,h Sₕtag(tobs)
Resonances and reconstruction of the decay products
Resonance h to a + b: Nobs,h = εrec,h Bh→ab Nh(tobs)
εrec,h = <Aab εa εb Pno-rescatt>phase space
For a resonance, εrec,h includes the kinematic acceptance of the product pair, the tracking and identification efficiencies, the branching ratio, and the probability that the products are not rescattered to the point of making the invariant mass unreconstructible. Regeneration modifies Nh(tobs) in the population equation; it must not be absorbed into εrec,h. Instrumental corrections are applied after the physical computation and do not serve to adjust ΘS.
Scientific status
This architecture is a phenomenological ORI-C proposal. Its minimal version deliberately reduces the number of parameters, closes the link between source and formation rate, and separates the survival of a state from the evolution of a population of resonances. It replaces neither QCD, nor lattice computations, nor transport models.
Its value will depend on its capacity to describe, with one and the same set of inputs, the multiplicity dependences, the stable abundance ratios and the resonance modifications. An improvement obtained solely by adding parameters proper to each channel would not constitute a validation.
Final synthesis
Hadronisation is described here as the passage from a collective medium to an individuated composite organisation. The state becomes possible, forms within an evolving medium, and then its persistence is assessed according to its status: survival of a stable unit or evolution of a population of resonances up to a fixed observation window.
Final formulation Level-1 stabilisation = structural accessibility + effective formation + observable persistence of the state or of the population formed.
Didier Daloze ORI-C framework Observation • Regulation • Integration • Coherence ori-c.be | Belgium | 2026
Compact glossary
Symbol Definition Unit or domain
h Hadronic channel or species studied dimensionless
T, T* Local temperature and anchor temperature MeV or GeV
μB, μq Baryon and quark chemical potentials MeV or GeV
Aₕ Weighted geometric mean of the flavour availabilities 0 to 1
Gₕ Optional correlation observable; equal to 1 in the minimal model 0 to 1
Cconf Normalised indicator of confinement 0 to 1
Bχ Normalised indicator of chiral breaking 0 to 1
Qₕ Structural accessibility of channel h 0 to 1
Ωₕ Energy window used for the spectral weight MeV or GeV
Γform,h Effective rate of formation fm⁻¹
Γdiss,h Rate of dissociation or transformation in the medium fm⁻¹
Γdecay,h Intrinsic width or decay rate fm⁻¹ or GeV
θ Expansion scalar of the medium fm⁻¹
Fₕ Bounded score of effective formation 0 to 1
τmin,h Minimum duration for establishing the channel fm/c
τform,h First instant at which the continuous formation criterion is met fm/c
Sₕtag Probability of survival of a tagged unit up to tobs 0 to 1
ΘQ, ΘF Global structural threshold and operationally fixed formation threshold 0 to 1
αₕ, βₕ Coefficients of the phenomenological test rates, tied to a prescription for npre,h according to the parameterisation
ηₕ Sensitivity coefficient of τmin,h dimensionless
x, p Position within the medium and momentum of the state fm ; GeV
χₙᴮ Baryon susceptibility of order n according to convention
λ Non-linearity parameter of the confinement–chirality coupling dimensionless
Γₕ Total width of the resonance h fm⁻¹ or GeV
n⃗ Vector of abundances in a multichannel network density
εdet,h Acceptance and overall detection efficiency of channel h 0 to 1
RQCD Global combination of the confinement and chirality indicators 0 to 1
Symbol Definition Unit or domain
wC Global weight of Cconf within RQCD 0 to 1
npre,h Density of the precursor configurations of channel h fm⁻³
Rform,h Volume source of formation, Γform,h npre,h fm⁻⁴ in natural units
tobs Observation time fixed by the physical scenario fm/c
Nₕ Integrated or comoving abundance of channel h according to the system
Mₕres Modification of the population of a resonance between two windows dimensionless
a_q Smoothed availability of flavour q, built from n_q and its anchor value 0 to 1
n_q* Reference density n_q(T*, μ_q*) used in the availability a_q fm⁻³
φₕ Conditional efficiency of formation, defined by Fₕ/Qₕ 0 to 1
Tch Chemical freeze-out temperature used for a test calibration MeV or GeV
Cpre,h Normalisation constant of the precursor density, fixed by the stationary condition at Tch
fm^{3(Nval,h-1)} multiplied by a ratio of rates
nₕeq Equilibrium density of channel h used for normalisation at Tch fm⁻³ Rregen,h Regeneration source of channel h through inverse reactions fm⁻⁴ in natural units σa+b→h Cross-section of the inverse reaction producing h fm² or GeV⁻² τrelax,h Relaxation time associated with channel h fm/c V(τ) Effective volume of the evolving system fm³ τR, ν Time scale and exponent of the finite-volume profile fm/c ; dimensionless δχB Normalised departure of the ratio χ4B/χ2B from a reference dimensionless gh Phenomenological coupling of critical slowing down to channel h dimensionless εrec,h Overall reconstruction efficiency of a resonance 0 to 1 Bh→ab Branching ratio of the reconstructed decay channel 0 to 1 Pno-rescatt Probability that the products remain reconstructible 0 to 1
Σfo Particlisation or freeze-out hypersurface used to convert the local sources into particles fm³ within the element pμdσμ
Wₕ(x,p) Local conversion weight relating the scores and the formation kernel to particlisation dimensionless or according to the prescription
ξₕ Ratio of the correlation time to the lifetime of the resonance dimensionless Rregen,h(c) Regeneration source of channel h through the microscopic route c fm⁻⁴ in natural units Γdiss,h(c) Contribution of channel c to the total dissociation rate of h fm⁻¹ VIF Variance inflation factor used as a diagnostic of collinearity dimensionless SO,pi Normalised logarithmic sensitivity of the observable O to the parameter pi dimensionless
Natural-unit benchmarks The equations use c = ℏ = 1. The following conversions allow a rapid passage from natural to customary units without modifying the definitions of the model.
Relation Value Use
ℏc 0.19732698 GeV·fm Energy–length conversion
1 fm⁻¹ 0.19732698 GeV Rate or width expressed as an energy 1 GeV 5.0677307 fm⁻¹ Energy expressed as an inverse rate
1 fm/c 3.33564 × 10⁻²⁴ s Proper time in seconds
αₕ: fm⁻¹·GeV⁻² → GeV⁻¹ multiply by 0.19732698 Conversion of the coefficient of Γform,h = αₕT²e⁻ᵐʰᐟᵀ βₕ: fm⁻¹·GeV⁻³ → GeV⁻² multiply by 0.19732698 Conversion of the coefficient of Γdiss,h = βₕT³
Scientific references
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