Thermal and dense media

Medium modes evaluate amplitudes in an equilibrium medium at finite temperature or density. The medium is at rest in the frame of the supplied momenta.

Scope and limitations

Medium modes generate fixed-order amplitudes only; cross sections are rejected. Connected vacuum graphs, which give the pressure, are the only amplitudes validated against literature values. Higher-point amplitudes are supported but remain experimental.

The following are also unsupported:

• local UV counterterms from expanded 4D integrands (local_uv_cts_from_expanded_4d_integrands); medium modes use 3D local subtraction;
• external momenta or distinct mass parameters with values equal up to sign that make otherwise distinct propagator poles coincide, leaving unresolved 0/0 thermal factors. Generation uses the derivative treatment below only when propagators share one mass parameter and the same momentum routing, including external shifts;
• at zero temperature, a Fermi-surface term in which another fermion’s occupation step depends on the localized loop momentum, so that the two Fermi surfaces intersect; generation reports it;
• standalone evaluator export of processes with Fermi-surface localization; keep the saved process state instead;
• evaluating medium expressions with the standalone evaluator of the diagnostic 3Drep command.

Select the medium

Set global.generation.medium.mode before generate:

• thermodynamic_equilibrium evaluates the distributions at the inverse temperature 1/T given by runtime.general.inverse_temperature (default 1, in inverse model energy units);
• zero_temperature_equilibrium takes their T -> 0 limit: the inverse temperature is unused, and the distributions become step functions at the chemical potentials.

The mode is fixed at generation; changing it requires regenerating the integrand. The inverse temperature and chemical potentials are runtime values that can change per integrand without regeneration. enable_thresholds defaults to true and is rejected in medium modes, so disable it explicitly:

[cli_settings.global.generation.medium]
mode = "thermodynamic_equilibrium"
vacuum_subtraction = true

[cli_settings.global.generation.threshold_subtraction]
enable_thresholds = false

[default_runtime_settings.general]
inverse_temperature = 10.0

[default_runtime_settings.model]
muB = [0.5, 0.0]

Chemical potentials

Chemical potentials are model parameters. Each particle may name one (chemical_potential in JSON and UFO models); a missing assignment or the literal ZERO means zero chemical potential. Restriction-fixed chemical potentials are canonicalized to ZERO, including derived expressions. A runtime value of zero does not erase an adjustable assignment. UFO import keeps these assignments and therefore requires ufo-model-loader>=1.0.0. In the bundled sm model, antiparticles use the negated minus_<name> parameter, and every particle’s value is derived from the external parameters muB, muQ, muLe, muLmu, and muLtau (LHA block CHEMICALPOTENTIAL) using its baryon number, electric charge, and lepton flavor. Assignments use explicit particle metadata, not the mu prefix: muH belongs to the Higgs potential. The other bundled models declare no chemical potentials.

For medium evaluation, a particle and its antiparticle must have equal and opposite chemical-potential values; a self-conjugate particle must have zero chemical potential. Warm-up checks these relations at the current model values, including after parameter updates or state reloads. Independent external parameters are allowed when their values satisfy the relations.

With the default --simplify-model=true, a restriction card turns every parameter it sets to zero into a constant. sm-default therefore fixes all five chemical potentials at zero. Import sm-thermal instead, which sets muB = 3 and keeps it adjustable. To vary other chemical potentials, create a custom restriction card based on thermal and give the desired parameters nonzero initial values. Keep MU = MD = MS = ymu = ymd = yms = 0.0 in that card when the light quarks should remain massless with vanishing Yukawa couplings. sm-thermal_s_mass matches sm-thermal but also keeps the strange-quark mass MS and Yukawa input yms adjustable, retaining the corresponding Higgs and Goldstone interactions. After import, set the desired runtime values, including zero, for example with set model muB=0.5.

Finite-temperature evaluation requires a finite, positive inverse temperature. Chemical potentials must be finite and real. Bosons require a finite, nonnegative real mass and |mu| < m, with the massless m = mu = 0 case also supported. Massive saturation (|mu| = m > 0) and boson condensation are unsupported. Fermions may have |mu| >= m. Finite-temperature fermions require a finite real mass; at zero temperature the same check applies when a fermion has a chemical potential and therefore an occupation step. These checks use the current model values and any DOT mass overrides when the integrand is warmed up, including after model updates or state reloads.

Amplitude warm-up also checks mixed-sign denominators in the selected generated orientations and rejects those that vanish for every loop momentum at the current masses and external momenta. For coincident poles with the same routing, regenerate with a shared mass parameter so generation can use the derivative treatment. The check uses exact equality; nearly coincident poles remain a numerical issue.

Vacuum subtraction and the pressure

With vacuum_subtraction = true, only the medium-dependent part is generated: each distribution weight is replaced by its difference from the vacuum limit, and UV counterterms covering all loops are omitted, including when bridges remain in the graph. This option requires a medium mode.

GammaLoop keeps equilibrium vacuum amplitudes in its Minkowski convention. For the complete connected vacuum-graph sum $𝐴$, including model factors and counterterms, convert to the pressure contribution by hand: $𝛿𝑝=-𝑖𝐴$. This holds at every loop order and also after vacuum subtraction. The phase-convention derivation follows from Wick rotating the single overall spacetime-volume factor; stripped integrals and arbitrary numerator replacements require their own conversion. Keep integrator.integrated_phase = "imag" and comparison targets in the Minkowski convention; the real pressure contribution is then the reported imaginary part of $𝐴$, with its sign and uncertainty preserved. The maintained NLO cold quark matter run card integrates such a contribution.

Fermi surfaces at zero temperature

Several propagators sharing one mass parameter and the same momentum up to an overall sign, including external shifts, produce energy derivatives of a distribution, as in self-energy insertions. In zero_temperature_equilibrium, those derivatives of a fermion with a nonzero chemical potential are delta functions on its Fermi surface. Bosons and fermions at zero chemical potential have no Fermi surface, so their derivatives vanish. The pointwise occupation is undefined (NaN) exactly on the Fermi surface, E = sigma*mu; deterministic inspect points should lie off that discontinuity. Localized delta terms are evaluated separately and do not assign a pointwise value to theta(0). GammaLoop localizes each delta function by rescaling the loop momentum and inserting the normalized positive-scale profile configured in runtime.h_function. The profile needs a positive sigma and a supported power; exponential_ct is rejected. Integrated results do not depend on that profile, but pointwise values do, so fix it explicitly when comparing sampled values. Positive-order Fermi-surface terms at the degenerate onset sigma*mu = |m| > 0 are unsupported unless their product already has empty support. This is checked when the selected term is evaluated, at that evaluation’s precision; ordinary order-zero occupation steps are not rejected by this check. The one-loop Fermi-surface run card illustrates regular-shell localization.