The leading-order cross section, in Float

An executable transcription of EpsilonEridani.QFT.Scattering.DIS.loNCdSigma, together with a toy parton density to evaluate it with.

What is proved, and where

Nothing here. The positivity of the cross section, the Callan–Gross reduction and the identity with the conventional 1 + (1-y)² form are theorems, and they live at the ℝ level in EpsilonEridani.QFT.Scattering.DIS.CrossSection. This file repeats those definitions in Float so they can be evaluated; the correspondence is by inspection.

The toy parton density is a toy

f2Toy is not a fitted parton distribution. It is a valence-plus-sea shape with round-number coefficients, no scale dependence at all (no DGLAP evolution, so F₂ is independent of Q²), no momentum sum rule imposed, and no uncertainty. It exists so the sampler has a non-trivial positive function to sample, and so the machinery can be tested against an analytic answer. Any physics conclusion drawn from it would be wrong. Replacing it with a real PDF set is Phase 4 of the plan.

Flavour content is u, d, s with their antiquarks, charges +2/3, -1/3, -1/3; c, b, t are omitted, which at √s = 60 GeV is a real omission for charm.

def EpsilonEridani.Generator.alphaEM : Float

The electromagnetic coupling at low scale. Running is not implemented.

Equations

• EpsilonEridani.Generator.alphaEM = 1.0 / 137.035999084

def EpsilonEridani.Generator.betaShape (a b x : Float) : Float

Valence-like shape A xᵃ (1-x)ᵇ, the betaShape of the library's PDF module in Float.

Equations

• EpsilonEridani.Generator.betaShape a b x = x.pow a * (1.0 - x).pow b

def EpsilonEridani.Generator.uValence (x : Float) : Float

Toy up-valence density u_v(x), normalised to roughly two valence quarks.

Equations

• EpsilonEridani.Generator.uValence x = 2.6 * EpsilonEridani.Generator.betaShape 0.5 3.0 x / x

def EpsilonEridani.Generator.dValence (x : Float) : Float

Toy down-valence density d_v(x), normalised to roughly one valence quark.

Equations

• EpsilonEridani.Generator.dValence x = 1.3 * EpsilonEridani.Generator.betaShape 0.5 4.0 x / x

def EpsilonEridani.Generator.seaDensity (x : Float) : Float

Toy sea density, the same for each of ū, d̄, s, s̄.

Equations

• EpsilonEridani.Generator.seaDensity x = 0.25 * EpsilonEridani.Generator.betaShape (-0.2) 8.0 x / x

def EpsilonEridani.Generator.f2Toy (x _q2 : Float) : Float

Toy F₂(x) at leading order, x ∑_f e_f² (f + f̄), with no Q² dependence.

The Q² argument is accepted and ignored, so that the signature is already the one a real, scale-dependent parton density will need.

Equations

• One or more equations did not get rendered due to their size.

def EpsilonEridani.Generator.yFactor (y : Float) : Float

The inelasticity factor 1 - y + y²/2, transcribing DIS.yFactor.

Bounded below by 1/2 and above by 1 on y ∈ [0,1]; the ℝ-level yFactor_ge_half proves the lower bound, and the sampler's overestimate relies on the upper one.

Equations

• EpsilonEridani.Generator.yFactor y = 1.0 - y + y * y / 2.0

def EpsilonEridani.Generator.dSigmaDxDQ2 (x q2v y : Float) : Float

d²σ/(dx dQ²) at leading order, transcribing DIS.loNCdSigma.

Units are natural: α dimensionless, Q² in GeV², the result in GeV⁻⁴ up to the overall conversion that a total cross section in picobarns would need.

Equations

• One or more equations did not get rendered due to their size.