Struct BetaFunction 

pub struct BetaFunction {
    pub b0: f64,
    pub b1: f64,
    pub order: LoopOrder,
    pub name: String,
}

Beta function for a single running coupling constant g.

The beta function β(g) = μ dg/dμ governs how the coupling constant evolves with the RG scale μ. Coefficients b0, b1 follow the convention β(g) = −b0 g² − b1 g³ (perturbative, Euclidean).

Fields

b0: f64

One-loop coefficient b₀ (≥ 0 for asymptotic freedom).

b1: f64

Two-loop coefficient b₁.

order: LoopOrder

Loop order used for evaluation.

name: String

Name tag for the theory (e.g. “QCD”, “phi4”).

Implementations

impl BetaFunction

pub fn new(b0: f64, b1: f64, order: LoopOrder, name: impl Into<String>) -> Self

Construct a new beta function with given coefficients and loop order.

pub fn evaluate(&self, g: f64) -> f64

Evaluate β(g) at coupling g.

One-loop: β(g) = −b₀ g² Two-loop: β(g) = −b₀ g² − b₁ g³

pub fn derivative(&self, g: f64) -> f64

Derivative dβ/dg at coupling g.

pub fn fixed_point(&self) -> Option<f64>

Find the non-trivial (Wilson-Fisher) fixed point g* where β(g*) = 0, g* ≠ 0.

Returns None if no real fixed point exists in the perturbative regime.

pub fn stability_exponent(&self) -> Option<f64>

Stability exponent ω = dβ/dg|_{g=g*}.

ω > 0: UV fixed point (IR stable); ω < 0: IR fixed point (UV stable).

pub fn run_coupling(&self, g0: f64, t_final: f64, steps: usize) -> f64

Integrate the RG equation dg/dt = β(g) from t=0 to t=t_final using simple Euler integration with steps steps.

pub fn running_coupling_one_loop(&self, g0: f64, t: f64) -> f64

One-loop running coupling: analytic solution g(t) = g₀ / (1 + b₀ g₀ t).

Derived from dg/dt = −b₀ g², which integrates to 1/g(t) = 1/g₀ + b₀ t.

pub fn landau_pole(&self, g0: f64) -> f64

Landau pole scale: t_L where g → ∞ under one-loop running.

Trait Implementations

impl Clone for BetaFunction

fn clone(&self) -> BetaFunction

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for BetaFunction

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.