Documentation

EpsilonEridani.Particles.Parton.TMD.Reduction

TMD Reduction and Cross-Checks #

This module formalizes TMD-to-PDF reduction interfaces and a factorization cross-check theorem.

source
noncomputable def EpsilonEridani.Particles.Parton.TMD.collinearFromTmd {Flavor : Type} (fTmd : Tmd Flavor) (ktMax ζ : ℝ) :
PDF.Pdf Flavor

Collinear proxy obtained from truncated integration over the transverse measure.

The reduction integrates d²k_T, i.e. carries the Jacobian 2π k_T; see EpsilonEridani.Particles.Parton.TMD.integrateTransverse. An earlier version of this definition used the plain k_T integral integrateKT, which is off by that weight and so did not produce the collinear density its name claims.

Equations
Instances For
    source
    theorem EpsilonEridani.Particles.Parton.TMD.collinearFromTmd_nonneg {Flavor : Type} (fTmd : Tmd Flavor) (h : Assumptions fTmd) (ktMax ζ : ℝ) (i : Flavor) (x Q2 : ℝ) (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
    0 ≤ collinearFromTmd fTmd ktMax ζ i x Q2

    The collinear proxy built from a TMD is nonnegative throughout the physical region.

    A collinear density that could go negative would not be a reduction of a positive TMD, so this is the minimum the definition has to satisfy to deserve its name. It follows from Assumptions.nonneg alone; no reduction hypothesis is involved.

    source
    def EpsilonEridani.Particles.Parton.TMD.IntegratesToPdf {Flavor : Type} (fTmd : Tmd Flavor) (fPdf : PDF.Pdf Flavor) (ktMax ζ : ℝ) :
    Prop

    Reduction interface stating that the transverse-measure integral of a TMD recovers a collinear PDF.

    Equations
    Instances For
      source
      theorem EpsilonEridani.Particles.Parton.TMD.tmd_to_pdf_reduction {Flavor : Type} (fTmd : Tmd Flavor) (fPdf : PDF.Pdf Flavor) (ktMax ζ : ℝ) (hRed : IntegratesToPdf fTmd fPdf ktMax ζ) (i : Flavor) (x Q2 : ℝ) :
      collinearFromTmd fTmd ktMax ζ i x Q2 = fPdf i x Q2

      Canonical reduction theorem wrapper.

      source
      theorem EpsilonEridani.Particles.Parton.TMD.loStructureFunction_tmdCrossCheck {Flavor : Type} [Fintype Flavor] (C : QFT.Factorization.DIS.HardKernel Flavor) (fTmd : Tmd Flavor) (fPdf : PDF.Pdf Flavor) (ktMax ζ x Q2 : ℝ) (hRed : IntegratesToPdf fTmd fPdf ktMax ζ) :
      QFT.Factorization.DIS.loStructureFunction C (collinearFromTmd fTmd ktMax ζ) x Q2 = QFT.Factorization.DIS.loStructureFunction C fPdf x Q2

      Observable cross-check: replacing a PDF by its integrated TMD proxy preserves the LO structure function under the reduction hypothesis.