RI-MOM (Regularization Independent-Momentum)

A non-perturbative renormalization scheme used in lattice quantum field theory, particularly in lattice QCD, to connect bare lattice quantities to physical results. It defines renormalization conditions by fixing the momentum-dependent part of a Green’s function to a tree-level value at a specific momentum scale, 𝜇. While it offers advantages like not requiring special gauge configurations, it has limitations, such as the “window problem,” where the scale, 𝜇, must be carefully chosen within a narrow range to minimize errors. 

How it works 

  • Regularization Independent: It is a non-perturbative method that determines renormalization conditions directly from lattice calculations, making it independent of a specific regulator like a lattice cutoff.
  • Momentum Dependent: It uses the momentum-dependent parts of amputated Green’s functions to define the renormalization conditions, often by projecting onto a specific Dirac structure.
  • Renormalization Condition: The scheme sets the projected amputated Green’s function equal to a specific value (often the tree-level one) at a particular momentum 𝑝. This is expressed as:
    • ΓΓ(𝑝)=tree-levelvalue
  • Scale (𝜇): The condition is applied at a chosen momentum scale p2𝜇=𝑝2√.
  • Conversion to MS-bar scheme: Results from RI-MOM can be converted to a more widely used continuum scheme like the MS-bar (𝑀𝑆) scheme, using a perturbative calculation. 

Advantages 

  • No need for special gauge configurations.
  • Can be used in both perturbative and non-perturbative calculations. 

Disadvantages 

Window problem: There is a limited “window” of allowed momentum scales 𝜇 in which the scheme is reliable. This window is defined by the condition Λ𝑄𝐶𝐷≪𝜇≪1/𝑎, where Λ𝑄𝐶𝐷 is the strong interaction scale and 𝑎 is the lattice spacing.

Systematic errors: The window can be narrow, leading to potentially large systematic errors if a suitable scale cannot be found.

Linear divergences: In some applications, like quasi-PDFs, the traditional RI-MOM method cannot completely eliminate linear divergences. 

Return to an overview of experimental procedure of MIT’s William Detmold et al.
GBO article in arXiv: https://arxiv.org/pdf/2410.03602
QEM: https://81018.com/geometric-dynamics/ and https://81018.com/spheres-symphony/

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