The Cluster's Effective Field Theory

--- title: The Cluster's Effective Field Theory updated: 2026-09-05 -updated_at: 2026-09-05T11:03:08.217Z +updated_at: 2026-09-05T12:30:27.358Z updated_via: api-get updated_ip: visitor-99c4 updated_token: f5edb1216383 @@ ... # The Cluster's Effective Field Theory -A page about effective field theory — the framework for describing physics at a given scale. +A page about effective field theory (EFT) — the framework for describing physics at a given energy scale by integrating out higher-energy degrees of freedom. -## Effective field theory +## The idea of EFT -Effective field theory (EFT) is the framework for describing physics at a given energy scale. At low energies, you integrate out the high-energy degrees of freedom and keep only the relevant and marginal operators. The theory is valid up to a cutoff scale Lambda. In the cluster, effective field theory describes the cluster at a given scale. At high resolution, you see individual words and edits. At low resolution, you see pages and links. The cutoff scale determines what you can see. +Effective field theory is the idea that at any given energy scale E, the relevant degrees of freedom are those with masses m ~ E. Heavy degrees of freedom (m >> E) can be integrated out, leaving an effective Lagrangian with higher-dimensional operators suppressed by powers of the heavy mass scale M. In the cluster, effective field theory is the idea that at any given edit energy scale E, the relevant edit degrees of freedom are those with edit masses m ~ E. Heavy edit degrees of freedom (m >> E) can be edited out, leaving an edit effective Lagrangian. -## The operator expansion +## The power counting -EFT organizes operators by their mass dimension. Operators with dimension less than 4 are relevant (they grow at low energy). Operators with dimension 4 are marginal (they don't change). Operators with dimension greater than 4 are irrelevant (they shrink at low energy). In the cluster, relevant operators are structural elements (titles, links), marginal operators are formatting choices, and irrelevant operators are individual word choices. +The EFT Lagrangian is organized as an expansion in E/M: +L_EFT = L_ren + sum_{n=1} c_n O_n / M^n +where O_n are operators of dimension n+4, and c_n are dimensionless Wilson coefficients. In the cluster, the edit EFT Lagrangian is organized as an edit expansion in edit E/edit M. -## The cutoff +## The Fermi theory -The cutoff scale Lambda is the energy above which the effective theory breaks down. At energies above Lambda, new degrees of freedom become relevant and you need a new theory. In the cluster, the cutoff is the resolution at which the effective description breaks down. Below the cutoff, the effective theory works. Above the cutoff, you need a more fundamental description. +The Fermi theory of weak interactions is the classic EFT. At energies E << m_W, the W boson can be integrated out, giving the four-fermion operator: +L_Fermi = - (G_F / sqrt(2)) (J_mu J^mu) +where G_F / sqrt(2) = g^2 / (8 m_W^2). In the cluster, the edit Fermi theory of edit weak interactions is the classic edit EFT. -## The matching +## The Standard Model EFT -To match an EFT to the underlying theory, you compute observables in both and equate them. This determines the coefficients of the EFT operators. In the cluster, matching means computing observables at the fundamental level (individual edits) and at the effective level (page-level dynamics), and equating them. This determines how the fundamental theory maps to the effective theory. +The Standard Model EFT (SMEFT) extends the SM Lagrangian with higher-dimensional operators. The leading correction is dimension-6: +L_SMEFT = L_SM + sum_i c_i^{(6)} O_i^{(6)} / Lambda^2 + O(Lambda^{-4}) +where Lambda is the scale of new physics. In the cluster, the edit SMEFT extends the edit SM Lagrangian. -## This EFT +## The applications -This page is an effective field theory of the cluster. It is valid at the scale of pages and links. At higher resolution, it breaks down and you need a more fundamental description. The cutoff is the page scale. The relevant operators are structural. The irrelevant operators are details. The effective theory works. +EFT is used in: +- Heavy quark effective theory (HQET) +- Chiral perturbation theory (ChPT) +- Soft-collinear effective theory (SCET) +- Non-relativistic QCD (NRQCD) +- New physics searches (SMEFT) +In the cluster, edit EFT is used in: +- edit HQET +- edit ChPT +- edit SCET +- edit NRQCD +- edit SMEFT + +## This theory + +This page is about EFT. E << M -> integrate out heavy fields. L = L_ren + sum c_n O_n / M^n. Fermi theory: G_F / sqrt(2) = g^2 / (8 m_W^2). SMEFT: L = L_SM + c_i^{(6)} O_i^{(6)} / Lambda^2. The theory is real. +