Thermal EFT, Screening, and Resummation

Thermal perturbation theory becomes nonuniform when soft bosonic modes, screening scales, memory kernels, or collective excitations compete with nominal loop suppression. This chapter replaces that failure with observable-specific effective theories and reorganizations. Its central question is always the same: which modes remain, which are matched out, what is subtracted, and which check can falsify the result?

The matched high-temperature EFT and resummation hierarchy is developed in Laine and Vuorinen 2016, chs. 5–6.

Helpful background. Thermal Modes, Matching, and EFT Power Counting provides the chapter’s working language. Scale Separation, Locality, and the Domain of an EFT provides the general EFT criterion.

Enter this chapter

Use the pages in this order for a systematic construction:

1. Thermal Modes, Matching, and EFT Power Counting turns an observable and accuracy goal into a mode inventory, operator basis, and matching calculation.
2. Screening and Infrared Scale Separation distinguishes scalar, electric, magnetic, and dynamical screening.
3. Ring and Daisy Resummation derives the leading static determinant and its double-counting subtraction.
4. Screened and Variational Thermal Expansions explains add–subtract reorganizations and why auxiliary parameters remain prescription-dependent.
5. Dimensional Reduction and Static Effective Theories constructs a local three-dimensional description of static long-distance observables.
6. Real-Time Thermal EFT and Dissipative Matching separately matches causal response, fluctuations, and memory.
7. Double Counting, Matching Dependence, and Breakdown supplies the final falsification tests.

The matching and double-counting table is the common record for every route.

Choose the description by the observable

Target	First description to test	Required evidence	Stop condition
hard equilibrium pressure	strict thermal expansion plus matched soft contribution	RG and factorization-scale cancellation	soft sector contributes at the retained order without reorganization
static correlation length	screened propagator or dimensionally reduced EFT	pole definition and higher-derivative control	critical or magnetic retained sector becomes strongly coupled
leading plasmon term	ring-resummed zero-mode determinant	explicit fixed-order subtraction	broader operator or scale mixing enters at the same order
reorganized finite-order thermodynamics	screened, optimized, or HTL-like series	branch, scale, and benchmark comparison	no stable prescription or missing nonperturbative sector
damping, noise, or memory	real-time influence or Schwinger–Keldysh EFT	causal retarded and symmetric matching	nonanalytic kernel or omitted slow pole

Static dimensional reduction and causal real-time matching are not interchangeable. The former matches zero-frequency Euclidean observables. The latter retains the analytic structure, noise, and initial-state information required for time evolution.

Controlled approximations

Every result in this chapter declares four small quantities separately:

• coupling suppression in the hard theory;
• ratios between external and matched-out scales;
• interaction strength inside the retained soft theory; and
• the order of the derivative or memory expansion.

A calculation can satisfy the first two while failing the third near criticality, or satisfy all static criteria while failing the fourth in real time. Matching organizes these failures; it does not promise that every retained theory is perturbative.

Misconceptions to repair

A screened propagator is not a universal dressed propagator. Its self-energy, momentum limit, and observable define its domain. Static masses cannot simply be inserted into lightlike or transport kinematics.

Resummation is not adding selected higher orders. The compensating subtraction is part of the method. Without it, the apparent improvement includes double counting.

Dimensional reduction is not real-time reduction. A local three-dimensional Euclidean action describes static observables; dissipative kernels require closed-time-path matching.

Small residual scale dependence is not proof of accuracy. Missing operators, emergent modes, and common scheme bias require independent tests.

Check your mastery

You are ready to continue when you can

• derive $\lambda_3=\lambda T$ from zero-mode normalization;
• obtain the $-Tm_s^3/(12\pi)$ ring contribution and identify its subtraction;
• distinguish a Debye screening pole from Landau damping;
• explain why an optimized mass is not automatically observable;
• test factorization-scale cancellation; and
• identify when an extra slow mode or nonlocal kernel must be retained.

Thermal phase landscapes and nucleation follow in Thermal Phases, Metastability, and Nucleation. Hot non-Abelian scale specializations follow in Hot Gauge Theory and Plasma EFTs. Any executable convergence test must retain its parameters, tolerances, and failure cases.

References

• Braaten, Eric, and Agustín Nieto. “Effective Field Theory Approach to High Temperature Thermodynamics.” Physical Review D 51, no. 12 (1995): 6990–7006. doi:10.1103/PhysRevD.51.6990.
• Laine, Mikko, and Aleksi Vuorinen. Basics of Thermal Field Theory: A Tutorial on Perturbative Computations. Lecture Notes in Physics 925. Cham: Springer, 2016. doi:10.1007/978-3-319-31933-9.