n-Particle irreducible effective action techniques for gauge theories

J. Berges

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n-Particle irreducible effective action techniques for gauge theories

J. Berges

High Energy Physics - Phenomenology · Submitted 23 Jan 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We point out an equivalence hierarchy for nPI effective actions, which allows one to obtain a self-consistently complete result in practice.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.38 · compares with prior results 0.96 · mentions code or data availability 0.02.

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A loop or coupling expansion of a so-called n-particle irreducible (nPI) generating functional provides a well-defined approximation scheme in terms of self-consistently dressed propagators and n-point vertices. A self-consistently complete description determines the functional for arbitrarily high n to a given order in the expansion. We point out an equivalence hierarchy for nPI effective actions, which allows one to obtain a self-consistently complete result in practice. The method is applied to a SU(N) gauge theory with fermions up to four-loop or O(g^6) corrections. For non-equilibrium we discuss the connection to kinetic theory. The leading-order on-shell results in g can be obtained from the three-loop effective action approximation, which already includes in particular all diagrams enhanced by the Landau Pomeranchuk Migdal effect. Furthermore, we compare the effective action approach with Schwinger-Dyson (SD) equations. By construction, SD equations are expressed in terms of loop diagrams including both classical and dressed vertices, which leads to ambiguities of whether classical or dressed ones should be employed at a given truncation order. We point out that these problems are absent using effective action techniques. We show that a wide class of truncations of SD equations cannot be obtained from the nPI effective action. In turn, our results can be used to resolve SD ambiguities of whether classical or dressed vertices should be employed at a given truncation order.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	A loop or coupling expansion of a so-called n-particle irreducible (nPI) generating functional provides a well-defined approximation scheme in terms of self-consistently dressed propagators and n-point vertices.	background 0.95	0.14	0.11	asserted 1.00
s1	A self-consistently complete description determines the functional for arbitrarily high n to a given order in the expansion.	background 0.61	0.28	0.13	asserted 1.00
s2	We point out an equivalence hierarchy for nPI effective actions, which allows one to obtain a self-consistently complete result in practice.	contribution 0.94	0.88	0.37	asserted 0.98
s3	The method is applied to a SU(N) gauge theory with fermions up to four-loop or O(g^6) corrections.	contribution 0.65	0.92	0.09	asserted 1.00
s4	For non-equilibrium we discuss the connection to kinetic theory.	contribution 0.93	0.71	0.04	asserted 0.67
s5	The leading-order on-shell results in g can be obtained from the three-loop effective action approximation, which already includes in particular all diagrams enhanced by the Landau Pomeranchuk Migdal effect.	result 0.98	0.80	0.14	asserted 0.95
s6	Furthermore, we compare the effective action approach with Schwinger-Dyson (SD) equations.	contribution 0.87	0.82	0.06	asserted 0.87
s7	By construction, SD equations are expressed in terms of loop diagrams including both classical and dressed vertices, which leads to ambiguities of whether classical or dressed ones should be employed at a given truncation order.	background 0.89	0.20	0.06	asserted 1.00
s8	We point out that these problems are absent using effective action techniques.	result 0.77	0.82	0.17	asserted 1.00
s9	We show that a wide class of truncations of SD equations cannot be obtained from the nPI effective action.	result 0.96	0.93	0.19	asserted 1.00
s10	In turn, our results can be used to resolve SD ambiguities of whether classical or dressed vertices should be employed at a given truncation order.	interpretation 0.57	0.90	0.23	asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.67 · evidence from proof or derivation 0.96.

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The paper

Original abstract

A loop or coupling expansion of a so-called n-particle irreducible (nPI) generating functional provides a well-defined approximation scheme in terms of self-consistently dressed propagators and n-point vertices. A self-consistently complete description determines the functional for arbitrarily high n to a given order in the expansion. We point out an equivalence hierarchy for nPI effective actions, which allows one to obtain a self-consistently complete result in practice. The method is applied to a SU(N) gauge theory with fermions up to four-loop or O(g^6) corrections. For non-equilibrium we discuss the connection to kinetic theory. The leading-order on-shell results in g can be obtained from the three-loop effective action approximation, which already includes in particular all diagrams enhanced by the Landau Pomeranchuk Migdal effect. Furthermore, we compare the effective action approach with Schwinger-Dyson (SD) equations. By construction, SD equations are expressed in terms of loop diagrams including both classical and dressed vertices, which leads to ambiguities of whether classical or dressed ones should be employed at a given truncation order. We point out that these problems are absent using effective action techniques. We show that a wide class of truncations of SD equations cannot be obtained from the nPI effective action. In turn, our results can be used to resolve SD ambiguities of whether classical or dressed vertices should be employed at a given truncation order.

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References

Forward citations

Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Correlation functions of Landau gauge Yang-Mills theory
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A parameter-free simultaneous solution of the coupled equations of motion for Landau gauge Yang-Mills correlation functions reproduces lattice propagators and vertices and removes the quadratic-divergence ambiguity in...
ALP Production from Abelian Gauge Bosons: Beyond Hard Thermal Loops
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ALP production from a hot Abelian plasma is computed with full 1PI-resummed propagators, giving positive rates at all momenta and revealing dominant timelike-timelike photon contributions at very soft ALP momenta.
Non-equilibrium dynamics of Goldstone excitation from holography
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Holographic simulations with dynamical pions show a prethermalization stage at non-critical temperatures and a fitted k^2 t scaling, claimed as evidence for a non-thermal fixed point.
On the influence of three-point functions on the propagators of Landau gauge Yang-Mills theory
hep-th 2012-11 conditional novelty 6.0 of 10

Including the ghost-gluon vertex dynamically and a new Bose-symmetric three-gluon vertex model brings Dyson-Schwinger propagators of Landau gauge Yang-Mills theory closer to lattice results in the mid-momentum regime.

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Algorithmic derivation of functional renormalization group equations and Dyson-Schwinger equations
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DoFun 2.0 automates symbolic derivation of Dyson-Schwinger and functional renormalization group equations and converts them into algebraic expressions for O(N) and Gross-Neveu models.
The quark-gluon vertex in Landau gauge QCD: Its role in dynamical chiral symmetry breaking and quark confinement
hep-ph 2008-04 conditional novelty 6.0 of 10

The quark-gluon vertex's infrared singularity, generated self-consistently by dynamical chiral symmetry breaking, produces a linear confining potential; restoring chiral symmetry removes the singularity and leaves a C...
Infrared singularities in Landau gauge Yang-Mills theory
hep-th 2008-01 conditional novelty 6.0 of 10

Landau gauge Yang-Mills theory admits two infrared fixed points: a ghost-dominated scaling solution with new kinematic vertex singularities, and a decoupling solution with a massive gluon and no vertex enhancement.
Nonperturbative properties of Yang-Mills theories
hep-ph 2018-08 accept novelty 3.0 of 10

Functional equations now give self-consistent gluon and ghost propagators and vertices in Landau gauge Yang-Mills theory, including two-loop diagrams, with truncation dependence that is quantitative rather than qualit...
Gluon mass scale through the Schwinger mechanism
hep-ph 2025-01 conditional novelty 2.0 of 10

A comprehensive review showing how massless composite poles in QCD vertices can generate the gluon mass scale, with a BSE-based computation reaching m=367 MeV against the 354 MeV lattice value.
Quantum Transport and Electroweak Baryogenesis
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The review consolidates the first-principles derivation of electroweak baryogenesis transport equations and surveys which models remain viable after EDM and LHC constraints.
A beginner's guide to functional methods in particle physics
hep-ph 2025-10 accept novelty 1.0 of 10

A pedagogical review showing how Dyson-Schwinger, 3PI, and Bethe-Salpeter equations can be chained together to compute glueball masses in pure Yang-Mills theory, matching lattice QCD.

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51 later papers cite it

Evidence: substantial. 49 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

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3 later papers by other authors are listed below: 2 used its method or tool and 1 compared its own results with it.

Uses its method or tool 2 of 15 papers with this label

Derived its system of equations from the 3PI effective action truncated at three loops as in this paper.

Gluonic vertices and the gluon propagator in Landau gauge Yang-Mills theory (2019) · 0.92 · paraphrased: the licence does not allow quoting
Followed this paper's method to derive the 4PI effective action for a scalar theory with cubic and quartic interactions and a non-vanishing field expectation value.

Next-to-Leading Order Transport Coefficients from the Four-Particle Irreducible Effective Action (2009) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 paper

Said its truncation is similar to the one from a 3-loop expansion of a 3PI action analyzed in this paper, but differs in that the DSEs have one bare vertex in every graph.

Infrared singularities in Landau gauge Yang-Mills theory (2008) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

38 arXiv papers cite this in 80 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 58 sentences without a confident label

Uses its method or tool 10 papers

We perform a semi-perturbative calculation of the quark-gluon vertex inspired from its equation of motion of the three-loop expanded 3PI effective action [ref] .

On the quark-gluon vertex at non-vanishing temperature · Setup · 0.93
We use the Bethe-Salpeter equation (BSE) for a two-gluon bound state as derived from the 3PI effective action truncated to three loops [ref] as described in [ref] .

Quenched glueballs in the DSE/BSE framework · Methodology · 1.00
We use the method of successive Legendre transformations [ref] and obtain the 4PI effective action to 4-loop order.

Next-to-Leading Order Transport Coefficients from the Four-Particle Irreducible Effective Action · the 4PI Effective Action · 0.99

Compares with it 1 paper

The truncation we study here is also similar to the one obtained from a 3-loop expansion of a 3PI action analyzed in [ref] but with the difference that in the DSEs there is one bare vertex in every graph.

Infrared singularities in Landau gauge Yang-Mills theory · Inclusion of soft singularities and masses · 0.99

Cites it as background 9 papers

These vacuum diagrams are part of a generating functional known as the 2PI effective action (cf. [ref]) [ref] .

Dark matter freeze-in from non-equilibrium QFT: towards a consistent treatment of thermal effects · Fundamentals of the CTP formalism · 0.91
The $n$PI effective action formalism [ref] is another approach in which the action is expressed as a functional of non-perturbative vertex functions which are determined through a self-consistent stationary equation after the effective action is expanded to some order in the loop or $1/N$ expansion.

4-point vertices from the 2PI and 4PI Effective Actions · Introduction · 0.95
For more details beyond the short overview presented here, see, e.g., [ref] .

Correlation functions of Landau gauge Yang-Mills theory · Equations of motion · 1.00

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