Renormalization Group Patterns and C-Theorem in More Than Two Dimensions

Andrea Cappelli; Jos\'e Ignacio Latorre; Xavier Vilasis-Cardona

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Renormalization Group Patterns and C-Theorem in More Than Two Dimensions

Andrea Cappelli, José Ignacio Latorre, Xavier Vilasis-Cardona

High Energy Physics - Theory · Submitted 24 Sep 1991

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

The authors' main claim We argue that for all kinds of perturbative flows the $c$-function is well-defined and the $c$-theorem holds in any dimension.

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arXiv IDhep-th/9109041 (version 1)

Pith IDN7JVPIZV, Pith's permanent record of this paper and its review

CitationAndrea Cappelli, Jos\'e Ignacio Latorre, Xavier Vilasis-Cardona. "Renormalization Group Patterns and C-Theorem in More Than Two Dimensions." arXiv:hep-th/9109041 (1991). https://pith.science/paper/N7JVPIZV

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.62 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.65 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow above two dimensions. This involves the construction of a monotonically decreasing $c$-function using a spectral representation. The missing step of the proof is a good definition of this function at the fixed points. We argue that for all kinds of perturbative flows the $c$-function is well-defined and the $c$-theorem holds in any dimension. We provide examples in multicritical and multicomponent scalar theories for dimension $2<d<4$. We also discuss the non-perturbative flows in the yet unsettled case of the $O(N)$ sigma-model for $2\leq d\leq 4$ and large $N$.

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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	We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow above two dimensions.	background 0.71	0.73	0.04	asserted 0.78
s1	This involves the construction of a monotonically decreasing $c$-function using a spectral representation.	method 0.86	0.63	0.05	asserted 0.97
s2	The missing step of the proof is a good definition of this function at the fixed points.	gap 1.00	0.12	0.05	asserted 0.98
s3	We argue that for all kinds of perturbative flows the $c$-function is well-defined and the $c$-theorem holds in any dimension.	result 0.89	0.73	0.88	asserted 0.97
s4	We provide examples in multicritical and multicomponent scalar theories for dimension $2<d<4$.	contribution 0.96	0.89	0.06	asserted 0.99
s5	We also discuss the non-perturbative flows in the yet unsettled case of the $O(N)$ sigma-model for $2\leq d\leq 4$ and large $N$.	contribution 0.88	0.71	0.04	unclear 0.62

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 0.60.

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Original abstract

We elaborate on a previous attempt to prove the irreversibility of the renormalization group flow above two dimensions. This involves the construction of a monotonically decreasing $c$-function using a spectral representation. The missing step of the proof is a good definition of this function at the fixed points. We argue that for all kinds of perturbative flows the $c$-function is well-defined and the $c$-theorem holds in any dimension. We provide examples in multicritical and multicomponent scalar theories for dimension $2<d<4$. We also discuss the non-perturbative flows in the yet unsettled case of the $O(N)$ sigma-model for $2\leq d\leq 4$ and large $N$.

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References

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Cited by 5 Pith papers

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

A scaling theory for the long-range to short-range crossover and an infrared duality
hep-th 2017-03 conditional novelty 8.0 of 10

Adding a decoupled gaussian field chi to the short-range Ising CFT makes the long-range to short-range crossover weakly coupled, resolving missing-operator puzzles and yielding a new infrared duality.
Long-range critical exponents near the short-range crossover
cond-mat.stat-mech 2017-03 conditional novelty 7.0 of 10

The long-range to short-range crossover in the Ising model is recast as a flow from a short-range Ising fixed point plus a decoupled gaussian field, making the crossover weakly coupled and yielding computable critical...
On the RG running of the entanglement entropy of a circle
hep-th 2012-02 accept novelty 7.0 of 10

Strong subadditivity plus Lorentz covariance implies the entanglement entropy of a circle in 2+1 dimensions is concave, so its constant term increases from UV to IR.
Holographic c-theorems in arbitrary dimensions
hep-th 2010-11 conditional novelty 7.0 of 10

A holographic flow function built from tuned higher-curvature gravity decreases monotonically and equals the A-type trace anomaly coefficient in even dimensions and a universal entanglement-entropy coefficient in all ...
Ground state entanglement in quantum spin chains
quant-ph 2003-04 accept novelty 7.0 of 10

The entanglement entropy of a spin block in a critical chain scales as (c/3) log L, with c the conformal central charge, and saturates off criticality at a value set by the correlation length.

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7 arXiv papers cite this in 9 sentences

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Papers citing it · 4 sentences without a confident label

Uses its method or tool 1 paper

The $β$ function generated by such a deformation can be obtained using the conformal perturbation theory [ref] [equation] in which the even order terms vanish due to the $O(2)×O(N)$ symmetry and the coefficient $A$ can be extracted from the four-point function of $_K$.

Conformality loss and short-range crossover in long-range conformal field theories · Introduction · 0.97

Cites it as background 4 papers

This set of conditions, known as “$a/c$-Theorems” of Zamolodchikov's and Cardy's [ref] , have been intensively studied by the standard Wilson Renormalization Group and by perturbative field-theoretical methods as well [ref] .

Holographic RG Flows from Quasi-Topological Gravity · On the content of $a/c$- Theorems · 0.98
For some recent interesting analytic and numerical results on the spectrum of the WF theory and related theories see [ref] .

Correlation Functions and Trace Anomalies in Weakly Relevant Flows · Introduction · 0.96
The standard framework to describe CFTs with turned on weakly relevant local perturbations is conformal perturbation theory (see e.g. [ref] for $d=2$ and [ref] for general $d$).

A scaling theory for the long-range to short-range crossover and an infrared duality · Beta-function · 0.98

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