REVIEW
Topological susceptibility at zero and finite $T$ in SU(3) Yang-Mills theory
High Energy Physics - Lattice · Submitted 11 May 1996 · 33 references
Pith's verdict: A solid, methodologically honest SU(3) measurement of chi at finite T; the qualitative drop across Tc holds, but the largest quoted magnitudes rest on 2–3 sigma differences at the top beta values.
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Referee report
The paper
The paper claims
Using a smeared topological charge operator, the paper argues that the vacuum's topological susceptibility in pure SU(3) gauge theory is about 175 MeV at zero temperature and collapses by an order of magnitude at the deconfining transition.
The reading
What carries the argument
The load-bearing device is the family of improved topological charge densities $Q^{(i)}_L(x)$, built from plaquettes whose links have been smeared $i$ times with the APE-style recursion of eq. (2.2), with a tunable parameter $c$. Smearing pushes $Z^{(i)}$ close to 1 and makes the additive mixing small, so that the physical signal is no longer buried under lattice artefacts. The second device is the non-perturbative subtraction: $Z^{(i)}$ is read from the plateau of $Q^{(i)}_L$ after heating a single-instanton configuration, and $M^{(i)}$ from the plateau of $χ^{(i)}_L$ after heating a flat ($Q=0$) configuration, with each heated configuration checked by cooling a copy to confirm that no instanton was created or destroyed. Together these convert a lattice correlator into a physical susceptibility without relying on the cooling plateau, which the paper argues collapses near $T_c$.
Extended reading notes
Core claim
The paper claims a first-principles lattice determination of the topological susceptibility of pure SU(3) Yang–Mills theory in which both the multiplicative renormalization $Z(β)$ and the additive mixing $M(β)$ are computed non-perturbatively, so that the physical $χ$ is extracted from the lattice correlator $χ_L = Z^2 a^4 χ + M$ without appeal to cooling plateaus. At $T=0$ on a $16^4$ lattice the three operator choices (0-, 1- and 2-smeared) agree, giving $χ^{1/4} = 175(5)$ MeV, consistent with earlier work and with the Witten–Veneziano prediction. At finite temperature on a $32^3 × 8$ lattice the susceptibility is constant below $T_c$ and drops by roughly an order of magnitude as $T$ crosses $T_c$, a drop the authors read as the disappearance of instanton-mediated topological activity in the deconfined phase. An appended erratum corrects arithmetic in Table I and in the last columns of Tables II and III, notes a marginal violation of asymptotic scaling, and restates the result as the scaling ratio $χ^{1/4}/T_c$, yielding $χ^{1/4} = 170(7)$ MeV (or $174(7)$ MeV with an alternative $T_c/√σ$); the authors state that the conclusions are unchanged.
Load-bearing premise
The subtraction of the additive mixing $M(β)$ assumes that heating a configuration from a known topological charge never creates or destroys an instanton, an assumption checked only by cooling a copy of each configuration and discarding those outside a hand-chosen cut on the charge.
Editorial extensions
If this is right
- The Witten–Veneziano relation, $2N_f χ/f_π^2 = m_η^2 + m_{η'}^2 - 2m_K^2$, is confirmed at the few-percent level by an independent lattice determination of $χ$.
- The $U_A(1)$ symmetry is effectively restored at the same temperature as chiral symmetry, since the topological susceptibility that controls the anomaly-induced $η'$ mass drops sharply at $T_c$.
- Instantons are strongly suppressed in the deconfined phase, consistent with Debye screening cutting off tunnelling between different chiral sectors.
- Because $χ$ is flat below $T_c$ and collapses above it, $χ$ can be used as an order-parameter-like diagnostic of the confinement transition in pure gauge theory.
- The smeared-operator plus non-perturbative-subtraction scheme is portable to other gauge groups and to full QCD with dynamical fermions.
Reading between the lines
- If the drop in $χ$ is genuine, then the $η'$ mass should fall toward its $N_c \to \infty$ Goldstone value as the temperature approaches $T_c$ from below, which is a statement that could be checked against screening-mass measurements in the same gauge theory.
- The paper's own erratum shows that the conversion from lattice units to MeV depends on the ratio $T_c/√σ$; a reader should treat the MeV numbers as conditional on that input, while the dimensionless ratio $χ^{1/4}/T_c$ is the more robust published quantity.
- The method's weak point is the cut $|Q| < δ$ used to discard heated configurations that acquired an instanton; a reanalysis with a different $δ$ or with a purely topological (geometric) charge on the same ensembles would test whether the finite-temperature drop survives the choice.
- Because the mixing to the gluon condensate was only estimated preliminarily, a full separation of $B(β) a^4 G^2$ from $P(β)$ across $T_c$ would sharpen the claim that the drop belongs to $χ$ and not to a change in the gluon condensate.
Original abstract
We determine the topological susceptibility $\chi$ at T=0 in pure SU(3) gauge theory and its behaviour at finite $T$ across the deconfining transition. We use an improved topological charge density operator. $\chi$ drops sharply by one order of magnitude at the deconfining temperature $T_c$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We determine the topological susceptibility $\chi$ at T=0 in pure SU(3) gauge theory and its behaviour at finite $T$ across the deconfining transition. We use an improved topological charge density operator. $\chi$ drops sharply by one order of magnitude at the deconfining temperature $T_c$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We determine the topological susceptibility $\chi$ at T=0 in pure SU(3) gauge theory and its behaviour at finite $T$ across the deconfining transition. | contribution 1.00 | 0.95 | 0.14 | asserted 0.95 |
| s1 | We use an improved topological charge density operator. | method 1.00 | 0.85 | 0.02 | asserted 0.98 |
| s2 | $\chi$ drops sharply by one order of magnitude at the deconfining temperature $T_c$. | result 1.00 | 0.91 | 0.81 | asserted 1.00 |
What was examined, and what was not
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Figures
Showing 2 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-lat/9605013 by the authors.
References
How the literature uses it
32 arXiv papers cite this in 43 sentences
Papers whose main result relates to this one's
States the same result 2 papers
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A drop by an order of magnitude is observed from the confined to the deconfined phase.
Topological susceptibility through the deconfining phase transition · 0.99
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The behaviour of the topological susceptibility at the deconfining phase transition has been determined.
New results on topological susceptibility in SU(3) gauge theory · 0.90
Papers citing it · 24 sentences without a confident label
Uses its data or values 2 papers
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Forming a dimensionless ratio between the topological susceptibility and the square of the string tension, at zero temperature $χ_top(0)/σ^2$, only varies by $≈10%$ between $N=3$ [ref] and higher $N$ [ref] .
Fractional topological charge in $SU(N)$ gauge theories without dynamical quarks · Introduction · 0.85
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Lattice QCD simulation shows that the susceptibility in the pure Yang–Mills theory is a constant; $χ_top = χ_pure ∼(170-180MeV)^4$ [ref] .
QCD $\theta$-vacua from the chiral limit to the quenched limit · Mass-dependence of the $θ$-vacuum · 0.98
Compares with it 2 papers
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Different from the lattice simulation for the $SU(3)$ Yang-Mills theory [ref] where the susceptibility drops down rapidly, the susceptibility remains still $30%$ of its vacuum value at high temperature $T=250$ MeV.
Functional Renormalization for Chiral and U_A(1) Symmetries at Finite Temperature · Numerical results · 0.97
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The topological observables have been analyzed on the lattice in many works, we compare our dyon results to those in Refs. [ref] , which cover a similar temperature range as we have studied.
Deconfinement Phase Transition in the $SU(3)$ Instanton-dyon Ensemble · The vacuum angle and moments of the topological charge · 0.89
Cites it as support 6 papers
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This conclusion is supported by the lattice simulations [ref] .
Phase Transitions, theta Behavior and Instantons in QCD and its Holographic Model · conclusion · 1.00
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Secondly, the proposal is supported by the numerical lattice results [ref] - [ref] , see also a review article [ref] , which unambiguously suggest that the topological fluctuations related to $θ$ (in particular the topological susceptibility) are strongly suppressed in deconfined phase, and this suppression becomes more severe with increasing $N$.
Conformal window in QCD for large numbers of colours and flavours · Introduction · 1.00
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This scenario is strongly supported by lattice computations: the topological susceptibility drops at $T_c$ [ref] , and it does so faster and faster as $N$ increases, pointing to a vanishing of $χ$ right after $T_c$ in the large-$N$ limit [ref] .
$\theta$-dependence and center symmetry in Yang-Mills theories · Introduction · 0.97
Cites it as background 7 papers
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While topological fluctuations are a topic of active focus for lattice calculations at zero-temperature [ref] , there have only been a handful of studies at finite temperature; first for Yang–Mills [ref] and more recently in full QCD using chiral discretizations [ref] .
Lattice QCD input for axion cosmology · Introduction · 0.88
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The rapid drop of $χ(T)$ near $T_c$ is well known from early lattice works [ref] .
Topological features of the deconfinement transition · The discontinuity of the topological susceptibility · 0.84
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In the four-dimensional $(N)$ Yang–Mills case, lattice simulations are the main tool to study the $θ$-dependence of the vacuum (or free) energy, and several lattice studies have been devoted to investigating different aspects of this subject [ref] .
The $\theta$-dependence of the Yang-Mills spectrum from analytic continuation · Introduction · 0.89
Discussion
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