High Energy Physics - Lattice · arXiv:1105.6229 v1 · submitted 31 May 2011
The Wilson Dirac Spectrum for QCD with Dynamical Quarks
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“All microscopic correlation functions of the spectrum of the Hermitian Wilson Dirac operator with any number of flavors with equal masses are computed.”
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Abstract
All microscopic correlation functions of the spectrum of the Hermitian Wilson Dirac operator with any number of flavors with equal masses are computed. In particular, we give explicit results for the spectral density in the physical case with two light quark flavors. The results include the leading effect in the discretization error and are given for fixed index of the Wilson Dirac operator. They have been obtained starting from chiral Lagrangians for the generating function of the Dirac spectrum. Microscopic correlation functions of the real eigenvalues of the Wilson Dirac operator are computed following the same approach.
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How the main claim is stated: asserted 1.00
What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.06 · mentions code or data availability 0.02
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This table labels the abstract only (arXiv listing abstract v1). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| All microscopic correlation functions of the spectrum of the Hermitian Wilson Dirac operator with any number of flavors with equal masses are computed. | contribution 0.76 | 0.93 | 0.62 | asserted 1.00 |
| In particular, we give explicit results for the spectral density in the physical case with two light quark flavors. | result 0.83 | 0.95 | 0.25 | asserted 1.00 |
| The results include the leading effect in the discretization error and are given for fixed index of the Wilson Dirac operator. | result 0.70 | 0.92 | 0.12 | asserted 1.00 |
| They have been obtained starting from chiral Lagrangians for the generating function of the Dirac spectrum. | method 1.00 | 0.83 | 0.05 | asserted 1.00 |
| Microscopic correlation functions of the real eigenvalues of the Wilson Dirac operator are computed following the same approach. | contribution 0.52 | 0.93 | 0.11 | asserted 1.00 |
What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,166 input tokens, $0.0002 reported.
Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. This record as JSON
How the literature uses it
21 arXiv papers cite this in 65 sentences
Papers citing it · 47 sentences without a confident label
Builds on it 2 papers
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Extending the results of [ref] to the twisted mass case we find Z^ν_2(iz_t,-iz_t;a) = i e^4a^2/z_t π(16a^2) ∫_-∞^∞∫_-∞^∞ds_1 ds_2 (is_1-is_2) ×e^-1/4a^2[(s_1-z_t)^2+ (s_2+z_t)^2] (is_1)^ν(is_2)^νZ^ν_2(s_1,s_2;a=0),
The Microscopic Twisted Mass Dirac Spectrum · The microscopic two flavor maximally twisted mass partition function · 1.00
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These results were extended to $N_f=1$ flavours in [ref] and to $N_f=2$ in [ref] .
Individual Eigenvalue Distributions for the Wilson Dirac Operator · Introduction · 0.85
Uses its method or tool 3 papers
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The approach to the Wilson Dirac spectrum followed in this paper has been applied previously in Refs. [ref] and results from these studies will be used.
The Realization of the Sharpe-Singleton Scenario · Introduction · 0.88
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We employ the parametrization [ref] to evaluate [equation] We employ the same trick as in Ref. [ref] to linearize the exponent in $U$ and $U^-1$ by introducing an auxiliary Gaussian integral over a supermatrix, i.e. [equation] with [equation] After plugging Eq. [ref] in Eq. [ref] we diagonalize $σ=V(s_1,s_2)V^†$ and integrate over $V∈(1/1)$.
Spectral Properties of the Wilson Dirac Operator and random matrix theory · The $Z_1/1^ν$–Partition Function · 1.00
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At zero lattice spacing ($a=0$) this partition function can be identified with the one considered in Sec. [ref], [equation] Considering again the microscopic limit 3.2.5, we trace the integral back to the $a=0$ result by introducing a $N_f×N_f$ Hermitian random matrix $σ$ similar to the calculation in Ref. [ref] , [equation] Notice that $σ$ is an ordinary matrix and not a supermatrix because we consider fermionic flavors, only.
Surprising Pfaffian factorizations in Random Matrix Theory with Dyson index $\beta=2$ · An application: Wilson-Dirac random matrix · 0.83
Compares with it 5 papers
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The difference with [ref] is that we now have the twisted mass instead of the standard mass.
The Microscopic Twisted Mass Dirac Spectrum · The microscopic spectrum with two maximally twisted flavors · 0.83
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Thus it exhibits a similar effect as already found for lattice QCD with three colors of the Wilson Dirac operator [ref] .
Dirac Spectrum of the Wilson Dirac Operator for QCD with Two Colors · Supersymmetric Partition Function and Quenched Theory · 0.98
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The starting point of the construction are the monomials [equation] With help of the general formula for the orthogonal polynomials of the one-point weight $g_1^(±)$ [ref] as a quotient of determinants we find [equation] in monic normalization, i.e. $p_l^(±)(z)=z^l+…$ Since $g_1$ is a shifted Gaussian in our case the polynomials $p_l^(±)$ are shifted Hermite polynomials $H_l$ in monic normalization, [equation] This agrees with Refs. [ref] where a mixing of the eigenvalue statistics with a finite dimensional GUE was found.
Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT · The polynomials · 0.82
Cites it as background 5 papers
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More recently, a complete description of the microscopic details of the Wilson-Dirac operator spectrum up to and including $a^2$ corrections has been presented for both quenched [ref] and unquenched cases [ref] .
Finite-Volume Scaling of the Wilson-Dirac Operator Spectrum · Introduction · 0.92
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Subsequently, a series of interesting statements have been made about the spectral density of the Wilson Dirac operator using WChiPT and/or WRMT for both $N_f=0$ (quenched) and $N_f=1$ QCD [ref] .
Spectral properties of the Wilson Dirac operator in the $\epsilon$-regime · Introduction · 0.95
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In the infrared limit this operator can be modeled by the Wilson-Dirac random matrix [ref] which is a $(2n+ν)×(2n+ν)$ Hermitian matrix [equation] distributed by the Gaussian [equation] The variable $a$ plays the role of the lattice spacing.
Surprising Pfaffian factorizations in Random Matrix Theory with Dyson index $\beta=2$ · An application: Wilson-Dirac random matrix · 0.87
What it cites · 33 sentences without a confident label
Uses the method or tool of 1 paper
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It can be obtained from the discontinuity of the spectral resolvent across the real axis (see section II of [ref] )
Spectrum of the Wilson Dirac Operator at Finite Lattice Spacings · The distribution of real modes · 0.81
Compares with 2 papers
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Using the identity 2(t-iϵ+m/t-iϵ-m )^ν/2 K_ν((m^2-(t-iϵ)^2)^1/2) =∫_-∞^∞ds e^-νs e^-imsinhs -i(t-iϵ) coshs we obtain after performing the Gaussian integration over $t$ and shifting the $s$-integration by $πi$ Z^ν_N_f=-1(m,z;a) = e^2a^2∫_-∞^∞ds e^-νs e^im sinhs +izcoshs-4a^2cosh^2s = ∫_-∞^∞ds e^-νs e^-im sinhs -izcoshs-2a^2cosh(2s) . which agrees with the bosonic part of the result obtained in [ref] .
Microscopic Spectrum of the Wilson Dirac Operator · Diffusive Partition Function · 0.95
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As for the quenched case we have checked numerically that the one flavor microscopic density, which follows from the imaginary part of $G^ν_2|1$, is identical to the result [ref] obtained from chiral perturbation theory using the standard supersymmetric method.
Effects of dynamical quarks on the spectrum of the Wilson Dirac operator · One Flavor · 1.00
Cites as background 3 papers
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The chiral Lagrangian for Wilson chiral perturbation theory to $O(a^2)$ was derived in [ref] .
Chiral perturbation theory for the Wilson lattice action · Spectral Properties of the Hermitian Wilson Dirac Operator · 0.96
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The chiral Lagrangian for Wilson chiral perturbation theory to $O(a^2)$ was derived in [ref] .
Chiral perturbation theory at O(a^2) for lattice QCD · Spectral Properties of the Hermitian Wilson Dirac Operator · 0.96
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We consider the eigenvalue density of the Wilson Dirac operator in the microscopic scaling limit [ref] where the product of the eigenvalues and the four-volume, $V$, as well as the product $a^2 V$ are kept fixed.
Dirac Operator Spectra from Finite-Volume Partition Functions · Introduction · 0.92
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