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Finite volume partition functions and Itzykson-Zuber integrals
High Energy Physics - Theory · Submitted 26 May 1996
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find the finite volume QCD partition function for arbitrary quark masses.
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The abstract, sentence by sentence
We find the finite volume QCD partition function for arbitrary quark masses. This is a generalization of a result obtained by Leutwyler and Smilga for equal quark masses. Our result is derived in the sector of zero topological charge using a generalization of the Itzykson-Zuber integral appropriate for arbitrary complex matrices. We present a conjecture regarding the result for arbitrary topological charge which reproduces the Leutwyler-Smilga result in the limit of equal quark masses. We derive a formula of the Itzykson-Zuber type for arbitrary {\em rectangular} complex matrices, extending the result of Guhr and Wettig obtained for {\em square} matrices.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We find the finite volume QCD partition function for arbitrary quark masses. | result 0.87 | 0.97 | 0.81 | asserted 1.00 |
| s1 | This is a generalization of a result obtained by Leutwyler and Smilga for equal quark masses. | background 0.45 | 0.39 | 0.10 | asserted 1.00 |
| s2 | Our result is derived in the sector of zero topological charge using a generalization of the Itzykson-Zuber integral appropriate for arbitrary complex matrices. | method 0.98 | 0.96 | 0.13 | asserted 1.00 |
| s3 | We present a conjecture regarding the result for arbitrary topological charge which reproduces the Leutwyler-Smilga result in the limit of equal quark masses. | contribution 0.65 | 0.88 | 0.14 | speculative 0.98 |
| s4 | We derive a formula of the Itzykson-Zuber type for arbitrary {\em rectangular} complex matrices, extending the result of Guhr and Wettig obtained for {\em square} matrices. | contribution 0.55 | 0.94 | 0.12 | asserted 1.00 |
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The paper
Original abstract
We find the finite volume QCD partition function for arbitrary quark masses. This is a generalization of a result obtained by Leutwyler and Smilga for equal quark masses. Our result is derived in the sector of zero topological charge using a generalization of the Itzykson-Zuber integral appropriate for arbitrary complex matrices. We present a conjecture regarding the result for arbitrary topological charge which reproduces the Leutwyler-Smilga result in the limit of equal quark masses. We derive a formula of the Itzykson-Zuber type for arbitrary {\em rectangular} complex matrices, extending the result of Guhr and Wettig obtained for {\em square} matrices.
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References
How the literature uses it
12 arXiv papers cite this in 14 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 3 papers
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Assuming $^∀μ_f∈$ we substitute the well-known analytic formula for the above unitary integral [ref] to obtain Z^K>4({μ_f}) = C_N_f exp( g_1/4h∑_f^μ_f^2 ) ∫_0^∞x x ^-x^2 1≤i,j≤N_f (x μ_i)^j-1 I_j-1(xμ_i) / Δ((xμ_1^)^2,…,(xμ_N_f^)^2) ,
Chiral symmetry breaking with no bilinear condensate revisited · Finite-volume partition function · 0.97
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(See also [ref] .) The procedure of this method consists of three steps [ref] : * Relate the joint distribution of the first $k$ eigenvalues to the partition function with $βk+β(ν+1)/2-1$ additional masses and a fixed topological charge $2/β+1$. * Replace the partition function by the microscopically-scaled form [ref] by taking the asymptotic limit ([ref]). * Integrate over the scaled variables $ζ_i$ ($i=1,…,k$) in a cell $0≤ζ_1≤⋯≤ζ_k-1≤ζ_k$.
Janossy densities for chiral random matrix ensembles and their applications to two-color QCD · Individual eigenvalue distributions · 0.90
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This integral can be analytically carried out [ref] and yields (for real $$) Z ∝ȷ_L I_0(2ȷ_R)I_1(2ȷ_L) - ȷ_R I_0(2ȷ_L)I_1(2ȷ_R)/ȷ_L^2-ȷ_R^2 .
Analysis of the QCD Kondo phase using random matrices · Low-energy effective theory · 0.85
Compares with it 1 paper
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For $N=2$ and $a=(m_1,m_2)$ it becomes [equation] which agrees with [ref] .
GUE-chGUE Transition preserving Chirality at finite Matrix Size · New Unitary Group Integral · 0.98
Cites it as background 3 papers
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where we have introduced the partition function for chGUE (cf. [ref] )
Heavy-tailed chiral random matrix theory · Nonzero masses · 0.85
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Such integrals were studied in Guhr and Wettig [ref] , and Jackson, Şener and Verbaarschot [ref] .
Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles · Joint probability density of the product of two correlated coupled matrices · 0.96
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The list of most well-studied unitary matrix integrals includes, but is not limited to, the Brezin-Gross-Witten integral [ref] (also known as the Leutwyler-Smilga integral [ref] ), the Harish-Chandra-Itzykson-Zuber integral [ref] , and the Berezin-Karpelevich integral [ref] .
Unitary matrix integral for QCD with real quarks and the GOE-GUE crossover · Introduction · 1.00
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