1,176 research outputs found
Individual Eigenvalue Distributions for the Wilson Dirac Operator
We derive the distributions of individual eigenvalues for the Hermitian
Wilson Dirac Operator D5 as well as for real eigenvalues of the Wilson Dirac
Operator DW. The framework we provide is valid in the epsilon regime of chiral
perturbation theory for any number of flavours Nf and for non-zero low energy
constants W6, W7, W8. It is given as a perturbative expansion in terms of the
k-point spectral density correlation functions and integrals thereof, which in
some cases reduces to a Fredholm Pfaffian. For the real eigenvalues of DW at
fixed chirality nu this expansion truncates after at most nu terms for small
lattice spacing "a". Explicit examples for the distribution of the first and
second eigenvalue are given in the microscopic domain as a truncated expansion
of the Fredholm Pfaffian for quenched D5, where all k-point densities are
explicitly known from random matrix theory. For the real eigenvalues of
quenched DW at small "a" we illustrate our method by the finite expansion of
the corresponding Fredholm determinant of size nu.Comment: 20 pages, 5 figures; v2: typos corrected, refs added and discussion
of W6 and W7 extende
Spectral Properties of the Overlap Dirac Operator in QCD
We discuss the eigenvalue distribution of the overlap Dirac operator in
quenched QCD on lattices of size 8^{4}, 10^{4} and 12^{4} at \beta = 5.85 and
\beta = 6. We distinguish the topological sectors and study the distributions
of the leading non-zero eigenvalues, which are stereographically mapped onto
the imaginary axis. Thus they can be compared to the predictions of random
matrix theory applied to the \epsilon-expansion of chiral perturbation theory.
We find a satisfactory agreement, if the physical volume exceeds about (1.2
fm)^{4}. For the unfolded level spacing distribution we find an accurate
agreement with the random matrix conjecture on all volumes that we considered.Comment: 16 pages, 8 figures, final version published in JHE
Smallest Dirac Eigenvalue Distribution from Random Matrix Theory
We derive the hole probability and the distribution of the smallest
eigenvalue of chiral hermitian random matrices corresponding to Dirac operators
coupled to massive quarks in QCD. They are expressed in terms of the QCD
partition function in the mesoscopic regime. Their universality is explicitly
related to that of the microscopic massive Bessel kernel.Comment: 4 pages, 1 figure, REVTeX. Minor typos in subscripts corrected.
Version to appear in Phys. Rev.
Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition
We introduce a random two-matrix model interpolating between a chiral
Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without
symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE)
and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n
limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit)
this theory is in one to one correspondence to the partition function of Wilson
chiral perturbation theory in the epsilon regime, such as the related two
matrix-model previously introduced in refs. [20,21]. For a generic number of
flavours and rectangular block matrices in the chGUE part we derive an
eigenvalue representation for the partition function displaying a Pfaffian
structure. In the quenched case with nu=0,1 we derive all spectral correlations
functions in our model for finite-n, given in terms of skew-orthogonal
polynomials. The latter are expressed as Gaussian integrals over standard
Laguerre polynomials. In the weakly non-chiral microscopic limit this yields
all corresponding quenched eigenvalue correlation functions of the Hermitian
Wilson operator.Comment: 27 pages, 4 figures; v2 typos corrected, published versio
The Several Guises of the BRST Symmetry
We present several forms in which the BRST transformations of QCD in
covariant gauges can be cast. They can be non-local and even not manifestly
covariant. These transformations may be obtained in the path integral formalism
by non standard integrations in the ghost sector or by performing changes of
ghost variables which leave the action and the path integral measure invariant.
For different changes of ghost variables in the BRST and anti-BRST
transformations these two transformations no longer anticommute.Comment: 3 pages, revte
Critical Behavior of Non Order-Parameter Fields
We show that all of the relevant features of a phase transition can be
determined using a non order parameter field which is a physical state of the
theory. This fact allows us to understand the deconfining transition of the
pure Yang-Mills theory via the physical excitations rather than using the
Polyakov loop.Comment: RevTeX, 4-pages, 1 figur
On Witten's global anomaly for higher SU(2) representations
The spectral flow of the overlap operator is computed numerically along a
particular path in gauge field space. The path connects two gauge equivalent
configurations which differ by a gauge transformation in the non-trivial class
of pi_4(SU(2)). The computation is done with the SU(2) gauge field in the
fundamental, the 3/2, and the 5/2 representation. The number of eigenvalue
pairs that change places along this path is established for these three
representations and an even-odd pattern predicted by Witten is verified.Comment: 24 pages, 12 eps figure
The Chiral Condensate in a Finite Volume
Chiral perturbation theory at finite four-volume V (=L^3T) is reconsidered
with a view towards finding a computational scheme that can deal with any value
of M_\pi L, where M_\pi is a generic Nambu-Goldstone mass. The momentum zero
modes that cause the usual p-expansion to fail in the chiral limit are treated
separately, and partly integrated out to all orders. In this way the theory
remains infrared finite in the perturbative expansion, and the chiral limit can
be considered at finite volume. We illustrate the technique by computing the
quark condensate in a finite volume, smoothly connecting standard results in
the p-regime for larger masses with those of the epsilon-regime for smaller
masses. From the partially quenched theory we also obtain the spectral density
of the Dirac operator, a smooth function from the microscopic region to the
bulk region of the p-regime.Comment: 33 pages, 10 figures, corrections in (4.7), (6.5), (6.8), additional
comment on (3.16
Deconfining Phase Transition as a Matrix Model of Renormalized Polyakov Loops
We discuss how to extract renormalized from bare Polyakov loops in SU(N)
lattice gauge theories at nonzero temperature in four spacetime dimensions.
Single loops in an irreducible representation are multiplicatively renormalized
without mixing, through a renormalization constant which depends upon both
representation and temperature. The values of renormalized loops in the four
lowest representations of SU(3) were measured numerically on small, coarse
lattices. We find that in magnitude, condensates for the sextet and octet loops
are approximately the square of the triplet loop. This agrees with a large
expansion, where factorization implies that the expectation values of loops in
adjoint and higher representations are just powers of fundamental and
anti-fundamental loops. For three colors, numerically the corrections to the
large relations are greatest for the sextet loop, ; these
represent corrections of for N=3. The values of the renormalized
triplet loop can be described by an SU(3) matrix model, with an effective
action dominated by the triplet loop. In several ways, the deconfining phase
transition for N=3 appears to be like that in the matrix model of
Gross and Witten.Comment: 24 pages, 7 figures; v2, 27 pages, 12 figures, extended discussion
for clarity, results unchange
Morphology of High-Multiplicity Events in Heavy Ion Collisions
We discuss opportunities that may arise from subjecting high-multiplicity
events in relativistic heavy ion collisions to an analysis similar to the one
used in cosmology for the study of fluctuations of the Cosmic Microwave
Background (CMB). To this end, we discuss examples of how pertinent features of
heavy ion collisions including global characteristics, signatures of collective
flow and event-wise fluctuations are visually represented in a Mollweide
projection commonly used in CMB analysis, and how they are statistically
analyzed in an expansion over spherical harmonic functions. If applied to the
characterization of purely azimuthal dependent phenomena such as collective
flow, the expansion coefficients of spherical harmonics are seen to contain
redundancies compared to the set of harmonic flow coefficients commonly used in
heavy ion collisions. Our exploratory study indicates, however, that these
redundancies may offer novel opportunities for a detailed characterization of
those event-wise fluctuations that remain after subtraction of the dominant
collective flow signatures. By construction, the proposed approach allows also
for the characterization of more complex collective phenomena like higher-order
flow and other sources of fluctuations, and it may be extended to the
characterization of phenomena of non-collective origin such as jets.Comment: Matches version accepted for publication in Physical Review C. 13
pages, 9 figure
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