3,751 research outputs found
A Solution of the Cauchy Problem for the Loop Equation in Turbulence
Under certain conditions, imposed on the viscosity of the fluid, initial data
and the class of contours under consideration, the Cauchy problem with finite
values of time for the loop equation in turbulence with Gaussian random forces
is solved by making use of the smearing procedure for the loop space functional
Laplacian. The solution obtained depends on the initial data and its functional
derivatives and on the potential of the random forces.Comment: 6 pages, LaTeX, no figure
Dynamics of the Universal Confining String Theory on the Loop Space
Starting with the representation of the Wilson average in the Euclidean 4D
compact QED as a partition function of the Universal Confining String Theory,
we derive for it the corresponding loop equation, alternative to the familiar
one.
In the functional momentum representation the obtained equation decouples
into two independent ones, which describe the dynamics of the transverse and
longitudinal components of the area derivative of the Wilson loop. At some
critical value of the momentum discontinuity, which can be determined from a
certain equation, the transverse component does not propagate.
Next, we derive the equation for the momentum Wilson loop, where on the
left-hand side stands the sum of the squares of the momentum discontinuities,
multiplied by the loop, which describes its free propagation, while the
right-hand side describes the interaction of the loop with the functional
vorticity tensor current.
Finally, using the method of inversion of the functional Laplacian, we obtain
for the Wilson loop in the coordinate representation a simple Volterra type-II
linear integral equation, which can be treated perturbatively.Comment: 7 pages, LaTeX, no figures, a remark is adde
Gluodynamics String as a Low-Energy Limit of the Universal Confining String Theory
An effective string theory emerging from the bilocal approximation to the
Method of Vacuum Correlators in gluodynamics is shown to be well described by
the 4D theory of the massive Abelian Kalb-Ramond field interacting with the
string, which is known to be the low-energy limit of the Universal Confining
String Theory. The mass of the Kalb-Ramond field in this approach plays the
role of the inverse correlation length of the vacuum, and it is shown that in
the massless limit string picture disappears. The background field method,
known in the theory of nonlinear sigma models, is applied to derivation of the
effective action, quadratic in quantum fluctuations around a given (e.g.
minimal) string world-sheet.Comment: 11 pages, LaTeX, no figures, 2 references are updated, to be
published in Phys. Lett.
Stochastic loop equations
Stochastic quantization is applied to derivation of the equations for the
Wilson loops and generating functionals of the Wilson loops in the large-N
limit. These equations are treated both in the coordinate and momentum
representations. In the first case the connection of the suggested approach
with the problem of random closed contours and supersymmetric quantum mechanics
is established, and the equation for the Quenched Master Field Wilson loop is
derived. The regularized version of one of the obtained equations is presented
and applied to derivation of the equation for the bilocal field correlator. The
momentum loop dynamics is also investigated.Comment: 13 page
String Representation for the 't Hooft Loop Average in the Abelian Higgs Model
Making use of the duality transformation, we derive in the Londons' limit of
the Abelian Higgs Model string representation for the 't Hooft loop average
defined on the string world-sheet, which yields the values of two coefficient
functions parametrizing the bilocal correlator of the dual field strength
tensors. The asymptotic behaviours of these functions agree with the ones
obtained within the Method of Vacuum Correlators in QCD in the lowest order of
perturbation theory. We demonstrate that the bilocal approximation to the
Method of Vacuum Correlators is an exact result in the Londons' limit, i.e. all
the higher cumulants in this limit vanish. We also show that at large
distances, apart from the integration over metrics, the obtained string
effective theory (which in this case reduces to the nonlinear massive axionic
sigma model) coincides with the low-energy limit of the dual version of 4D
compact QED, the so-called Universal Confining String Theory. We derive string
tension of the Nambu-Goto term and the coupling constant of the rigidity term
for the obtained string effective theory and demonstrate that the latter one is
always negative, which means the stability of strings, while the positiveness
of the former is confirmed by the present lattice data. These data enable us to
find the Higgs boson charge and the vacuum expectation value of the Higgs
field, which model QCD best of all. We also study dynamics of the weight factor
of the obtained string representation for the 't Hooft average in the loop
space. In conclusion, we obtain string representation for the partition
function of the correlators of an arbitrary number of Higgs currents, by virtue
of which we rederive the structure of the bilocal correlator of the dual field
strength tensors, which yields the surface term in the string effective action.Comment: 11 pages, LaTeX, no figures, references are adde
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