7,469 research outputs found
On the cusp anomalous dimension in the ladder limit of SYM
We analyze the cusp anomalous dimension in the (leading) ladder limit of
SYM and present new results for its higher-order perturbative
expansion. We study two different limits with respect to the cusp angle .
The first is the light-like regime where . This limit is
characterised by a non-trivial expansion of the cusp anomaly as a sum of powers
of , where the maximum exponent increases with the loop order. The
coefficients of this expansion have remarkable transcendentality features and
can be expressed by products of single zeta values. We show that the whole
logarithmic expansion is fully captured by a solvable Woods-Saxon like
one-dimensional potential. From the exact solution, we extract generating
functions for the cusp anomaly as well as for the various specific
transcendental structures appearing therein. The second limit that we discuss
is the regime of small cusp angle. In this somewhat simpler case, we show how
to organise the quantum mechanical perturbation theory in a novel efficient way
by means of a suitable all-order Ansatz for the ground state of the associated
Schr\"odinger problem. Our perturbative setup allows to systematically derive
higher-order perturbative corrections in powers of the cusp angle as explicit
non-perturbative functions of the effective coupling. This series approximation
is compared with the numerical solution of the Schr\"odinger equation to show
that we can achieve very good accuracy over the whole range of coupling and
cusp angle. Our results have been obtained by relatively simple techniques.
Nevertheless, they provide several non-trivial tests useful to check the
application of Quantum Spectral Curve methods to the ladder approximation at
non zero , in the two limits we studied.Comment: 21 pages, 3 figure
Calibrating the Na\"ive Cornell Model with NRQCD
Along the years, the Cornell Model has been extraordinarily successful in
describing hadronic phenomenology, in particular in physical situations for
which an effective theory of the strong interactions such as NRQCD cannot be
applied. As a consequence of its achievements, a relevant question is whether
its model parameters can somehow be related to fundamental constants of QCD. We
shall give a first answer in this article by comparing the predictions of both
approaches. Building on results from a previous study on heavy meson
spectroscopy, we calibrate the Cornell model employing NRQCD predictions for
the lowest-lying bottomonium states up to NLO, in which the bottom mass is
varied within a wide range. We find that the Cornell model mass parameter can
be identified, within perturbative uncertainties, with the MSR mass at the
scale GeV. This identification holds for any value of or
the bottom mass, and for all perturbative orders investigated. Furthermore, we
show that: a) the "string tension" parameter is independent of the bottom mass,
and b) the Coulomb strength of the Cornell model can be related to the
QCD strong coupling constant at a characteristic non-relativistic
scale. We also show how to remove the renormalon of the static QCD
potential and sum-up large logs related to the renormalon subtraction by
switching to the low-scale, short-distance MSR mass, and using R-evolution. Our
R-improved expression for the static potential remains independent of the heavy
quark mass value and agrees with lattice QCD results for values of the radius
as large as fm, and with the Cornell model potential at long distances.
Finally we show that for moderate values of , the R-improved NRQCD and
Cornell static potentials are in head-on agreement.Comment: 22 pages, 13 figures, 3 table
Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new
approach to optimization problems with a large number of interacting agents.
The description of such models split in two parts, one describing the evolution
of the density of players in some parameter space, the other the value of a
cost functional each player tries to minimize for himself, anticipating on the
rational behavior of the others.
Quadratic Mean Field Games form a particular class among these systems, in
which the dynamics of each player is governed by a controlled Langevin equation
with an associated cost functional quadratic in the control parameter. In such
cases, there exists a deep relationship with the non-linear Schr\"odinger
equation in imaginary time, connexion which lead to effective approximation
schemes as well as a better understanding of the behavior of Mean Field Games.
The aim of this paper is to serve as an introduction to Quadratic Mean Field
Games and their connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
The Dirac equation, the concept of quanta, and the description of interactions in quantum electrodynamics
In this article the Dirac equation is used as a guideline to the historical emergence of the concept of quanta, associated with the quantum field. In Pascual Jordanâs approach, electrons as quanta result from the quantization of a classical field described by the Dirac equation. With this quantization procedure â also used for the electromagnetic field â the concept of quanta becomes a central piece in quantum electrodynamics. This does not seem to avoid the apparent impossibility of using the concept of quanta when interacting fields are considered together as a closed system. In this article it is defended that the type of analysis that leads to so drastic conclusions is avoidable if we look beyond the mathematical structure of the theory and take into account the physical ideas inscribed in this mathematical structure. In this case we see that in quantum electrodynamics we are not considering a closed system of interacting fields, what we have is a description of the interactions between distinct fields. In this situation the concept of quanta is central, the Fock space being the natural mathematical structure that permits maintaining the concept of quanta when considering the interaction between the fields
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