132,322 research outputs found
Parametrization-free determination of the shape parameters for the pion electromagnetic form factor
Recent data from high statistics experiments that have measured the modulus
of the pion electromagnetic form factor from threshold to relatively high
energies are used as input in a suitable mathematical framework of analytic
continuation to find stringent constraints on the shape parameters of the form
factor at . The method uses also as input a precise description of the
phase of the form factor in the elastic region based on Fermi-Watson theorem
and the analysis of the scattering amplitude with dispersive Roy
equations, and some information on the spacelike region coming from recent high
precision experiments. Our analysis confirms the inconsistencies of several
data on the modulus, especially from low energies, with analyticity and the
input phase, noted in our earlier work. Using the data on the modulus from
energies above , we obtain, with no specific parametrization,
the prediction for the charge
radius. The same formalism leads also to very narrow allowed ranges for the
higher-order shape parameters at , with a strong correlation among them.Comment: v2 is 11 pages long using EPJ style files, and has 8 figures;
Compared to v1, number of figures has been reduced, discussion has been
improved significantly, minor errors have been corrected, references have
added, and the manuscript has been significantly revised; this version has
been accepted for publication in EPJ
Central Charge Bounds in 4D Conformal Field Theory
We derive model-independent lower bounds on the stress tensor central charge
C_T in terms of the operator content of a 4-dimensional Conformal Field Theory.
More precisely, C_T is bounded from below by a universal function of the
dimensions of the lowest and second-lowest scalars present in the CFT. The
method uses the crossing symmetry constraint of the 4-point function, analyzed
by means of the conformal block decomposition.Comment: 16 pages, 6 figure
The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
Conformal field theories have been long known to describe the fascinating
universal physics of scale invariant critical points. They describe continuous
phase transitions in fluids, magnets, and numerous other materials, while at
the same time sit at the heart of our modern understanding of quantum field
theory. For decades it has been a dream to study these intricate strongly
coupled theories nonperturbatively using symmetries and other consistency
conditions. This idea, called the conformal bootstrap, saw some successes in
two dimensions but it is only in the last ten years that it has been fully
realized in three, four, and other dimensions of interest. This renaissance has
been possible both due to significant analytical progress in understanding how
to set up the bootstrap equations and the development of numerical techniques
for finding or constraining their solutions. These developments have led to a
number of groundbreaking results, including world record determinations of
critical exponents and correlation function coefficients in the Ising and
models in three dimensions. This article will review these exciting
developments for newcomers to the bootstrap, giving an introduction to
conformal field theories and the theory of conformal blocks, describing
numerical techniques for the bootstrap based on convex optimization, and
summarizing in detail their applications to fixed points in three and four
dimensions with no or minimal supersymmetry.Comment: 81 pages, double column, 58 figures; v3: updated references, minor
typos correcte
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Carving Out the Space of 4D CFTs
We introduce a new numerical algorithm based on semidefinite programming to
efficiently compute bounds on operator dimensions, central charges, and OPE
coefficients in 4D conformal and N=1 superconformal field theories. Using our
algorithm, we dramatically improve previous bounds on a number of CFT
quantities, particularly for theories with global symmetries. In the case of
SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal
technicolor. In N=1 superconformal theories, we place strong bounds on
dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the
line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive
anomalous dimensions in this region. We also place novel upper and lower bounds
on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we
find examples of lower bounds on central charges and flavor current two-point
functions that scale with the size of global symmetry representations. In the
case of N=1 theories with an SU(N) flavor symmetry, our bounds on current
two-point functions lie within an O(1) factor of the values realized in
supersymmetric QCD in the conformal window.Comment: 60 pages, 22 figure
Bounds for State Degeneracies in 2D Conformal Field Theory
In this note we explore the application of modular invariance in
2-dimensional CFT to derive universal bounds for quantities describing certain
state degeneracies, such as the thermodynamic entropy, or the number of
marginal operators. We show that the entropy at inverse temperature 2 pi
satisfies a universal lower bound, and we enumerate the principal obstacles to
deriving upper bounds on entropies or quantum mechanical degeneracies for fully
general CFTs. We then restrict our attention to infrared stable CFT with
moderately low central charge, in addition to the usual assumptions of modular
invariance, unitarity and discrete operator spectrum. For CFT in the range
c_left + c_right < 48 with no relevant operators, we are able to prove an upper
bound on the thermodynamic entropy at inverse temperature 2 pi. Under the same
conditions we also prove that a CFT can have a number of marginal deformations
no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.Comment: 23 pages, LaTeX, minor change
Five dimensional -symmetric CFTs from conformal bootstrap
We investigate the conformal bootstrap approach to symmetric CFTs in
five dimension with particular emphasis on the lower bound on the current
central charge. The bound has a local minimum for all , and in the large
limit we propose that the minimum is saturated by the critical
vector model at the UV fixed point, the existence of which has been recently
argued by Fei, Giombi, and Klebanov. The location of the minimum is generically
different from the minimum of the lower bound of the energy-momentum tensor
central charge when it exists for smaller .
To better understand the situation, we examine the lower bounds of the
current central charge of symmetric CFTs in three dimension to compare.
We find the similar agreement in the large limit but the discrepancy for
smaller with the other sectors of the conformal bootstrap.Comment: 5 pages, 6 figures. v2: minor change
Solving the 3D Ising Model with the Conformal Bootstrap
We study the constraints of crossing symmetry and unitarity in general 3D
Conformal Field Theories. In doing so we derive new results for conformal
blocks appearing in four-point functions of scalars and present an efficient
method for their computation in arbitrary space-time dimension. Comparing the
resulting bounds on operator dimensions and OPE coefficients in 3D to known
results, we find that the 3D Ising model lies at a corner point on the boundary
of the allowed parameter space. We also derive general upper bounds on the
dimensions of higher spin operators, relevant in the context of theories with
weakly broken higher spin symmetries.Comment: 32 pages, 11 figures; v2: refs added, small changes in Section 5.3,
Fig. 7 replaced; v3: ref added, fits redone in Section 5.
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