179 research outputs found
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
Bounds in 4D Conformal Field Theories with Global Symmetry
We explore the constraining power of OPE associativity in 4D Conformal Field
Theory with a continuous global symmetry group. We give a general analysis of
crossing symmetry constraints in the 4-point function ,
where Phi is a primary scalar operator in a given representation R. These
constraints take the form of 'vectorial sum rules' for conformal blocks of
operators whose representations appear in R x R and R x Rbar. The coefficients
in these sum rules are related to the Fierz transformation matrices for the R x
R x Rbar x Rbar invariant tensors. We show that the number of equations is
always equal to the number of symmetry channels to be constrained. We also
analyze in detail two cases - the fundamental of SO(N) and the fundamental of
SU(N). We derive the vectorial sum rules explicitly, and use them to study the
dimension of the lowest singlet scalar in the Phi x Phi* OPE. We prove the
existence of an upper bound on the dimension of this scalar. The bound depends
on the conformal dimension of Phi and approaches 2 in the limit dim(Phi)-->1.
For several small groups, we compute the behavior of the bound at dim(Phi)>1.
We discuss implications of our bound for the Conformal Technicolor scenario of
electroweak symmetry breaking.Comment: 30 page
Improved bounds for CFT's with global symmetries
The four point function of Conformal Field Theories (CFT's) with global
symmetry gives rise to multiple crossing symmetry constraints. We explicitly
study the correlator of four scalar operators transforming in the fundamental
representation of a global SO(N) and the correlator of chiral and anti-chiral
superfields in a superconformal field theory. In both cases the constraints
take the form of a triple sum rule, whose feasibility can be translated into
restrictions on the CFT spectrum and interactions. In the case of SO(N) global
symmetry we derive bounds for the first scalar singlet operator entering the
Operator Product Expansion (OPE) of two fundamental representations for
different value of N. Bounds for the first scalar traceless-symmetric
representation of the global symmetry are computed as well. Results for
superconformal field theories improve previous investigations due to the use of
the full set of constraints. Our analysis only assumes unitarity of the CFT,
crossing symmetry of the four point function and existence of an OPE for
scalars.Comment: 24 pages, 6 figure
The ABC (in any D) of Logarithmic CFT
Logarithmic conformal field theories have a vast range of applications, from
critical percolation to systems with quenched disorder. In this paper we
thoroughly examine the structure of these theories based on their symmetry
properties. Our analysis is model-independent and holds for any spacetime
dimension. Our results include a determination of the general form of
correlation functions and conformal block decompositions, clearing the path for
future bootstrap applications. Several examples are discussed in detail,
including logarithmic generalized free fields, holographic models,
self-avoiding random walks and critical percolation.Comment: 55 pages + appendice
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
Monojet versus rest of the world I: t-channel Models
Monojet searches using Effective Field Theory (EFT) operators are usually
interpreted as a robust and model independent constraint on direct detection
(DD) scattering cross-sections. At the same time, a mediator particle must be
present to produce the dark matter (DM) at the LHC. This mediator particle may
be produced on shell, so that direct searches for the mediating particle can
constrain the effective operator being applied to monojet constraints. In this
first paper, we do a case study on t-channel models in monojet searches, where
the (Standard Model singlet) DM is pair produced via a t-channel mediating
particle, whose supersymmetric analogue is the squark. We compare monojet
constraints to direct constraints on single or pair production of the mediator
from multi-jets plus missing energy searches and we identify the regions where
the latter dominate over the former. We show that computing bounds using
supersymmetric simplified models and in the narrow width approximation, as done
in previous work in the literature, misses important quantitative effects. We
perform a full event simulation and statistical analysis, and we compute the
effects of both on- and off-shell production of the mediating particle, showing
that for both the monojet and multi-jets plus missing energy searches,
previously derived bounds provided more conservative bounds than what can be
extracted by including all relevant processes in the simulation. Monojets and
searches for supersymmetry (SUSY) provide comparable bounds on a wide range of
the parameter space, with SUSY searches usually providing stronger bounds,
except in the regions where the DM particle and the mediator are very mass
degenerate. The EFT approximation rarely is able to reproduce the actual
limits. In a second paper to follow, we consider the case of s-channel
mediators.Comment: 22 pages + appendices, 10 figure
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
Bootstrapping the O(N) Archipelago
We study 3d CFTs with an global symmetry using the conformal bootstrap
for a system of mixed correlators. Specifically, we consider all nonvanishing
scalar four-point functions containing the lowest dimension vector
and the lowest dimension singlet , assumed to be the only
relevant operators in their symmetry representations. The constraints of
crossing symmetry and unitarity for these four-point functions force the
scaling dimensions to lie inside small islands. We
also make rigorous determinations of current two-point functions in the
and models, with applications to transport in condensed matter systems.Comment: 32 pages, 13 figures; updated Fig.2, added references and minor
corrections in Sec.3.
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