We use numerical bootstrap techniques to study correlation functions of scalars transforming in the adjoint representation of $SU(N)$ in three dimensions. We obtain upper bounds on operator dimensions for various representations and study their depen
dence on $N$. We discover new families of kinks, one of which could be related to bosonic QED${}_3$. We then specialize to the cases $N=3,4$, which have been conjectured to describe a phase transition respectively in the ferromagnetic complex projective model $CP^2$ and the antiferromagnetic complex projective model $ACP^{3}$. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to small regions overlapping with the lattice predictions.
We study a supersymmetric tensor model with four supercharges and $O(N)^3$ global symmetry. The model is based on a chiral scalar superfield with three indices and quartic tetrahedral interaction in the superpotential, which is relevant below three d
imensions. In the large-$N$ limit the model is dominated by melonic diagrams. We solve the Dyson-Schwinger equations in superspace for generic $d$ and extract the dimension of the chiral field and the dimensions of bilinear operators transforming in various representations of $O(N)^3$. We find that all operator dimensions are real and above the unitarity bound for $1<d<3$. Our results also agree with perturbative results in $3-varepsilon$ expansion. Finally, we extract the large spin behaviour of bilinear operators and discuss the connection with lightcone bootstrap.
We use numerical bootstrap techniques to study correlation functions of a traceless symmetric tensors of $O(N)$ with two indexes $t_{ij}$. We obtain upper bounds on operator dimensions for all the relevant representations and several values of $N$. W
e discover several families of kinks, which do not correspond to any known model and we discuss possible candidates. We then specialize to the case $N=4$, which has been conjectured to describe a phase transition in the antiferromagnetic real projective model $ARP^{3}$. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to a closed region overlapping with the lattice prediction. The region is still present after pushing the numerics in the single correlator case or when considering a mixed system involving $t$ and the lowest dimension scalar singlet.
We perform a numerical bootstrap study of the mixed correlator system containing the half-BPS operators of dimension two and three in $mathcal N = 4$ Super Yang-Mills. This setup improves on previous works in the literature that only considered singl
e correlators of one or the other operator. We obtain upper bounds on the leading twist in a given representation of the R-symmetry by imposing gaps on the twist of all operators rather than the dimension of a single one. As a result we find a tension between the large $N$ supergravity predictions and the numerical finite $N$ results already at $Nsim 100$. A few possible solutions are discussed: the extremal spectrum suggests that at large but finite $N$, in addition to the double trace operators, there exists a second tower of states with smaller dimension. We also obtain new bounds on the dimension of operators which were not accessible with a single correlator setup. Finally we consider bounds on the OPE coefficients of various operators. The results obtained for the OPE coefficient of the lightest scalar singlet show evidences of a two dimensional conformal manifold.
We study the mixed system of correlation functions involving a scalar field charged under a global $U(1)$ symmetry and the associated conserved spin-1 current $J_mu$. Using numerical bootstrap techniques we obtain bounds on new observables not access
ible in the usual scalar bootstrap. We then specialize to the $O(2)$ model and extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the $O(2)$ model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors.
We explore consequences of the Averaged Null Energy Condition (ANEC) for scaling dimensions $Delta$ of operators in four-dimensional $mathcal{N}=1$ superconformal field theories. We show that in many cases the ANEC bounds are stronger than the corres
ponding unitarity bounds on $Delta$. We analyze in detail chiral operators in the $(frac12 j,0)$ Lorentz representation and prove that the ANEC implies the lower bound $Deltagefrac32j$, which is stronger than the corresponding unitarity bound for $j>1$. We also derive ANEC bounds on $(frac12 j,0)$ operators obeying other possible shortening conditions, as well as general $(frac12 j,0)$ operators not obeying any shortening condition. In both cases we find that they are typically stronger than the corresponding unitarity bounds. Finally, we elucidate operator-dimension constraints that follow from our $mathcal{N}=1$ results for multiplets of $mathcal{N}=2,4$ superconformal theories in four dimensions. By recasting the ANEC as a convex optimization problem and using standard semidefinite programming methods we are able to improve on previous analyses in the literature pertaining to the nonsupersymmetric case.
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 $O(N)$ 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.
In 4d $mathcal{N}=1$ superconformal field theories (SCFTs) the R-symmetry current, the stress-energy tensor, and the supersymmetry currents are grouped into a single object, the Ferrara-Zumino multiplet. In this work we study the most general form of
three-point functions involving two Ferrara-Zumino multiplets and a third generic multiplet. We solve the constraints imposed by conservation in superspace and show that non-trivial solutions can only be found if the third multiplet is R-neutral and transforms in suitable Lorentz representations. In the process we give a prescription for counting independent tensor structures in superconformal three-point functions. Finally, we set the Grassmann coordinates of the Ferrara-Zumino multiplets to zero and extract all three-point functions involving two R-currents and a third conformal primary. Our results pave the way for bootstrapping the correlation function of four R-currents in 4d $mathcal{N}=1$ SCFTs.
We consider simplified models for dark matter (DM) at the LHC, focused on mono-Higgs, -Z, or -b produced in the final state. Our primary purpose is to study the LHC reach of a relatively complete set of simplified models for these final states, while
comparing the reach of the mono-X DM search against direct searches for the mediating particle. We find that direct searches for the mediating particle, whether in di-jets, jets+MET, multi-b+MET, or di-boson+MET, are usually stronger. We draw attention to the cases that the mono-X search is strongest, which include regions of parameter space in inelastic DM, two Higgs doublet, and squark mediated production models with a compressed spectrum.
