There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Collider physics at the precision frontier

Simple singularities and simple algebraic groups

We report on a new fully differential calculation of the next-to-next-to-leading-order (NNLO) QCD radiative corrections to the production of top-quark pairs at hadron colliders. The calculation is performed by using the qT subtraction formalism to handle and cancel infrared singularities in real and virtual contributions. The computation is implemented in the Matrix framework, thereby allowing us to efficiently compute arbitrary infrared-safe observables for stable top quarks. We present NNLO predictions for several single- and double-differential kinematical distributions in pp collisions at the centre-of-mass energy $$ \sqrt{s}=13 $$
 TeV, and we compare them with recent LHC data by the CMS collaboration.

/pdf/top-quark-pair-production-at-the-lhc-fully-differential-qcd-ty6w4f61sw.pdf

Top-quark pair production at the LHC: fully differential QCD predictions at NNLO

We study the O(N) and Gross-Neveu models at large N on AdSd+1 background. Thanks to the isometries of AdS, the observables in these theories are constrained by the SO(d, 2) conformal group even in the presence of mass deformations, as was discussed by Callan and Wilczek [1], and provide an interesting two-parameter family of quantities which interpolate between the S-matrices in flat space and the correlators in CFT with a boundary. For the actual computation, we judiciously use the spectral representation to resum loop diagrams in the bulk. After the resummation, the AdS 4-particle scattering amplitude is given in terms of a single unknown function of the spectral parameter. We then “bootstrap” the unknown function by requiring the absence of double-trace operators in the boundary OPE. Our results are at leading nontrivial order in $$ \frac{1}{N} $$
 , and include the full dependence on the quartic coupling, the mass parameters, and the AdS radius. In the bosonic O(N) model we study both the massive phase and the symmetry-breaking phase, which exists even in AdS2 evading Coleman’s theorem, and identify the AdS analogue of a resonance in flat space. We then propose that symmetry breaking in AdS implies the existence of a conformal manifold in the boundary conformal theory. We also provide evidence for the existence of a critical point with bulk conformal symmetry, matching existing results and finding new ones for the conformal boundary conditions of the critical theories. For the Gross-Neveu model we find a bound state, which interpolates between the familiar bound state in flat space and the displacement operator at the critical point.

/pdf/a-study-of-quantum-field-theories-in-ads-at-finite-coupling-257ogbrwyc.pdf

A study of quantum field theories in AdS at finite coupling

We consider the scattering matrices of massive quantum field theories with no bound states and a global O(N) symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass m transforming in the vector representation as restricted by the general conditions of unitarity, crossing, analyticity and O(N) symmetry. We found a rich structure in that space by using convex maximization and in particular its convex dual minimization problem. At the boundary of the allowed space special geometric points such as vertices were found to correspond to integrable models. The dual convex minimization problem provides a novel and useful approach to the problem allowing, for example, to prove that generically the S-matrices so obtained saturate unitarity and, in some cases, that they are at vertices of the allowed space.

/pdf/the-o-n-s-matrix-monolith-2x3c88xjvj.pdf

The O(N) S-matrix monolith

In this work we apply the S-matrix bootstrap maximization program to the 2d bosonic O(N) integrable model which has N species of scalar particles of mass m and no bound states. Since in previous studies theories were defined by maximizing the coupling between particles and their bound states, the main problem appears to be to find what other functional can be used to define this model. Instead, we argue that the defining property of this integrable model is that it resides at a vertex of the convex space determined by the unitarity and crossing constraints. Thus, the integrable model can be found by maximizing any linear functional whose gradient points in the general direction of the vertex, namely within a cone determined by the normals to the faces intersecting at the vertex. This is a standard problem in applied mathematics, related to semi-definite programming and solvable by fast available numerical algorithms. The information provided by the numerical solution is enough to reproduce the known analytical solution without using integrability, namely the Yang-Baxter equation. This situation seems quite generic so we expect that other theories without continuous parameters can also be found by maximizing linear functionals in the convex space of allowed S-matrices.

/pdf/a-note-on-the-s-matrix-bootstrap-for-the-2d-o-n-bosonic-39xgg2wxex.pdf

A note on the S-matrix bootstrap for the 2d O(N) bosonic model

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kl(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fl(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

/pdf/s-matrix-bootstrap-in-3-1-dimensions-regularization-and-dual-2qdn3ia5xy.pdf

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the $2\rightarrow 2$ scattering matrix $S_{2\rightarrow 2}$ such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider $3+1$ dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves $k_\ell(s)$ that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves $f_\ell(s)$, for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

/pdf/s-matrix-bootstrap-in-3-1-dimensions-regularization-and-dual-v8zsk079tf.pdf

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem.

Based on the spectrum identified in our earlier work [arXiv:1809.02191], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the $Q$-state Potts model. Crucial in our approach is the existence of "interchiral conformal blocks", which arise from the degeneracy of fields with conformal weight $h_{r,1}$, with $r\in\mathbb{N}^*$, and are related to the underlying presence of the "interchiral algebra" introduced in [arXiv:1207.6334]. We also find evidence for the existence of "renormalized" recursions, replacing those that follow from the degeneracy of the field $\Phi_{12}^D$ in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

/pdf/geometrical-four-point-functions-in-the-two-dimensional-npilrm8wv3.pdf

Yifei He

Papers

The O(N) S-matrix monolith

A note on the S-matrix bootstrap for the 2d O(N) bosonic model

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem.

Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap