The Quantum Focusing Conjecture (QFC) lies at the foundation of holography and semiclassical gravity. The QFC implies the Bousso bound and the Quantum Null Energy Condition (QNEC). The QFC also ensures the consistency of the quantum extremal surface prescription and bulk reconstruction in AdS/CFT.

United States

Raphael Bousso

Elisa Tabor

Discrete Max-Focusing

We present a new and fully general algorithm for the automated construction of the integrands of two-loop scattering amplitudes. This is achieved through a generalisation of the open-loops method to two loops. The core of the algorithm consists of a numerical recursion, where the various building blocks of two-loop diagrams are connected to each other through process-independent operations that depend only on the Feynman rules of the model at hand. This recursion is implemented in terms of tensor coefficients that encode the polynomial dependence of loop numerators on the two independent loop momenta. The resulting coefficients are ready to be combined with corresponding tensor integrals to form scattering probability densities at two loops. To optimise CPU efficiency we have compared several algorithmic options identifying one that outperforms naive solutions by two orders of magnitude. This new algorithm is implemented in the OpenLoops framework in a fully automated way for two-loop QED and QCD corrections to any Standard Model process. The technical performance is discussed in detail for several 2 → 2 and 2 → 3 processes with up to order 105 two-loop diagrams. We find that the CPU cost scales linearly with the number of two-loop diagrams and is comparable to the cost of corresponding real-virtual ingredients in a NNLO calculation. This new algorithm constitutes a key building block for the construction of an automated generator of scattering amplitudes at two loops.

Switzerland

Stefano Pozzorini

Natalie Schär

Max Zoller

Two-loop tensor integral coefficients in OpenLoops

Rational counterterms are a key ingredient for the automation of loop calculations through numerical methods. Building on the recently established properties of rational terms of UV origin at two loops, in this paper we present a systematic method for the determination of rational counterterms within spontaneously broken theories. In particular we introduce a generalised vev-expansion approach that makes it possible to obtain the rational counterterms of UV origin for a spontaneously broken theory by means of calculations in the unbroken phase. The drastic simplifications that result from the underlying symmetry open the door to the efficient determination of rational counterterms for the full Standard Model at two loops. The renormalisation-scheme dependence is analysed in detail, and we show that rational counterterms need to be determined only once and for all in a generic renormalisation scheme for the symmetric phase and, a posteriori, they can be easily adapted to a wide range of physical renormalisation schemes for the spontaneously broken phase. As a first application we determine the full set of   <math> <mi>O</mi> <mfenced> <msubsup> <mi>α</mi> <mi>S</mi> <mn>2</mn> </msubsup> </mfenced> </math>  $$ \mathcal{O}\left({\alpha}_{\mathrm{S}}^2\right) $$  rational counterterms of UV origin for the full Standard Model, i.e. for all superficially UV-divergent two-loop vertex functions involving combinations of gluons, quarks, electroweak vector bosons and scalar bosons.

Jean-Nicolas Lang

Hantian Zhang

Two-loop rational terms for spontaneously broken theories

Novel methods to reconstruct the slant depth of the maximum of the longitudinal profile ( $$X_{\mathrm{max}}$$  <math> <msub> <mi>X</mi> <mi>max</mi> </msub> </math>  ) of high-energy showers initiated by gamma-rays as well as their energy ( $$E_0$$  <math> <msub> <mi>E</mi> <mn>0</mn> </msub> </math>  ) are presented. The methods were developed for gamma rays with energies ranging from a few hundred GeV to  $$\sim 10$$  <math> <mrow> <mo>∼</mo> <mn>10</mn> </mrow> </math>   TeV. An estimator of  $$X_{\mathrm{max}}$$  <math> <msub> <mi>X</mi> <mi>max</mi> </msub> </math>   is obtained, event-by-event, from its correlation with the distribution of the arrival time of the particles at the ground, or the signal at the ground for lower energies. An estimator of  $$E_0$$  <math> <msub> <mi>E</mi> <mn>0</mn> </msub> </math>   is obtained, event-by-event, using a parametrization that has as inputs the total measured energy at the ground, the amount of energy contained in a region near to the shower core and the estimated  $$X_{\mathrm{max}}$$  <math> <msub> <mi>X</mi> <mi>max</mi> </msub> </math>  . Resolutions about  $$40 \, (20)\,\mathrm{g/cm^2}$$  <math> <mrow> <mn>40</mn> <mspace width="0.166667em"></mspace> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> <mspace width="0.166667em"></mspace> <mrow> <mi>g</mi> <mo>/</mo> <msup> <mi>cm</mi> <mn>2</mn> </msup> </mrow> </mrow> </math>   and about  $$30 \, (20)\%$$  <math> <mrow> <mn>30</mn> <mspace width="0.166667em"></mspace> <mo>(</mo> <mn>20</mn> <mo>)</mo> <mo>%</mo> </mrow> </math>   for, respectively,  $$X_{\mathrm{max}}$$  <math> <msub> <mi>X</mi> <mi>max</mi> </msub> </math>   and  $$E_0$$  <math> <msub> <mi>E</mi> <mn>0</mn> </msub> </math>   at  $$1 \, (10) \ \mathrm {TeV}$$  <math> <mrow> <mn>1</mn> <mspace width="0.166667em"></mspace> <mo>(</mo> <mn>10</mn> <mo>)</mo> <mspace width="4pt"></mspace> <mi>TeV</mi> </mrow> </math>   energies are obtained, considering vertical showers. The obtained results are auspicious and can lead to the opening of new physics avenues for large wide field-of-view gamma-ray observatories. The dependence of the resolutions with experimental conditions is discussed.

Portugal

R. Conceição

L. Peres

M. Pimenta

B. Tomé

New methods to reconstruct  $$X_{\mathrm{max}}$$  <math> <msub> <mi>X</mi> <mi>max</mi> </msub> </math>   and the energy of gamma-ray air showers with high accuracy in large wide-field observatories

Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to n f fermions with arbitrary masses.

Two-loop rational terms in Yang-Mills theories

The advent of efficient numerical algorithms for the construction of one-loop amplitudes has played a crucial role in the automation of NLO calculations, and the development of similar algorithms at two loops is a natural strategy for NNLO automation. Within a numerical framework the numerator of loop integrals is usually constructed in four dimensions, and the missing rational terms, which arise from the interplay of the ( 4)-dimensional parts of the loop numerator with 1 ( 4) poles in dimensions, are reconstructed separately. At one loop, such rational terms arise only from UV divergences and can be restored through process-independent local counterterms. In this paper we investigate the behaviour of rational terms of UV origin at two loops. The main result is a general formula that combines the subtraction of UV poles with the reconstruction of the associated rational parts at two loops. This formula has the same structure as the R-operation, and all poles and rational parts are described through a finite set of process-independent local counterterms. We also present a general formula for the calculation of all relevant two-loop rational counterterms in any renormalisable theory based on one-scale tadpole integrals. As a first application, we derive the full set of two-loop rational counterterms for QED in the -gauge.

Rational terms of UV origin at two loops

The atmospheric depth where the energy deposit profile of secondary particles from extensive air showers (EAS) reaches its maximum, $$X_{\mathrm{max}}$$ <math><msub><mi>X</mi><mi>max</mi></msub></math> , is related to the primary particle mass. The mass composition of the ultra-high energy cosmic rays (UHECRs) can be inferred from measurements of $$X_{\mathrm{max}}$$ <math><msub><mi>X</mi><mi>max</mi></msub></math>  distributions in each energy interval, by fitting these distributions with Monte Carlo (MC) templates for four primary species (p, He, N and Fe). On the basis of simulations, we show that a high abundance of some intermediate elements in the $$X_{\mathrm{max}}$$ <math><msub><mi>X</mi><mi>max</mi></msub></math>  distributions, e.g. Ne or Si, may affect the quality of the fit and also the reconstructed fractions of different species with respect to their true values. We propose a method for finding the “best combination” of elements in each energy interval from a larger set of primaries (p, He, C, N, O, Ne, Si and Fe) which best describes the $$X_{\mathrm{max}}$$ <math><msub><mi>X</mi><mi>max</mi></msub></math>  distributions. Applying this method to the $$X_{\mathrm{max}}$$ <math><msub><mi>X</mi><mi>max</mi></msub></math>  distributions measured by the Pierre Auger Observatory (2014), we found that the “best combination” of elements which best describe the data suggest the presence of Ne or Si in some low energy bins for the EPOS-LHC model.

Romania

Nicusor Arsene

Octavian Sima

UHECRs mass composition from $$X_{\mathrm{max}}$$ <math><msub><mi>X</mi><mi>max</mi></msub></math>  distributions

Theoretical predictions for t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  production are of crucial importance for t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  H measurements in the H → b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  channel at the LHC. To address the large uncertainties associated with the modelling of extra QCD radiation in t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  events, in this paper we present a calculation of pp → t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  j at NLO QCD. The behaviour of NLO corrections is analysed in a variety of observables, and to assess theoretical uncertainties we use factor- two rescalings as well as different dynamic scales. In this context, we propose a systematic alignment of dynamic scales that makes it possible to disentangle normalisation and shape uncertainties in a transparent way. Scale uncertainties at NLO are typically at the level of 20–30% in integrated cross sections, and below 10% for the shapes of distributions. The kinematics of QCD radiation is investigated in detail, including the effects of its recoil on the objects of the t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  system. In particular, we discuss various azimuthal correlations that allow one to characterise the QCD recoil pattern in a precise and transparent way. In general, the calculation at hand provides a variety of precise benchmarks that can be used to validate the modelling of QCD radiation in t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  generators. Moreover, as we will argue, pp → t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  j at NLO entails information that can be used to gain insights into the perturbative convergence of the inclusive t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  cross section beyond NLO. Based on this idea, we address the issue of the large NLO K-factor observed in   <math> <msub> <mi>σ</mi> <mrow> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>b</mi> <mover> <mi>b</mi> <mo>¯</mo> </mover> </mrow> </msub> </math>  $$ {\sigma}_{t\overline{t}b\overline{b}} $$ , and we provide evidence that supports the reduction of this K-factor through a mild adjustment of the QCD scales that are conventionally used for this process. The presented 2 → 5 NLO calculations have been carried out using OpenLoops 2 in combination with Sherpa and Munich.

Federico Buccioni

Italy

Stefan Kallweit

NLO QCD predictions for t   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  b   <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$  production in association with a light jet at the LHC

We present the new version of OpenLoops, an automated generator of tree and one-loop scattering amplitudes based on the open-loop recursion. One main novelty of OpenLoops 2 is the extension of the original algorithm from NLO QCD to the full Standard Model, including electroweak (EW) corrections from gauge, Higgs and Yukawa interactions. In this context, among several new features, we discuss the systematic bookkeeping of QCD–EW interferences, a flexible implementation of the complex-mass scheme for processes with on-shell and off-shell unstable particles, a special treatment of on-shell and off-shell external photons, and efficient scale variations. The other main novelty is the implementation of the recently proposed on-the-fly reduction algorithm, which supersedes the usage of external reduction libraries for the calculation of tree–loop interferences. This new algorithm is equipped with an automated system that avoids Gram-determinant instabilities through analytic methods in combination with a new hybrid-precision approach based on a highly targeted usage of quadruple precision with minimal CPU overhead. The resulting significant speed and stability improvements are especially relevant for challenging NLO multi-leg calculations and for NNLO applications.

United Kingdom

Jonas Lindert

Germany

Philipp Maierhöfer

OpenLoops 2

In the context of massive (bi-)gravity, non-minimal matter couplings have been proposed. These couplings are special in the sense that they are free of the Boulware-Deser ghost below the strong coupling scale and can be used consistently as an effective field theory. Furthermore, they enrich the phenomenology of massive gravity. We consider these couplings in the framework of bimetric gravity and study the cosmological implications for background and linear tensor, vector, and scalar Previous works have investigated special branches of solutions. Here we perform a complete perturbation analysis for the general background equations of motion, completing previous analyses.

Japan

China

Xian Gao

Lavinia Heisenberg

Doubly coupled matter fields in massive bigravitySupported by the Chinese National Youth Thousand Talents Program (71000-41180003), JSPS Grant-in-Aid for Scientific Research (15H02082, 25287054, 26610062), L. H. Acknowledges Financial Support from Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation