Cross-ratio Function Research Articles

Embedding space approach to Lorentzian CFT amplitudes and causal spherical functions

Conformal field theory in a Minkowski setting is discussed in an embedding space approach, paying special attention to causality constraints for four-point amplitudes. The physics of dilatation and Lorentz boost is emphasized in specifying the noncompact maximal Abelian subgroup of SO(d,2). Reduction of a conformal field theory four-point amplitudes as functions of cross ratios is shown to be equivalent to enforcing H bi-invariance, i.e., F(hgh′)=F(g), with g∈SO(d,2) and H an appropriate subgroup. Causality is imposed by introducing appropriate semigroups. Causal zonal spherical functions are constructed, making contact with Minkowski conformal blocks introduced previously. Published by the American Physical Society 2024

Bootstrap of the defect 1/2 BPS Wilson lines in N=4 Chern-Simons-matter theories

We compute correlation functions of local operator insertions on the 1/2 Bogomol’nyi-Prasad-Sommerfield Wilson lines of N=4 Chern-Simons-matter theories in three dimensions. We study the algebra preserved by the defect conformal field theory supported on the line, identify the superdisplacement multiplet, and discuss some of its weak-coupling realizations. By employing a superspace description, we present the four-point functions of the superdisplacement and show how they are determined by functions of cross ratios. Within an analytic bootstrap approach, we derive these functions at leading and next-to-leading order at strong coupling, obtaining a result in agreement with appropriate orbifolds of the ABJM case considered in Bianchi [Analytic bootstrap and Witten diagrams for the ABJM Wilson line as defect CFT1, ]. Published by the American Physical Society 2024

Bernstein-based estimation of the cross ratio function

Local association measures provide useful insights in time-varying changes in association, especially between time-to-event variables. Such local dependence between two correlated random variables can be measured using the cross ratio function. The cross ratio function is defined as the ratio of conditional hazard functions which have been estimated using Bernstein polynomials before. Alternatively, the cross ratio function can be expressed in terms of (derivatives of) the joint survival function of the two random variables. In this paper, we discuss an alternative Bernstein-based plug-in estimator of the cross ratio function in which each of the ingredients is estimated separately. Next to asymptotic normality of the nonparametric estimator, a simulation study is used to assess its finite-sample performance. Finally, the novel estimator is applied to a real-life data application.

Wilson networks in AdS and global conformal blocks

We develop the relation between gravitational Wilson line networks, defined as a particular product of Wilson line operators averaged over the cap states, and conformal correlators in the context of the AdS2/CFT1 correspondence. The n-point sl(2,R) comb channel global conformal block in CFT1 is explicitly calculated by means of the extrapolate dictionary relation from the gravitational Wilson line network with n boundary endpoints stretched in AdS2. Remarkably, the Wilson line calculation directly yields the conformal block in a particularly simple form which up to the leg factor is given by the comb function of cross-ratios. It is also found that the comb channel structure constants are expressed in terms of factorials and triangle functions of conformal weights which form determines fusion rules for a given 3-valent vertex. We obtain analytic expressions for the Wilson line matrix elements in AdS2 which are building blocks of the Wilson line networks. We analyze general cap states and specify those which lead to asymptotic values of the Wilson line networks interpreted as boundary correlators of CFT1 primary operators. The cases of (in)finite-dimensional sl(2,R) modules carried by Wilson lines are treated on equal footing that boils down to consideration of singular submodules and their contributions to the Wilson line matrix elements.

Shift operators from the simplex representation in momentum-space CFT

We derive parametric integral representations for the general n-point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman integral with the topology of an (n − 1)-simplex, featuring an arbitrary function of momentum-space cross ratios. Here, we show all graph polynomials for this integral can be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of weight-shifting operators, expressible as determinants, that connect n-point functions in spacetime dimensions differing by two. Moreover, the action of all previously known weight-shifting operators preserving the spacetime dimension is manifest. Finally, the new parametric representations enable the validity of the conformal Ward identities to be established directly, without recourse to recursion in the number of points.

Conformal correlators as simplex integrals in momentum space

We find the general solution of the conformal Ward identities for scalar n-point functions in momentum space and in general dimension. The solution is given in terms of integrals over (n − 1)-simplices in momentum space. The n operators are inserted at the n vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where n-point functions are built in terms of (n − 1)-point functions. To illustrate our discussion, we derive the simplex representation of n-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves (n − 2) integrations, which is an improvement (when n > 4) relative to the Mellin representation that involves n(n − 3)/2 integrations.

Conformal group theory of tensor structures

The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are functions of cross ratios only, and the correlation functions that depend on insertion points in the d-dimensional Euclidean space. Here we develop an entirely group theoretic approach to tensor structures, based on the Cartan decomposition of the conformal group. It provides us with a new universal formula for tensor structures and thereby a systematic derivation of crossing equations. Our approach applies to a ‘gauge’ in which the conformal blocks are wave functions of Calogero-Sutherland models rather than solutions of the more standard Casimir equations. Through this ab initio construction of tensor structures we complete the Calogero-Sutherland approach to conformal correlators, at least for four-point functions of local operators in non-supersymmetric models. An extension to defects and superconformal symmetry is possible.

Nonparametric estimation of the cross ratio function

The cross ratio function (CRF) is a commonly used tool to describe local dependence between two correlated variables. Being a ratio of conditional hazards, the CRF can be rewritten in terms of (first and second derivatives of) the survival copula of these variables. Bernstein estimators for (the derivatives of) this survival copula are used to define a nonparametric estimator of the cross ratio, and asymptotic normality thereof is established. We consider simulations to study the finite sample performance of our estimator for copulas with different types of local dependency. A real dataset is used to investigate the dependence between food expenditure and net income. The estimated CRF reveals that families with a low net income relative to the mean net income will spend less money to buy food compared to families with larger net incomes. This dependence, however, disappears when the net income is large compared to the mean income.

Proportional cross-ratio model.

Cross-ratio is an important local measure of the strength of dependence among correlated failure times. If a covariate is available, it may be of scientific interest to understand how the cross-ratio varies with the covariate as well as time components. Motivated by the Tremin study, where the dependence between age at a marker event reflecting early lengthening of menstrual cycles and age at menopause may be affected by age at menarche, we propose a proportional cross-ratio model through a baseline cross-ratio function and a multiplicative covariate effect. Assuming a parametric model for the baseline cross-ratio, we generalize the pseudo-partial likelihood approach of Hu et al. (Biometrika 98:341-354, 2011) to the joint estimation of the baseline cross-ratio and the covariate effect. We show that the proposed parameter estimator is consistent and asymptotically normal. The performance of the proposed technique in finite samples is examined using simulation studies. In addition, the proposed method is applied to the Tremin study for the dependence between age at a marker event and age at menopause adjusting for age at menarche. The method is also applied to the Australian twin data for the estimation of zygosity effect on cross-ratio for age at appendicitis between twin pairs.

Reliability modelling incorporating load share and frailty

The stochastic behaviour of lifetimes of a two component system is often primarily influenced by the system structure and by the covariates shared by the components. Any meaningful attempt to model the lifetimes must take into consideration the factors affecting their stochastic behaviour. In particular, for a load share system, we describe a reliability model incorporating both the load share dependence and the effect of observed and unobserved covariates. The model includes a bivariate Weibull to characterize load share, a positive stable distribution to describe frailty, and also incorporates effects of observed covariates. We investigate various interesting reliability properties of this model using cross ratio functions and conditional survivor functions. We implement maximum likelihood estimation of the model parameters and discuss model adequacy and selection. We illustrate our approach using a simulation study. For a real data situation, we demonstrate the superiority of the proposed model that incorporates both load share and frailty effects over competing models that incorporate just one of these effects. An attractive and computationally simple cross‐validation technique is introduced to reconfirm the claim. We conclude with a summary and discussion.

On the shape of the cross-ratio function in bivariate survival models induced by truncated and folded normal frailty distributions

In shared frailty models for bivariate survival data the frailty is identifiable through the cross-ratio function (CRF), which provides a convenient measure of association for correlated survival variables. The CRF may be used to compare patterns of dependence across models and data sets. We explore the shape of the CRF for the families of one-sided truncated normal and folded normal frailty distributions.

On the conditional probability for assessing time dependence of association in shared frailty models with bivariate current status data

Shared frailty models are frequently used for inducing dependence between survival times. In this paper, we consider bivariate current status data that are reasonable to model by shared frailty models. A time-dependent association measure that has a conditional probability interpretation is revisited for its potential application to such data. We propose a method of estimation and derive asymptotic standard errors for this measure. Its small sample performance and its performance in assessing the temporal variation in the strength of association in realistic scenarios is investigated by means of experiments. We show that the measure based on the conditional probability can vary with time even in the absence of any time-dependent effects. Furthermore, we give evidence that it lacks interpretability in suggesting appropriate frailty models. We provide an illustration with multivariate current status data arising from a community-based study of cardiovascular diseases in Taiwan. We compare the observed patterns of association with the ones obtained by employing a fairly new time-varying association measure that is relevant for shared frailty models, owing to its connection to the cross-ratio function, and which serves as a diagnostic tool for suggesting classes of frailty distributions with constant, increasing or decreasing association over time.

Dynamic frailty models based on compound birth-death processes.

Frailty models are used in survival analysis to model unobserved heterogeneity. They accommodate such heterogeneity by the inclusion of a random term, the frailty, which is assumed to multiply the hazard of a subject (individual frailty) or the hazards of all subjects in a cluster (shared frailty). Typically, the frailty term is assumed to be constant over time. This is a restrictive assumption and extensions to allow for time-varying or dynamic frailties are of interest. In this paper, we extend the auto-correlated frailty models of Henderson and Shimakura and of Fiocco, Putter and van Houwelingen, developed for longitudinal count data and discrete survival data, to continuous survival data. We present a rigorous construction of the frailty processes in continuous time based on compound birth-death processes. When the frailty processes are used as mixtures in models for survival data, we derive the marginal hazards and survival functions and the marginal bivariate survival functions and cross-ratio function. We derive distributional properties of the processes, conditional on observed data, and show how to obtain the maximum likelihood estimators of the parameters of the model using a (stochastic) expectation-maximization algorithm. The methods are applied to a publicly available data set.

Cross ratio coordinates for the deformation spaces of a marked Möbius group

We introduce a new kind of coordinate systems for the deformation space of a finitely generated free Möbius group by using cross ratio functions induced by the fixed points of Möbius transformations. As an application, we give a new complete distance on the Schottky space by using such functions, which is not greater than the Teichmüller distance.

Structure of infrared singularities of gauge-theory amplitudes at three and four loops

The infrared divergences of massless n-parton scattering amplitudes can be derived from the anomalous dimension of n-jet operators in soft-collinear effective theory. Up to three-loop order, the latter has been shown to have a very simple structure: it contains pairwise color-dipole interactions among the external partons, governed by the cusp anomalous dimension and a logarithm of the kinematic invariants s_{ij}, plus a possible three-loop correlation involving four particles, which is described by a yet unknown function of conformal cross ratios of kinematic invariants. This function is constrained by two-particle collinear limits and by the known behavior of amplitudes in the high-energy limit. We construct a class of relatively simple functions satisfying these constraints. We also extend the analysis to four-loop order, finding that three additional four-particle correlations and a single five-particle correlation appear, which again are governed by functions of conformal cross ratios. Our results suggest that the dipole conjecture, which states that only two-particle color-dipole correlations appear in the anomalous dimension, may need to be generalized. We present a weaker form of the conjecture, stating that to all orders in perturbation theory corrections to the dipole formula are governed by functions of conformal cross ratios, and are O(1/N_c^2) suppressed relative to the dipole term. If true, this conjecture implies that the cusp anomalous dimension obeys Casimir scaling to all orders in perturbation theory.

The relative frailty variance and shared frailty models

SummaryThe relative frailty variance among survivors provides a readily interpretable measure of how the heterogeneity of a population, as represented by a frailty model, evolves over time. We discuss the properties of the relative frailty variance, show that it characterizes frailty distributions and that, suitably rescaled, it may be used to compare patterns of dependence across models and data sets. In shared frailty models, the relative frailty variance is closely related to the cross-ratio function, which is estimable from bivariate survival data. We investigate the possible shapes of the relative frailty variance function for the purpose of model selection, and we review available frailty distribution families in this context. We introduce several new families with contrasting properties, including simple but flexible time varying frailty models. The benefits of the approach that we propose are illustrated with two applications to bivariate current status data obtained from serological surveys.

Infinitesimal Liouville currents, cross-ratios and intersection numbers

Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichm\"uller space of complex structures on S. When two cross-ratio functions are sufficiently regular, they have a geometric intersection number, which generalizes the intersection number of two closed curves. In the case of the cross-ratio functions associated to tangent vectors to the Teichm\"uller space, we show that two such cross-ratio functions have a well-defined geometric intersection number, and that this intersection number is equal to the Weil-Petersson scalar product of the corresponding vectors.

Equivalence of Wilson loops inN=6super Chern-Simons matter theory andN=4SYM theory

In previous investigations, it was found that four-sided polygonal lightlike Wilson loops in $\mathcal{N}=6$ super Chern-Simons matter (ABJM) theory calculated to two-loop order have the same form as the corresponding Wilson loop in $\mathcal{N}=4$ SYM at one-loop order. Here we study lightlike polygonal Wilson loops with $n$ cusps in planar three-dimensional Chern-Simons and ABJM theory to two loops. Remarkably, the result in ABJM theory precisely agrees with the corresponding Wilson loop in $\mathcal{N}=4$ SYM at one-loop order for arbitrary $n$. In particular, anomalous conformal Ward identities allow for a so-called remainder function of conformal cross ratios for $n\ensuremath{\ge}6$, which is found to be trivial at two loops in ABJM theory in the same way as it is trivial in $\mathcal{N}=4$ SYM at one-loop order. Furthermore, the result for arbitrary $n$ obtained here, allows for a further investigation of a Wilson loop/amplitude duality in ABJM theory, for which nontrivial evidence was recently found by a calculation of four-point amplitudes that match the Wilson loop in ABJM theory.

Analyticity for multi-Regge limits of the Bern–Dixon–Smirnov amplitudes

As a consequence of the AdS/CFT correspondence, planar N = 4 super-Yang–Mills SU ( N ) theory is expected to exhibit stringy behavior and multi-Regge asymptotic. In this paper we extend our recent investigation to consider issues of analyticity, a central feature of Regge asymptotics. We contrast flat-space open string theory in the planar limit with the N = 4 super-Yang–Mills theory, as represented by the Bern, Dixon and Smirnov (2005) [1] (BDS) conjecture for n-gluon scattering, believed to be exact for n = 4 , 5 and modified only by a function of cross-ratios for n ⩾ 6 . It is emphasized that multi-Regge factorization should be applied to trajectories with definite signature. A variety of analyticity and factorization constraints realized in flat space string theory are not satisfied by the BDS conjecture, at least when the exponential factors are truncate in the infra-red regulator below O ( ϵ ) .

On the structure of infrared singularities of gauge-theory amplitudes

A closed formula is obtained for the infrared singularities of dimensionally regularized, massless gauge-theory scattering amplitudes with an arbitrary number of legs and loops. It follows from an all-order conjecture for the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory. We show that the form of this anomalous dimension is severely constrained by soft-collinear factorization, non-abelian exponentiation, and the behavior of amplitudes in collinear limits. Using a diagrammatic analysis, we demonstrate that these constraints imply that to three-loop order the anomalous dimension involves only two-parton correlations, with the possible exception of a single color structure multiplying a function of conformal cross ratios depending on the momenta of four external partons, which would have to vanish in all two-particle collinear limits. We argue that such a function does not appear at three-loop order, and that the same is true in higher orders. Our formula predicts Casimir scaling of the cusp anomalous dimension to all orders in perturbation theory, and we explicitly check that the constraints exclude the appearance of higher Casimir invariants at four loops. Using known results for the quark and gluon form factors, we derive the three-loop coefficients of the 1/epsilon^n pole terms (with n=1,...,6) for an arbitrary n-parton scattering amplitude in massless QCD. This generalizes Catani's two-loop formula proposed in 1998.