Point Function Research Articles

Entanglement entropy in scalar field theory and ZM gauge theory on Feynman diagrams

Entanglement entropy (EE) in interacting field theories has two important issues: renormalization of UV divergences and non-Gaussianity of the vacuum. In this letter, we investigate them in the framework of the two-particle irreducible formalism. In particular, we consider EE of a half space in an interacting scalar field theory. It is formulated as $\mathbb{Z}_M$ gauge theory on Feynman diagrams: $\mathbb{Z}_M$ fluxes are assigned on plaquettes and summed to obtain EE. Some configurations of fluxes are interpreted as twists of propagators and vertices. The former gives a Gaussian part of EE written in terms of a renormalized 2-point function while the latter reflects non-Gaussianity of the vacuum.

Primordial non-Gaussianity from biased tracers: likelihood analysis of real-space power spectrum and bispectrum

Upcoming galaxy redshift surveys promise to significantly improve current limits on primordial non-Gaussianity (PNG) through measurements of 2- and 3-point correlation functions in Fourier space. However, realizing the full potential of this dataset is contingent upon having both accurate theoretical models and optimized analysis methods. Focusing on the local model of PNG, parameterized by f NL, we perform a Monte-Carlo Markov Chain analysis to confront perturbation theory predictions of the halo power spectrum and bispectrum in real space against a suite of N-body simulations. We model the halo bispectrum at tree-level, including all contributions linear and quadratic in f NL, and the halo power spectrum at 1-loop, including tree-level terms up to quadratic order in f NL, and all loops induced by local PNG linear in f NL. Keeping the cosmological parameters fixed, we examine the effect of informative priors on the linear non-Gaussian bias parameter on the statistical inference of f NL. A conservative analysis of the combined power spectrum and bispectrum, in which only loose priors are imposed and all parameters are marginalized over, can improve the constraint on by more than a factor of 5 relative to the power spectrum-only measurement. Imposing a strong prior on b ϕ, or assuming bias relations for both b ϕ and b ϕδ (motivated by a universal mass function assumption), improves the constraints further by a factor of few. In this case, however, we find a significant systematic shift in the inferred value of f NL if the same range of wavenumber is used. Likewise, a Poisson noise assumption can lead to significant systematics, and it is thus essential to leave all the stochastic amplitudes free.

Conformal partial waves in momentum space

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension d \geq 3d≥3 (including non-integer dd). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in d = 4d=4 dimensions.

From VOAs to Short Star Products in SCFT

We build a bridge between two algebraic structures in superconformal field theories (SCFT): a vertex operator algebra (VOA) in the Schur sector of 4d $$\mathcal {N}=2$$ theories and an associative algebra in the Higgs sector of 3d $$\mathcal {N}=4$$ . The natural setting is a 4d $$\mathcal {N}=2$$ SCFT placed on $$S^3\times S^1$$ : by sending the radius of $$S^1$$ to zero, we recover the 3d $$\mathcal {N}=4$$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: (1) the Higgs branch operators remain in the cohomology; (2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the $$S^1$$ ; (3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient $$\mathcal {A}_H = \mathrm{Zhu}_{s}(V)/N$$ , where $$\mathrm{Zhu}_{s}(V)$$ is the non-commutative Zhu algebra of the VOA V (for $${s}\in \mathrm{Aut}(V)$$ ), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map $$T_{s}: \mathrm{Zhu}_{s}(V) \rightarrow \mathbb {C}$$ determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips $$\mathcal {A}_H$$ with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map $$T_{s}$$ is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non- $$C_2$$ -cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas.

Non-analyticity of the Correlation Length in Systems with Exponentially Decaying Interactions

We consider a variety of lattice spin systems (including Ising, Potts and XY models) on mathbb {Z}^d with long-range interactions of the form J_x = psi (x) e^{-|x|}, where psi (x) = e^{{mathsf o}(|x|)} and |cdot | is an arbitrary norm. We characterize explicitly the prefactors psi that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature beta , magnetic field h, etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever psi is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein–Zernike theory of correlations.

Multiwavelength mock galaxy catalogues of the low-redshift Universe

ABSTRACT We present a new suite of mock galaxy catalogues mimicking the low-redshift Universe, based on an updated halo occupation distribution (HOD) model and a scaling relation between optical properties and the neutral hydrogen (H i) content of galaxies. Our algorithm is constrained by observations of the luminosity function and luminosity- and colour-dependent clustering of Sloan Digital Sky Survey (SDSS) galaxies, as well as the H i mass function and H i-dependent clustering of massive H i-selected galaxies in the Arecibo Legacy Fast ALFA (ALFALFA) survey. Mock central and satellite galaxies with realistic values of r-band luminosity, g − r and u − r colour, stellar mass and H i mass are populated in an N-body simulation, inheriting a number of properties of the density and tidal environment of their host haloes. The host halo of each central galaxy is also ‘baryonified’ with realistic spatial distributions of stars as well as hot and cold gas, along with the corresponding rotation curve. Our default HOD assumes that galaxy properties are a function of group halo mass alone, and can optionally include effects such as galactic conformity and colour-dependent galaxy assembly bias. The mocks predict the relation between the stellar mass and H i mass of massive H i galaxies, as well as the 2-point cross-correlation function of spatially co-located optical and H i-selected samples. They enable novel null tests for galaxy assembly bias, provide predictions for the H i velocity width function, and clarify the origin and universality of the radial acceleration relation in the Lambda cold dark matter framework.

Galaxy skew-spectra in redshift-space

Modern galaxy surveys focus on the galaxy power spectrum or 2-point correlation function to test and constrain cosmological models.Additional information comes from higher-order N-point functions, but their analysis is challenging. A simple solution is to compute the cross-power spectrum between the squared galaxy density and the galaxy density. Being simple to measure and to plot, this skew-spectrum shares many of the familiar useful properties of the standard galaxy power spectrum. We show that by computing multiple quadratic fields and correlating them with the density, all contributions to the tree-level redshift-space galaxy bispectrum can be captured with skew-spectra. Using synthetic datasets, we show that our measurement pipeline matches analytical predictions, and that their dependence on galaxy bias parameters and the logarithmic growth rate is as expected theoretically.

Defect CFT techniques in the 6d mathcal{N} = (2, 0) theory

Surface operators are among the most important observables of the 6d mathcal{N} = (2, 0) theory. Here we apply the tools of defect CFT to study local operator insertions into the 1/2-BPS plane. We first relate the 2-point function of the displacement operator to the expectation value of the bulk stress tensor and translate this relation into a constraint on the anomaly coefficients associated with the defect. Secondly, we study the defect operator expansion of the stress tensor multiplet and identify several new operators of the defect CFT. Technical results derived along the way include the explicit supersymmetry tranformations of the stress tensor multiplet and the classification of unitary representations of the superconformal algebra preserved by the defect.

Redshift-space distortions in Lagrangian perturbation theory

We present the one-loop 2-point function of biased tracers in redshift space computed with Lagrangian perturbation theory, including a full resummation of both long-wavelength (infrared) displacements and associated velocities. The resulting model accurately predicts the power spectrum and correlation function of halos and mock galaxies from two different sets of N-body simulations at the percent level for quasi-linear scales, including the damping of the baryon acoustic oscillation signal due to the bulk motions of galaxies. We compare this full resummation with other, approximate, techniques including the moment expansion and Gaussian streaming model. We discuss infrared resummation in detail and compare our Lagrangian formulation with the Eulerian theory augmented by an infrared resummation based on splitting the input power spectrum into “wiggle” and “no-wiggle” components. We show that our model is able to recover unbiased cosmological parameters in mock data encompassing a volume much larger than what will be available to future galaxy surveys. We demonstrate how to efficiently compute the resulting expressions numerically, making available a fast Python code capable of rapidly computing these statistics in both configuration and Fourier space.

Stress-energy tensor correlators from the world-sheet

The large N limit of symmetric orbifold theories was recently argued to have an AdS/CFT dual world-sheet description in terms of an sl(2, ℝ) WZW model. In previous work the world-sheet state corresponding to the symmetric orbifold stress-energy tensor was identified. We calculate certain 2- and 3-point functions of the corresponding vertex operator on the world-sheet, and demonstrate that these amplitudes reproduce exactly what one expects from the dual symmetric orbifold perspective.

Searching for gauge theories with the conformal bootstrap

Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO(N) vector 4-point functions in general dimension D. In the large N limit, upper bounds on the scaling dimensions of the lowest SO(N) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O(N) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1/2 at large N. We show that the bootstrap bounds also admit another infinite family of kinks {mathcal{T}}_D , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks {mathcal{T}}_D appear in general dimensions with a D-dependent critical N* below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO(N) vectors, SU(N) fundamentals, and SU(N) × SU(N) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of {mathcal{T}}_D are subgroups of SO(N), and we speculate that the kinks {mathcal{T}}_D relate to the fixed points of gauge theories coupled to fermions.

Two-point function of the energy-momentum tensor and generalised conformal structure

Theories with generalised conformal structure contain a dimensionful parameter, which appears as an overall multiplicative factor in the action. Examples of such theories are gauge theories coupled to massless scalars and fermions with Yukawa interactions and quartic couplings for the scalars in spacetime dimensions other than 4. Many properties of such theories are similar to that of conformal field theories (CFT), and in particular their 2-point functions take the same form as in CFT but with the normalisation constant now replaced by a function of the effective dimensionless coupling g constructed from the dimensionful parameter and the distance separating the two operators. Such theories appear in holographic dualities involving non-conformal branes and this behaviour of the correlators has already been observed at strong coupling. Here we present a perturbative computation of the two-point function of the energy-momentum tensor to two loops in dimensions d= 3, 5, confirming the expected structure and determining the corresponding functions of g to this order, including the effects of renormalisation. We also discuss the hbox {d}=4 case for comparison. The results for d=3 are relevant for holographic cosmology, and in this case we also study the effect of a Phi ^6 coupling, which while marginal in the usual sense it is irrelevant from the perspective of the generalised conformal structure. Indeed, the effect of such coupling in the 2-point function is washed out in the IR but it modifies the UV.

Complex Sachdev-Ye-Kitaev model in the double scaling limit

We solve for the exact energy spectrum, 2-point and 4-point functions of the complex SYK model, in the double scaling limit at all energy scales. This model has a U(1) global symmetry. The analysis shows how to incorporate a chemical potential in the chord diagram picture, and we present results for the various observables also at a given fixed charge sector. In addition to matching to the spectral asymmetry, we consider an analogous asymmetry measure of the 2-point function obeying a non-trivial dependence on the operator’s dimension. We also provide the chord diagram structure for an SYK-like model that has a U(M) global symmetry at any disorder realization. We then show how to exactly compute the effect of inserting very heavy operators, with formally infinite conformal dimension. The latter separate the gravitational spacetime into several parts connected by an interface, whose properties are exactly computable at all scales. In particular, light enough states can still go between the spaces. This behavior has a simple description in the chord diagram picture.

Supersymmetry anomaly in the superconformal Wess-Zumino model

We present a comprehensive analysis of supersymmetry anomalies in the free and massless Wess-Zumino (WZ) model in perturbation theory. At the classical level the model possesses mathcal{N} = 1 superconformal symmetry, which is partially broken by quantum anomalies. The form of the anomalies and the part of the symmetry they break depend on the multiplet of conserved currents used. It was previously shown that the R-symmetry anomaly of the conformal current multiplet induces an anomaly in Q-supersymmetry, which appears first in 4-point functions. Here we confirm this result by an explicit 1-loop computation using a supersymmetric Pauli-Villars regulator.The conformal current multiplet does not exist in the regulated theory because the regulator breaks conformal invariance, R-symmetry and S-supersymmetry explicitly. The minimal massive multiplet is the Ferrara-Zumino (FZ) one and the supersymmetry preserved by the regulator is a specific field dependent combination of Q- and S- supersymmetry of the conformal multiplet. While this supersymmetry is non anomalous, conformal invariance, R-symmetry and the original Q- and S-supersymmetries are explicitly broken by finite contact terms, both in the regulated and renormalized theories.A conformal current multiplet does exist for the renormalized theory and may be obtained from the FZ multiplet by a set of finite local counterterms that eliminate the explicit symmetry breaking, thus restoring superconformal invariance up to anomalies. However, this necessarily renders both Q- and S-supersymmetries anomalous, as is manifest starting at 4-point functions of conformal multiplet currents. The paper contains a detailed discussion of a number of issues and subtleties related to Ward identities that may be useful in a wider context.

Universal correlators and novel cosets in 2d RCFT

The two-character level-1 WZW models corresponding to Lie algebras in the Cvitanović-Deligne series A1, A2, G2, D4, F4, E6, E7 have been argued to form coset pairs with respect to the meromorphic E8,1 CFT. Evidence for this has taken the form of holomorphic bilinear relations between the characters. We propose that suitable 4-point functions of primaries in these models also obey bilinear relations that combine them into current correlators for E8,1, and provide strong evidence that these relations hold in each case. Different cases work out due to special identities involving tensor invariants of the algebra or hypergeometric functions. In particular these results verify previous calculations of correlators for exceptional WZW models, which have rather subtle features. We also find evidence that the intermediate vertex operator algebras A0.5 and E7.5, as well as the three-character A4,1 theory, also appear to satisfy the novel coset relation.

Generalisations of the Harer–Zagier recursion for 1-point functions

Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer–Zagier recursion, but our methodology also applies to the enumeration of dessins d’enfant, to Bousquet-Mélou–Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

Field sparsening for the construction of the correlation functions in lattice QCD

Two field-sparsening methods, namely the sparse-grid method and the random field selection method, are used in this paper for the construction of the 2-point and 3-point correlation functions in lattice QCD. We argue that, due to the high correlation among the lattice correlators at different field points associated with source, current, and sink locations, one can save a lot of computational time by performing the summation over a subset of the lattice sites. Furthermore, with this strategy, one only needs to store a small fraction of the full quark propagators. It is found that the number of field points can be reduced by a factor of $\sim$100 for the point-source operator and a factor of $\sim$1000 for the Gaussian-smeared operator, while the uncertainties of the correlators only increase by $\sim$15\%. Therefore, with a modest cost of the computational resources, one can approach the precision of the all-to-all correlators using the field-sparsening methods.

Analyticity of off-shell Green\u2019s functions in superstring field theory

We consider the off-shell momentum space Green’s functions in closed superstring field theory. Recently in [1], the off-shell Green’s functions — after explicitly removing contributions of massless states — have been shown to be analytic on a domain (to be called the LES domain) in complex external momenta variables. Analyticity of off-shell Green’s functions in local QFTs without massless states in the primitive domain is a well-known result. Using complex Lorentz transformations and Bochner’s theorem allow to extend the LES domain to a larger subset of the primitive domain. For the 2-, 3- and 4-point functions, the full primitive domain is recovered. For the 5-point function, we are not able to obtain the full primitive domain analytically, only a large part of it is recovered. While this problem arises also for higher-point functions, it is expected to be only a technical issue.

Hyperbolic cylinders and entanglement entropy: gravitons, higher spins, p-forms

We show that the entanglement entropy of D = 4 linearized gravitons across a sphere recently computed by Benedetti and Casini coincides with that obtained using the Kaluza-Klein tower of traceless transverse massive spin-2 fields on S1× AdS3. The mass of the constant mode on S1 saturates the Brietenholer-Freedman bound in AdS3. This condition also ensures that the entanglement entropy of higher spins determined from partition functions on the hyperbolic cylinder coincides with their recent conjecture. Starting from the action of the 2-form on S1× AdS5 and fixing gauge, we evaluate the entanglement entropy across a sphere as well as the dimensions of the corresponding twist operator. We demonstrate that the conformal dimensions of the corresponding twist operator agrees with that obtained using the expectation value of the stress tensor on the replica cone. For conformal p-forms in even dimensions it obeys the expected relations with the coefficients determining the 3-point function of the stress tensor of these fields.

Cross-bispectra constraints on modified gravity theories from the Nancy Grace Roman Space Telescope and the Rubin Observatory Legacy Survey of Space and Time

One major goal of upcoming large-scale-structure surveys is to constrain dark energy and modified gravity theories. In particular, galaxy clustering and gravitational lensing convergence are probes sensitive to modifications of general relativity. While the standard analysis for these surveys typically includes power spectra or 2-point correlation functions, it is known that the bispectrum contains more information, and could offer improved constraints on parameters when combined with the power spectra. However, the use of bispectra has been limited so far to one single probe, e.g. the lensing convergence bispectrum or the galaxy bispectrum. In this paper, we extend the formalism to explore the power of cross-bispectra between different probes, and exploit their ability to break parameter degeneracies and improve constraints. We study this on a test case of lensing convergence and galaxy density auto- and cross-bispectra, for a particular sub-class of Horndeski theories parametrized by $c_M$ and $c_B$. Using the 2000 deg$^2$ notional survey of the Nancy Grace Roman Space Telescope with overlapping photometry from the Rubin Observatory Legacy Survey of Space and Time, we find that a joint power spectra and bispectra analysis with three redshift bins at $l_{\rm max} = 1000$ yields $\sigma_{c_M} = 1.0$ and $\sigma_{c_B} = 0.3$, both a factor of $\sim$1.2 better than the power spectra results; this would be further improved to $\sigma_{c_M} = 0.7$ and $\sigma_{c_B} = 0.2$ if $l_{\rm max} = 3000$ is taken. Furthermore, we find that using all possible cross-bispectra between the two probes in different tomographic bins improves upon auto-bispectra results by a factor of 1.3 in $\sigma_{c_M}$, 1.1 in $\sigma_{c_B}$ and 1.3 in $\sigma_{\Omega_m}$. We expect that similar benefits of using cross-bispectra between probes could apply to other science cases and surveys.