Five-point Function Research Articles

Two-loop five-point integrals: light, heavy and large-spin correlators

We evaluated all two-loop conformal integrals appearing in five-point correlation functions of protected operators of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super Yang-Mills in several kinematical regimes. Starting from the correlation function of the lightest operators of the theory, we were able to extract structure constants of up to two spinning operators for small and large values of spin and the tensor structure label (polarization). We conjectured an universal all-loop behaviour for the large-spin, small-polarization structure constants and comment on the subtleties of analytically continuing it from finite to large spin. We also consider correlation functions of heavier operators that get factorized in the more fundamental object called decagon. We fixed this object at two loops in general kinematics and studied its physical properties under OPE and null limits.

The five-point bootstrap

We study five-point correlation functions of scalar operators in d-dimensional conformal field theories. We develop a new approach to computing the five-point conformal blocks for exchanged primary operators of arbitrary spin by introducing a generalization of radial coordinates, using an appropriate ansatz, and perturbatively solving two quadratic Casimir differential equations. We then study five-point correlators 〈σσϵσσ〉 in the critical 3d Ising model. We truncate the operator product expansions (OPEs) in the correlator by including a finite number of primary operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We then compute several OPE coefficients involving ϵ and two spinning operators by demanding that the truncated correlator approximately satisfies the crossing relation.

Conformal multi-Regge theory

We propose and explore the Regge limit for correlation functions of five local primary operators in conformal field theories. After reviewing some features of Regge theory for flat-space scattering amplitudes, we analyze the analytic structure of conformal blocks both in position and Mellin space in the Regge limit and propose an extension of conformal Regge theory for five-point functions. As a byproduct of our analysis we also introduce a new basis of three-point correlation functions for operators with spin and the associated Euclidean conformal blocks.

Kaluza-Klein five-point functions from AdS5×S5 supergravity

We continue to explore the bootstrap approach to five-point correlation functions for IIB supergravity on AdS5 × S5. Building on the result of [1], we develop an improved algorithm that allows us to more efficiently compute correlators of higher Kaluza-Klein modes. The new method uses only factorization and a superconformal twist, and is entirely within Mellin space where the analytic structure of holographic correlators is simpler. Using this method, we obtain in a closed form all five-point functions of the form 〈pp222〉, extending the earlier result for p = 2. As a byproduct of our analysis, we also obtain explicit results for spinning four-point functions of higher Kaluza-Klein modes.

Higher-point integrands in $\mathcal{N} = 4$ super Yang-Mills theory

We compute the integrands of five-, six-, and seven-point correlation functions of twenty-prime operators with general polarizations at the two-loop order in \mathcal{N}=4𝒩=4 super Yang-Mills theory. In addition, we compute the integrand of the five-point function at three-loop order. Using the operator product expansion, we extract the two-loop four-point function of one Konishi operator and three twenty-prime operators. Two methods were used for computing the integrands. The first method is based on constructing an ansatz, and then numerically fitting for the coefficients using the twistor-space reformulation of \mathcal{N}=4𝒩=4 super Yang-Mills theory. The second method is based on the OPE decomposition. Only very few correlator integrands for more than four points were known before. Our results can be used to test conjectures, and to make progresses on the integrability-based hexagonalization approach for correlation functions.

Worldsheet computation of heavy-light correlators

We compute a large collection of string worldsheet correlators describing light probes interacting with heavy black hole microstates. The heavy states consist of NS5 branes carrying momentum and/or fundamental string charge. In the fivebrane decoupling limit, worldsheet string theory on a family of such backgrounds is given by exactly solvable null-gauged WZW models. We construct physical vertex operators in these cosets, including all massless fluctuations. We compute a large class of novel heavy-light-light-heavy correlators in the AdS3 limit, where the light operators include those dual to chiral primaries of the holographically dual CFT. We compare a subset of these correlators to the holographic CFT at the symmetric product orbifold point, and find precise agreement in all cases, including for light operators in twisted sectors of the orbifold CFT. The agreement is highly non-trivial, and includes amplitudes that describe the analogue of Hawking radiation for these microstates. We further derive a formula for worldsheet correlators consisting of n light insertions on these backgrounds, and discuss which subset of these correlators are likely to be protected. As a test, we compute a heavy-light five-point function, obtaining precisely the same result both from the worldsheet and the symmetric orbifold CFT. This paper is a companion to and extension of [1].

Rebooting quarter-BPS operators in mathcal{N} = 4 super Yang-Mills

We start a systematic study of quarter-BPS operators in four-dimensional mathcal{N} = 4 Super Yang-Mills with gauge group SU(N) making use of recently developed tools in conformal field theory. We adapt the technology of embedding space tensor structures in four dimensions to the problem of computing R-symmetry tensor structures, and we use the underlying chiral algebra to obtain the superconformal Ward identities. This allows us to fix the protected part of the four-point correlators, up to few ambiguities. As applications, we use the Lorentzian inversion formula to study the leading order OPE data in the large N supergravity limit and we make contact with the OPE limit of the five-point function of half-BPS operators.

Three-Dimensional Anisotropy and Scaling Properties of Solar Wind Turbulence at Kinetic Scales in the Inner Heliosphere: Parker Solar Probe Observations

Abstract We utilize the data from the Parker Solar Probe mission at its first perihelion to investigate the three-dimensional (3D) anisotropies and scalings of solar wind turbulence for the total, perpendicular, and parallel magnetic-field fluctuations at kinetic scales in the inner heliosphere. By calculating the five-point second-order structure functions, we find that the three characteristic lengths of turbulence eddies for the total and the perpendicular magnetic-field fluctuations in the local reference frame ( L ˆ ⊥ , l ˆ ⊥ , l ˆ ∣ ∣ ) defined with respect to the local mean magnetic field B local feature as l ∣∣ > L ⊥ > l ⊥ in both the transition range and the ion-to-electron scales, but l ∣∣ > L ⊥ ≈ l ⊥ for the parallel magnetic-field fluctuations. For the total magnetic-field fluctuations, the wave-vector anisotropy scalings are characterized by l ∣ ∣ ∝ l ⊥ 0.78 and L ⊥ ∝ l ⊥ 1.02 in the transition range, and they feature as l ∣ ∣ ∝ l ⊥ 0.44 and L ⊥ ∝ l ⊥ 0.73 in the ion-to-electron scales. Still, we need more complete kinetic-scale turbulence models to explain all these observational results.

The Veneziano amplitude via mostly BRST exact operator

The Veneziano amplitude is derived from fixing one degree of freedom of PSL(2,R) symmetry by the insertion of a mostly BRST exact operator. Evaluating the five-point function which consists of four open string tachyons and this gauge fixing operator, we find it equals the Veneziano amplitude up to a sign factor. The sign factor is interpreted as a signed intersection number. The result implies that the mostly BRST exact operator, which is originally used to provide two-point string amplitudes, correctly fixes the PSL(2,R) gauge symmetry for general amplitudes. We conjecture an expression for general n-point tree amplitudes with an insertion of this gauge fixing operator.

Cutting cosmological correlators

The initial conditions of our universe appear to us in the form of a classical probability distribution that we probe with cosmological observations. In the current leading paradigm, this probability distribution arises from a quantum mechanical wavefunction of the universe. Here we ask what the imprint of quantum mechanics is on the late time observables. We show that the requirement of unitary time evolution, colloquially the conservation of probabilities, fixes the analytic structure of the wavefunction and of all the cosmological correlators it encodes. In particular, we derive in perturbation theory an infinite set of single-cut rules that generalize the Cosmological Optical Theorem and relate a certain discontinuity of any tree-level n-point function to that of lower-point functions. Our rules are closely related to, but distinct from the recently derived Cosmological Cutting Rules. They follow from the choice of the Bunch-Davies vacuum and a simple property of the (bulk-to-bulk) propagator and are astoundingly general: we prove that they are valid for fields with a linear dispersion relation and any mass, any integer spin and arbitrary local interactions with any number of derivatives. They also apply to general FLRW spacetimes admitting a Bunch-Davies vacuum, including de Sitter, slow-roll inflation, power-law cosmologies and even resonant oscillations in axion monodromy. We verify the single-cut rules in a number of non-trivial examples, including four massless scalars exchanging a massive scalar, as relevant for cosmological collider physics, four gravitons exchanging a graviton, and a scalar five-point function.

Momentum space spinning correlators and higher spin equations in three dimensions

In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator overline{psi}psi is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of overline{psi}psi operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.

Multi-particle finite-volume effects for hexagon tessellations

Correlation functions of gauge-invariant composite operators in mathcal{N} = 4 super Yang-Mills theory can be computed by integrability using triangulations. The elementary tile in this process is the hexagon, which should be glued by appropriately inserting resolutions of the identity involving virtual (“mirror”) magnons. We consider this problem for five-point functions of protected operators. At one-loop in the ’t Hooft coupling, it is necessary to glue three adjacent tiles which involves two virtual magnons scattering among each other. We show that the result can be simplified by using an adapted mirror rotation and employing appropriate summation techniques. The mirror-particle contributions then yield hyperlogarithms of weight two. Finally, we use these results to investigate braiding prescriptions introduced in earlier work on the problem.

Observational Quantification of Three-dimensional Anisotropies and Scalings of Space Plasma Turbulence at Kinetic Scales

Abstract A statistical survey of spectral anisotropy of space plasma turbulence is performed using five years of measurements from the Magnetospheric Multiscale mission in the magnetosheath. By measuring the five-point second-order structure functions of the magnetic field, we have for the first time quantified the three-dimensional anisotropies and scalings at sub-ion scales (<100 km). In the local reference frame defined with respect to local mean magnetic field , the “statistical eddies” are found to be mostly elongated along and shortened in the direction perpendicular to both and local field fluctuations. From several d i (ion inertial length) toward ∼0.05 d i , the ratio between eddies’ parallel and perpendicular lengths features a trend of rise then fall, whereas the anisotropy in the perpendicular plane appears scale-invariant. Specifically, the anisotropy relations for the total magnetic field at 0.1–1.0 d i are obtained as , and , respectively. Our results provide new observational evidence to compare with phenomenological models and numerical simulations, which may help to better understand the nature of kinetic-scale turbulence.

20\u2032 five-point function from AdS5\xd7 S5 supergravity

We develop new techniques to compute five-point correlation functions from IIB supergravity on AdS5 × S5. Our methods rely entirely on symmetry and general con- sistency conditions, and eschew detailed knowledge of the supergravity effective action. We demonstrate our methods by computing the five-point function of the 20′ operator, which is the superconformal primary of the stress tensor multiplet. We also develop systematic methods to compute the five-point conformal blocks in series expansions. Using the ex- plicit expressions of the conformal blocks, we perform a Euclidean OPE analysis of the 20′ five-point function. We find expected agreement with non-renormalized quantities and also extract new CFT data at strong coupling.

Derivation of spontaneously broken gauge symmetry from the consistency of effective field theory II: Scalar field self-interactions and the electromagnetic interaction

We extend our study of deriving the local gauge invariance with spontaneous symmetry breaking in the context of an effective field theory by considering self-interactions of the scalar field and inclusion of the electromagnetic interaction. By analyzing renormalizability and the scale separation conditions of three-, four- and five-point vertex functions of the scalar field, we fix the two couplings of the scalar field self-interactions of the leading order Lagrangian. Next we add the electromagnetic interaction and derive conditions relating the magnetic moment of the charged vector boson to its charge and the masses of the charged and neutral massive vector bosons to each other and the two independent couplings of the theory. We obtain the bosonic part of the Lagrangian of the electroweak Standard Model as a unique solution to the conditions imposed by the self-consistency conditions of the considered effective field theory.

Hexagonalization of correlation functions II: two-particle contributions

In this work, we compute one-loop planar five-point functions in mathcal{N}=4 super-Yang-Mills using integrability. As in the previous work, we decompose the correlation functions into hexagon form factors and glue them using the weight factors which depend on the cross-ratios. The main new ingredient in the computation, as compared to the four-point functions studied in the previous paper, is the two-particle mirror contribution. We develop techniques to evaluate it and find agreement with the perturbative results in all the cases we analyzed. In addition, we consider next-to-extremal four-point functions, which are known to be protected, and show that the sum of one-particle and two-particle contributions at one loop adds up to zero as expected. The tools developed in this work would be useful for computing higher-particle contributions which would be relevant for more complicated quantities such as higher-loop corrections and non-planar correlators.

Anomalous dimensions of spinning operators from conformal symmetry

We compute, to the first non-trivial order in the ϵ-expansion of a perturbed scalar field theory, the anomalous dimensions of an infinite class of primary operators with arbitrary spin ℓ = 0, 1, . . . , including as a particular case the weakly broken higher-spin currents, using only constraints from conformal symmetry. Following the bootstrap philosophy, no reference is made to any Lagrangian, equations of motion or coupling constants. Even the space dimensions d are left free. The interaction is implicitly turned on through the local operators by letting them acquire anomalous dimensions.When matching certain four-point and five-point functions with the corresponding quantities of the free field theory in the ϵ → 0 limit, no free parameter remains. It turns out that only the expected discrete d values are permitted and the ensuing anomalous dimensions reproduce known results for the weakly broken higher-spin currents and provide new results for the other spinning operators.

Comparative study of four-bar hyperbolic function generation mechanism with four and five accuracy points

Kinematic mechanisms are synthesized for a task. Function generation provides precise displacement at output links that obeys a given functional relations. This article describes the synthesis of four-bar mechanism for the hyperbolic function generation with four and five accuracy point, which is further optimized using least square method. This research is concerned with development of mathematical formulation based on Freudenstein–Chebyshev approximation theory, extended to four- and five-point synthesis function generation problem. The objective function is analyzed for the structural error between the generated function and the desired function. Resulting nonlinear equations are converted into set of linear equations applying the compatibility conditions and are solved using Gauss elimination method. The formulation is proposed for five position synthesis for algebraic and trigonometric function generation problem. Associated structural error is estimated. Comparison of estimated error through the formulation is carried out with the reported errors through graphical method. The error for hyperbolic function is estimated. Attempt is made to minimize the error through simple of least square technique. The results obtained are compared with Freudenstein–Chebyshev approximation method. Three hyperbolic functions, namely sinh(x), cosh(x) and tanh(x), are used to demonstrate the effectiveness of the proposed synthesis method.

Thyroid Status, Quality of Life, and Mental Health in Patients on Hemodialysis.

In the general population, there is increasing recognition of the effect of thyroid function on patient-centered outcomes, including health-related quality of life and depression. Although hypothyroidism is highly prevalent in hemodialysis patients, it is unknown whether thyroid status is a risk factor for impaired health-related quality of life or mental health in this population. We examined the association of thyroid status, defined by serum thyrotropin, with health-related quality of life and depressive symptoms over time in a prospective cohort of 450 patients on hemodialysis from 17 outpatient dialysis facilities from May of 2013 to May of 2015 who underwent protocolized thyrotropin testing, Short-Form 36 surveys, and Beck Depression Inventory-II questionnaires every 6 months. We examined the association of baseline and time-dependent thyrotropin categorized as tertiles and continuous variables with eight Short-Form 36 domains and Beck Depression Inventory-II scores using expanded case mix plus laboratory adjusted linear mixed effects models. In categorical analyses, the highest baseline thyrotropin tertile was associated with a five-point lower Short-Form 36 domain score for energy/fatigue (P=0.04); the highest time-dependent tertile was associated with a five-point lower physical function score (P=0.03; reference: lowest tertile). In continuous analyses, higher baseline serum thyrotropin levels (+Δ1 mIU/L) were associated with lower role limitations due to physical health (β=-1.3; P=0.04), energy/fatigue (β=-0.8; P=0.03), and pain scores (β=-1.4; P=0.002), equivalent to five-, three-, and five-point lower scores, respectively, for every 1-SD higher thyrotropin. Higher time-dependent thyrotropin levels were associated with lower role limitations due to physical health scores (β=-1.0; P=0.03), equivalent to a three-point decline for every 1-SD higher thyrotropin. Baseline and time-dependent thyrotropin were not associated with Beck Depression Inventory-II scores. In patients on hemodialysis, higher serum thyrotropin levels are associated with impaired health-related quality of life across energy/fatigue, physical function, and pain domains. Studies are needed to determine if thyroid-modulating therapy improves the health-related quality of life of hemodialysis patients with thyroid dysfunction.

A probabilistic angle on one-loop scalar integrals

Recasting the N-point one-loop scalar integral as a probabilistic problem allows the derivation of integral recurrence relations, as well as exact analytical expressions in the most common cases. ϵ expansions are derived by writing a formula that relates an N-point function in decimal dimensions to an N-point function in integer dimensions. As an example, we give relations for the massive five-point function in and dimensions. The reduction of tensor integrals of rank two with N = 5 is achieved showing the method’s potential. No hypergeometric functions are involved. Results are expressed as integrals of arcsine functions, whose analytical continuation is well known.