Higher Loop Research Articles

Lifting integrable models and long-range interactions

In this paper we discuss a constructive approach to check whether a constant Hamiltonian is Yang-Baxter integrable. We then apply our method to long-range interactions and find the Lax operator and R-matrix of the three-loop SU(2) sector in N=4 SYM. We show that all known integrable long-range deformations of the 6-vertex models of this type can be obtained from a Lax operator and an R-matrix. Finally we discuss what happens at higher loops and highlight some general structures that these models exhibit.

Universality of loop corrected soft theorems in 4d

In [1], logarithmic correction to subleading soft photon and soft graviton theorems have been derived in four spacetime dimensions from the ratio of IR-finite S-matrices. This has been achieved after factoring out IR-divergent components from the traditional electromagnetic and gravitational S-matrices using Grammer-Yennie prescription. Although the loop corrected subleading soft theorems are derived from one-loop scattering amplitudes involving scalar particles in a minimally coupled theory with scalar contact interaction, it has been conjectured that the soft factors are universal (theory independent) and one-loop exact (don’t receive corrections from higher loops).This paper extends the analysis conducted in [1] to encompass general spinning particle scattering with non-minimal couplings permitted by gauge invariance and general coordinate invariance. By re-deriving the ln ω soft factors in this generic setup, we establish their universal nature. Furthermore, we summarize the results of loop corrected soft photon and graviton theorems up to sub-subleading order, which follows from the analysis of one and two loop QED and quantum gravity S-matrices. While the classical versions of these soft factors have already been derived in the literature, we put forth conjectures regarding the quantum soft factors and outline potential strategies for their derivation.

Stability of φ^{4}-vector model: Four-loop ε expansion study.

In most papers, φ^{4}-field theory with the vector (d-component) field φ_{α} is considered as a particular case of the n-component field model for n=d and O(n) symmetry. However, in such a model the symmetry O(d) admits an addition to the action of a term proportional to the squared divergence of the field ∼h(∂_{α}φ_{α})^{2}. From the point of view of renormalization group analysis, it requires a separate consideration, because it may well change the nature of the critical behavior of the system. Therefore, this frequently neglected term in the action requires a detailed and accurate study on the issue of the existence of new fixed points and their stability. It is known that within the lower order of perturbation theory the only infrared stable fixed point with h=0 exists but the corresponding positive value of stability exponent ω_{h} is tiny. This led us to analyze this constant in higher orders of perturbation theory by calculating the four-loop renormalization group contributions for ω_{h} in d=4-2ɛ within the minimal subtraction scheme, which should be enough to infer positivity or negativity of this exponent. The value turned out to be undoubtedly positive, although still small even in higher loops: 0.0156(3). These results cause the corresponding term to be neglected in the action when analyzing the critical behavior of the O(n)-symmetric model. At the same time, the small value of ω_{h} shows that the corresponding corrections to the critical scaling are significant in a wide range.

QCD cusp anomalous dimension: Current status

Calculation results for the HQET field anomalous dimension and the QCD cusp anomalous dimension, as well as their properties, are reviewed. The HQET field anomalous dimension [Formula: see text] is known up to four loops. The cusp anomalous dimension [Formula: see text] is known up to three loops, and its small-angle and large-angle asymptotics up to four loops. Some (but not all) color structures at four loops are known with the full [Formula: see text]-dependence. Some simple contributions are known at higher loops. For the [Formula: see text] asymptotics of [Formula: see text] (the light-like cusp anomalous dimension) and the [Formula: see text]-term of the small-[Formula: see text] expansion (the Bremsstrahlung function), the [Formula: see text] SYM results are equal to the highest-weight parts of the QCD results. There is an interesting conjecture about the structure of [Formula: see text] which holds up to three loops; at four loops it holds for some color structures and breaks down for other ones. In the cases when it holds, it related highly nontrivial functions of [Formula: see text], and it cannot be accidental; however, the reasons of this conjecture and its failures are not understood. The cusp anomalous dimension at the Euclidean angle [Formula: see text] is related to the static quark–antiquark potential due to conformal symmetry; in QCD, this relation is broken by an anomalous term proportional to the [Formula: see text]-function. Some new results are also presented. Using the recent four-loop result for [Formula: see text], here we obtain analytical expressions for some terms in the four-loop on-shell renormalization constant of the massive quark field [Formula: see text] which were previously known only numerically. We also present two new contributions to [Formula: see text], [Formula: see text] at five loops and to the quark–antiquark potential at four loops.

Spectral functions of gauge theories with Banks-Zaks fixed points

We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalization group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a K\"all\'en-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation of complex conjugated branch cuts renders a causal representation impossible. We also derive relations for scaling exponents that determine the presence or absence of propagator nonanalyticities. Further results include spectral functions for all fields up to five loop order, bounds on the conformal window, and an algorithm to find running gauge coupling analytically at higher loops. Implications of our findings and extensions to other theories are discussed.

Point-of-care prediction model of loop gain in patients with obstructive sleep apnea: development and validation

BackgroundHigh loop gain (unstable ventilatory control) is an important—but difficult to measure—contributor to obstructive sleep apnea (OSA) pathogenesis, predicting OSA sequelae and/or treatment response. Our objective was to develop and validate a clinical prediction tool of loop gain.MethodsA retrospective cohort of consecutive adults with OSA (apnea–hypopnea index, AHI > 5/hour) based on in-laboratory polysomnography 01/2017–12/2018 was randomly split into a training and test-set (3:1-ratio). Using a customized algorithm (“reference standard”) loop gain was quantified from raw polysomnography signals on a continuous scale and additionally dichotomized (high > 0.7). Candidate predictors included general patient characteristics and routine polysomnography data. The model was developed (training-set) using linear regression with backward selection (tenfold cross-validated mean square errors); the predicted loop gain of the final linear regression model was used to predict loop gain class. More complex, alternative models including lasso regression or random forests were considered but did not meet pre-specified superiority-criteria. Final model performance was validated on the test-set.ResultsThe total cohort included 1055 patients (33% high loop gain). Based on the final model, higher AHI (beta = 0.0016; P < .001) and lower hypopnea-percentage (beta = −0.0019; P < .001) predicted higher loop gain values. The predicted loop gain showed moderate-to-high correlation with the reference loop gain (r = 0.48; 95% CI 0.38–0.57) and moderate discrimination of patients with high versus low loop gain (area under the curve = 0.73; 95% CI 0.67–0.80).ConclusionTo our knowledge this is the first prediction model of loop gain based on readily-available clinical data, which may facilitate retrospective analyses of existing datasets, better patient selection for clinical trials and eventually clinical practice.

Duality invariant string beta functions at two loops

We compute, for cosmological backgrounds, the O(d, d; ℝ) invariant beta functions for the sigma model of the bosonic string at two loops. This yields an independent first-principle derivation of the order α′ corrections to the cosmological target-space equations. To this end we revisit the quantum consistency of Tseytlin’s duality invariant formulation of the worldsheet theory. While we confirm the absence of gravitational (and hence Lorentz) anomalies, our results show that the minimal subtraction scheme is not applicable, implying significant technical complications at higher loops. To circumvent these we then change gears and use the Polyakov action for cosmological backgrounds, applying a suitable perturbation scheme that, although not O(d, d; ℝ) invariant, allows one to efficiently determine the O(d, d; ℝ) invariant beta functions.

Beta functions for the duality-invariant sigma model

The O(d, d) invariant worldsheet theory for bosonic string theory with d abelian isometries is employed to compute the beta functions and Weyl anomaly at one-loop. We show that vanishing of the Weyl anomaly coefficients implies the equations of motion of the Maharana-Schwarz action. We give a self-contained introduction into the required techniques, including beta functions, the Weyl anomaly for two-dimensional sigma models and the background field method. This sets the stage for a sequel to this paper on generalizations to higher loops and α′ corrections.

Assessment of Anterior Loop of Inferior Alveolar Nerve and Its Anatomic Variations with Age, Gender, and Dentition Status in Indian Population: A CBCT Study.

Background The posterior region of the mandible is more often related to iatrogenic errors, but the interforaminal region is also not spared for neurovascular complications. This study aimed to use CBCT images to evaluate the prevalence of anterior nerve looping and its variations with age, gender, and dentition status. Methods This retrospective study was carried out by studying 600 CBCT scans retrieved from archival records of a CBCT center in Lucknow. The scans were inspected by two trained investigators. The length of the anterior loop was measured using the measuring tool of Carestream 3D imaging software. Descriptive and analytical tests were performed. Results The prevalence of the anterior loop of the inferior alveolar nerve was found to be 56%. The prevalence was found to be more on the right side (29.0%) compared to the left side (27.0%). The most common anterior looping of the inferior alveolar nerve was type 3 followed by type 1. Males were found to have significantly higher loops compared to females. The number of loops was found to decrease significantly with age. The mean length of the loop was found to vary from 1.14 to 1.61 mm. Conclusion The anterior looping of IAN is very much prevalent in the Lucknow population. The use of the CBCT technique and appropriate preplanning prior to surgery or implant placement should be performed to prevent nerve injury.

A three-point form factor through five loops

We bootstrap the three-point form factor of the chiral part of the stress­tensor supermultiplet in planar mathcal{N} = 4 super-Yang-Mills theory, obtaining new results at three, four, and five loops. Our construction employs known conditions on the first, second, and final entries of the symbol, combined with new multiple-final-entry conditions, “extended-Steinmann-like” conditions, and near-collinear data from the recently-developed form factor operator product expansion. Our results are expected to give the maximally transcendental parts of the gg → Hg and H → ggg amplitudes in the heavy-top limit of QCD. At two loops, the extended-Steinmann-like space of functions we describe contains all transcendental functions required for four-point amplitudes with one massive and three massless external legs, and all massless internal lines, including processes such as gg → Hg and γ* → qoverline{q}g . We expect the extended-Steinmann-like space to contain these amplitudes at higher loops as well, although not to arbitrarily high loop order. We present evidence that the planar mathcal{N} = 4 three-point form factor can be placed in an even smaller space of functions, with no independent ζ values at weights two and three.

Quantum origin of inflation in the geometric inflation model

In this paper we investigate the cosmological dynamics of geometric inflation by means of the tools of the dynamical systems theory. We focus in the study of two explicit models where it is possible to sum the infinite series of higher curvature corrections that arises in the formalism. These would be very interesting possibilities since, if regard gravity as a quantum effective theory, a key feature is that higher powers of the curvature invariants are involved at higher loops. Hence, naively, consideration of the whole infinite tower of curvature invariants amounts to consideration of all of the higher order loops. The global dynamics of these toy models in the phase space is discussed and the quantum origin of primordial inflation is exposed.

Solar Type U Burst Associated with a High Coronal Loop

An inverted U burst with equally developed ascending and descending branches observed by Giant Ukrainian Radio Telescope (GURT) on 18 April 2017 in meter wavelengths band is discussed. This U burst was attributed to the high coronal loop above the limb active region NOAA 12651. Under the assumption that, associated with the burst, the coronal loop confines isothermal plasma stratified according to a Boltzmann density relation, the geometrical and physical parameters of the loop were estimated. According to our model coronal loops may contain plasma which is up to 20 times denser than the surrounding coronal plasma. In general the proposed model gives the relation between the plasma temperature and the height of the loop in such a way that under the given parameters of the associated U burst, higher loops contain cooler plasma and vice versa. An alternative method of coronal loop height determination was suggested. Assuming that the observed U burst and the preceding type III burst were generated by the same exciter we define the height of the loop from the delay of the former with respect to the latter at certain frequency. We show that defining of coronal loops heights by another independent method, e.g. interferometric or tied-array imaging may solve the uncertainty of the inside-the-loop plasma temperature determination.

Diffeomorphism invariance demands conformal anomalies

We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2\phi}(-d\eta^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2\phi}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving momentum in cosmology. It is confirmed by calculating the effective action of QED in such a curved spacetime at the 3-loop level using dimensional regularization. The same applies to the case of QCD, in which we show that the effective action can be summarized in the form of the reciprocal of a running coupling constant squared described by the physical momentum. We also see that the same holds for renormalizable quantum conformal gravity and that conformal anomalies are indispensable for formulating the theory.

Gravitational shock waves and scattering amplitudes

We study gravitational shock waves using scattering amplitude techniques. After first reviewing the derivation in General Relativity as an ultrarelativistic boost of a Schwarzschild solution, we provide an alternative derivation by exploiting a novel relation between scattering amplitudes and solutions to Einstein field equations. We prove that gravitational shock waves arise from the classical part of a three point function with two massless scalars and a graviton. The region where radiation is localized has a distributional profile and it is now recovered in a natural way, thus bypassing the introduction of singular coordinate transformations as used in General Relativity. The computation is easily generalized to arbitrary dimensions and we show how the exactness of the classical solution follows from the absence of classical contributions at higher loops. A classical double copy between gravitational and electromagnetic shock waves is also provided and for a spinning source, using the exponential form of three point amplitudes, we infer a remarkable relation between gravitational shock waves and spinning ones, also known as gyratons. Using this property, we infer a family of exact solutions describing gravitational shock waves with spin. We then compute the phase shift of a particle in a background of shock waves finding agreement with an earlier computation by Amati, Ciafaloni and Veneziano for particles in the high energy limit. Applied to a gyraton, it provides a result for the scattering angle to all orders in spin.

On the Vilkovisky-DeWitt approach and renormalization group in effective quantum gravity

The effective action in quantum general relativity is strongly dependent on the gauge-fixing and parametrization of the quantum metric. As a consequence, in the effective approach to quantum gravity, there is no possibility to introduce the renormalization-group framework in a consistent way. On the other hand, the version of effective action proposed by Vilkovisky and DeWitt does not depend on the gauge-fixing and parametrization off- shell, opening the way to explore the running of the cosmological and Newton constants as well as the coefficients of the higher-derivative terms of the total action. We argue that in the effective framework the one-loop beta functions for the zero-, two- and four-derivative terms can be regarded as exact, that means, free from corrections coming from the higher loops. In this perspective, the running describes the renormalization group flow between the present-day Hubble scale in the IR and the Planck scale in the UV.

Traintrack Calabi-Yaus from twistor geometry

We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in ℙ3. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in ℙ1× ℙ1. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction.

Decagon at two loops

We have computed the simplest five point function in mathcal{N} = 4 SYM at two loops using the hexagonalization approach to correlation functions. Along the way we have determined all two-particle mirror contributions at two loops and we have computed all the integrals involved in the final result. As a test of our results we computed a few four-point functions and they agree with the perturbative results computed previously. We have also obtained l loop results for some parts of the two-particle contributions with l arbitrary. We also derive differential equations for a class of integrals that should appear at higher loops in the five point function.

AdS/CFT unitarity at higher loops: high-energy string scattering

What is the space of weakly-coupled, gravitational theories which contain massive, higher-spin particles? This class of theories is highly constrained and it is conjectured their ultraviolet completion must be string theory. We provide more evidence for this conjecture by studying the Regge limit in large N , 4d CFTs with single-trace operators of unbounded spin. We show that in the Regge limit, these theories have bulk scattering amplitudes which are consistent with the string theory prediction to all orders in 1/N for large, but finite, coupling. In the language of Regge theory, we show Pomeron exchange naturally exponentiates in the 1/N expansion. To do this, we solve the bootstrap equations at tree-level and then use the Lorentzian inversion formula to find the one-loop correlator in the Regge limit. This is a unitarity method for AdS/CFT which can be repeated iteratively to make all orders statements. We also explain under what conditions the tree-level result exponentiates in the 1/N expansion at arbitrary coupling. Finally, we comment on further inelastic effects and show they give subleading contributions at large coupling. As a consistency check, we recover results from bulk Einstein gravity in the limit where all higher-spin particles decouple.

Sign flip triangulations of the amplituhedron

We present new triangulations of the m = 4 amplituhedron relevant for scattering amplitudes in planar mathcal{N} = 4 super-Yang-Mills, obtained directly from the combinatorial definition of the geometry. Using the “sign flip” characterization of the amplituhedron, we reproduce the canonical forms for the all-multiplicity next-to-maximally helicity violating (NMHV) and next-to-next-to-maximally helicity violating (N2MHV) tree-level as well as the NMHV one-loop cases, without using any input from traditional amplitudes methods. Our results provide strong evidence for the equivalence of the original definition of the amplituhedron [1] and the topological one [2], and suggest a new path forward for computing higher loop amplitudes geometrically. In particular, we realize the NMHV one-loop amplituhedron as the intersection of two amplituhedra of lower dimensionality, which is reflected in the novel structure of the corresponding canonical form.

Nonrenormalization and Operator Mixing via On-Shell Methods

Using on-shell methods, we present a new perturbative nonrenormalization theorem for operator mixing in massless four-dimensional quantum field theories. By examining how unitarity cuts of form factors encode anomalous dimensions, we show that longer operators are often restricted from renormalizing shorter operators at the first order where Feynman diagrams exist. The theorem applies quite generally and depends only on the field content of the operators involved. We apply our theorem to operators of dimension five through seven in the standard model effective field theory, including examples of nontrivial zeros in the anomalous-dimension matrix at one through four loops. The zeros at two and higher loops go beyond those previously explained using helicity selection rules. We also include explicit sample calculations at two loops.