Higher Loop Research Articles

Logarithmic terms in entropy of Schwarzschild black holes in higher loops

The Bekenstein-Hawking (BH) entropy is expected to be modified by certain correction terms in the quantum loop expansion. As is well known the logarithmic terms in the entropy of black holes appear as a one-loop addition to the classical BH entropy. In this note we study the further modifications of the logarithmic terms in the entropy of the Schwarzschild black holes due to higher quantum loops: up to three loops in a general renormalisable theory of gauge fields, scalars and fermions and two loops in quantum gravity. For a large class of field multiplets (including that of the Standard Model) that include graviton and for a certain range in the values of the couplings these modifications manifest themselves in cooling down the black holes at later stages of evaporation and, respectively, in increasing the life time of the black hole. If this picture persists to even higher loops, then the small black holes formed in the early stages of the cosmic evolution do not evaporate completely by now as is predicted in the standard picture. Instead, their long-lived (Planckian mass) remnants are present in abundance in today's Universe.

Thin loop groups

We verify that for a finite simplicial complex X and for piecewise linear loops on X, the “thin” loop space is a topological group of the same homotopy type as the space of continuous loops. This turns out not to be the case for the higher loops.

The effective field theory of large scale structure at three loops

We study the power spectrum of dark matter density fluctuations in the framework of the Effective Field Theory of Large Scale Structures (EFTofLSS) up to three loop orders. We show that the extra coefficients in the EFT are sufficient to match numerical simulations with percent accuracy when a generic renormalization prescription is implemented (allowing for running of the individual counter-terms). We show that the level of accuracy increases with respect to the two loop results, up to k ≃ 0.4 h Mpc−1 at redshift z=0, although the overall improvement is somewhat marginal. At the same time, we argue there is evidence that the behavior of the loop expansion in the EFTofLSS is typical of an asymptotic series, already on the brink of its maximum predictive power (at z=0). Hence, the inclusion of higher orders will likely deteriorate the matching to data, even at moderate values of k. Part of the reason for this behavior is due to large contributions to the (renormalized) power spectrum at three loop order from mildly non-linear scales, even after the UV counter-terms are included. In conclusion, the EFTofLSS to three loop orders provides the best approximation to the (deterministic part of the) power spectrum in the weakly non-linear regime at z=0, and higher loops are not expected to improve our level of accuracy.

P1636Causes and risk factors for non-cardiovascular hospital readmissions among heart failure patients

Abstract Background/Introduction Despite better patient-management and technological improvements, patients with heart failure (HF) have poor prognosis and impaired health-related quality of life (HRQoL). HF patients also experience significant non-cardiovascular morbidity leading to higher disease burden and recurrent hospitalizations, which also put a heavy burden on healthcare systems. Purpose To describe causes and risk factors for non-cardiovascular hospital readmissions among HF patients. Methods We analyzed 3,172 hospital discharge summaries for 927 patients enrolled in a clinical trial testing the efficacy of a disease management program compared to usual care between August 2007 and June 2011. The study intervention had no effect on hospital admission and mortality rates. Baseline information on patient characteristics included age, gender, LVEF, NYHA classification, 6-minute walk test, BNP level, BMI, chronic comorbidity and HRQoL score. Analysis of hospital discharge summaries included identification and coding of the primary hospital admission cause. These causes were further grouped to HF-related, other cardiovascular disease-related and non-cardiovascular causes. Information on follow-up prescription purchases of HF drugs was also collected. Factors associated with non-cardiovascular hospital admission rate were tested in multivariable models, adjusted for study group, study period and year of recruitment. Results Non-cardiovascular hospital admissions (of which 77% were non-elective) accounted for 45% of all hospital readmissions. Of these hospitalizations, 23% were due to infectious diseases, 13% due to gastrointestinal causes, 12% due to respiratory causes and 8% due to falls and associated trauma. Patients with two or more non-cardiovascular hospital admissions during follow-up were significantly older, more commonly female, had higher proportion of preserved LVEF (≥50%), worse baseline functional capacity and HRQoL score, and higher co-morbidity score, BNP level and mortality rates compared to other patients. Factors significantly and independently associated with higher acute non-cardiovascular hospitalization rate included [rate ratios (RRs), 95% confidence intervals (CIs)]: non-preserved vs. reduced LVEF [1.48 (1.14, 1.92)]; higher baseline comorbidity score [1.14 (1.08, 1.21) per 1-point increment]; and higher loop diuretics, beta-adrenergic receptor blockers and ACE-I/ARBs treatment level during follow-up; 4th vs. 1st quartiles: 4.13 (2.93, 6.32), 1.51 (1.07, 2.13) and 1.32 (0.96, 1.82), respectively. Baseline overweight/obesity and greater 6-min. walking distance [0.93 (0.89, 0.93) per 50m increment] were found protective. Conclusion This study indicates a higher disease burden and mortality rate among HF patients with recurrent non-cardiovascular hospital admissions. It provides pertinent information needed for the design and implementation of effective interventions to reduce disease burden in HF patients.

ABJM flux-tube and scattering amplitudes

There is a number of indications that scattering amplitudes in the Aharony-Bergman-Jafferis-Maldacena theory might have a dual description in terms of a holonomy of a supergauge connection on a null polygonal contour in a way analogous to the four-dimensional maximally supersymmetric Yang-Mills theory. However, so far its explicit implementations evaded a successful completion. The difficulty is intimately tied to the lack of the T-self-duality of the sigma model on the string side of the gauge/string correspondence. Unscathed by the last misfortune, we initiate with this study an application of the pentagon paradigm to scattering amplitudes of the theory. With the language being democratic and nondiscriminatory to whether one considers a Wilson loop expectation value or an amplitude, the success in the application of the program points towards a possible unified observable on the field theory side. Our present consideration is focused on two-loop perturbation theory in the planar limit, begging for higher loop data in order to bootstrap current analysis to all orders in the ’t Hooft coupling.

Extreme-ultraviolet Late Phase Caused by Magnetic Reconnection over Quadrupolar Magnetic Configuration in a Solar Flare

Abstract A second emission enhancement in warm coronal extreme-ultraviolet (EUV) lines (about 2–7 MK) during some solar flares is known as the EUV late phase. Imaging observations confirm that the late-phase emission originates from a set of longer or higher loops than the main flare loops. Nevertheless, some questions remain controversial: What is the relationship between these two loop systems? What is the heating source of late-phase emission? Does heating accompany the main-phase heating or does it occur quite later? In this paper, we present clear evidence for a heating source in a late-phase solar flare: magnetic reconnection of the overlying field in a quadrupolar magnetic configuration. The event is triggered by an erupted core structure that eventually leads to a coronal mass ejection. A cusp feature and its shrinkage motion high in the late-phase emission region are manifestations of the later phase reconnection following the main flare reconnection. Using the enthalpy-based thermal evolution of loops model, we reasonably reproduce the late-phase emissions in some EUV lines. We suggest that continuous additional heating is responsible for the appearance of the elongated EUV late phase.

Causality and Loop-Tree Duality at Higher Loops.

We relate an l-loop Feynman integral to a sum of phase space integrals, where the integrands are determined by the spanning trees of the original l-loop graph. Causality requires that the propagators of the trees have a modified iδ prescription, and we present a simple formula for the correct iδ prescription.

The Bremsstrahlung function of mathcal{N} = 2 SCQCD

For SU(N) superconformal QCD we perform a three-loop calculation of the cusp anomalous dimension for a generalized Maldacena-Wilson operator, using HQET formalism. We obtain an expression that is valid at generic geometric and internal angles and finite gauge group rank N. For equal and opposite angles this expression vanishes, proving that at these points the cusp becomes BPS. From its small angle expansion we derive the corresponding Bremsstrahlung function at three loops, matching the matrix model prediction given in terms of derivatives of the Wilson loop on the ellipsoid. Finally, we discuss possible scenarios at higher loops, with respect to the existence of a universal effective coupling in an integrable subsector of the model.

Tensor O(N) model near six dimensions: Fixed points and conformal windows from four loops

In search of nontrivial field theories in high dimensions, we study further the tensor representation of the $O(N)$-symmetric ${\ensuremath{\phi}}^{4}$ field theory introduced by Herbut and Janssen [Phys. Rev. D 93, 085005 (2016)] by using four-loop perturbation theory in two cubic interaction coupling constants near six dimensions. For infinitesimal values of the parameter $\ensuremath{\epsilon}=(6\ensuremath{-}d)/2$ we find an infrared-stable fixed point with two relevant quadratic operators for $N$ within the conformal windows $1<N<2.653$ and $2.999<N<4$, and compute critical exponents at this fixed point to the order ${\ensuremath{\epsilon}}^{4}$. Taking the four-loop beta functions at their face value we determine the higher-order corrections to the edges of the above conformal windows at finite $\ensuremath{\epsilon}$, to find both intervals to shrink to zero above $\ensuremath{\epsilon}\ensuremath{\approx}0.15$. The disappearance of the conformal windows with the increase of $\ensuremath{\epsilon}$ is due to the collision of the Wilson-Fisher $\mathcal{O}(\ensuremath{\epsilon})$ infrared fixed point with the $\mathcal{O}(1)$ mixed-stable fixed point that appears at two and persists at higher loops. The latter may be understood as a Banks-Zaks type fixed point that becomes weakly coupled near the right edge of either conformal window. The consequences and issues raised by such an evolution of the flow with dimension are discussed. It is also shown both within the perturbation theory and exactly that the tensor representation at $N=3$ and right at the $\mathcal{O}(\ensuremath{\epsilon})$ infrared-stable fixed point exhibits an emergent $U(3)$ symmetry. A role of this enlarged symmetry in possible protection of the infrared fixed point at $N=3$ is noted.

Ultraviolet properties of N=8 supergravity at five loops

We use the recently developed generalized double-copy construction to obtain an improved representation of the five-loop four-point integrand of $N = 8$ supergravity whose leading ultraviolet behavior we analyze using state of the art loop-integral expansion and reduction methods. We find that the five-loop critical dimension where ultraviolet divergences first occur is $D_c=24/5$, corresponding to a $D^8 R^4$ counterterm. This ultraviolet behavior stands in contrast to the cases of four-dimensional $N = 4$ supergravity at three loops and $N = 5$ supergravity at four loops whose improved ultraviolet behavior demonstrates enhanced cancellations beyond implications from standard-symmetry considerations. We express this $D_c=24/5$ divergence in terms of two relatively simple positive-definite integrals reminiscent of vacuum integrals, excluding any additional ultraviolet cancellations at this loop-order. We note nontrivial relations between the integrals describing this leading ultraviolet behavior and integrals describing lower-loop behavior. This observation suggests not only a path towards greatly simplifying future calculations at higher loops, but may even allow us to directly investigate ultraviolet behavior in terms of simplified integrals, avoiding the construction of complete integrands.

Self-duality, helicity conservation, and normal ordering in nonlinear QED

We give a proof of the equivalence of the electric-magnetic duality on one side and helicity conservation of the tree-level amplitudes on the other side within general models of nonlinear electrodynamics. Using modified Feynman rules derived from a generalized normal ordered Lagrangian, we discuss the interrelation of the above two properties of the theory also at higher loops. As an illustration we present two explicit examples; namely we find the generalized normal ordered Lagrangian for the Born-Infeld theory and derive a semiclosed expression for the Lagrangian of the Bossard-Nicolai model (in terms of the weak field expansion with explicitly known coefficients) from its normal ordered form.

ChPT loops for the lattice: pion mass and decay constant, HVP at finite volume and nn̅-oscillations

I present higher loop order results for several calculations in Chiral perturbation Theory. 1) Two-loop results at finite volume for hadronic vacuum polarization. 2) A three-loop calculation of the pion mass and decay constant in two-flavour ChPT. For the pion mass all needed auxiliary parameters can be determined from lattice calculations of ππ-scattering. 3) Chiral corrections to neutron-anti-neutron oscillations.

Higher-Loop Amplitude Monodromy Relations in String and Gauge Theory.

The monodromy relations in string theory provide a powerful and elegant formalism to understand some of the deepest properties of tree-level field theory amplitudes, like the color-kinematics duality. This duality has been instrumental in tremendous progress on the computations of loop amplitudes in quantum field theory, but a higher-loop generalization of the monodromy construction was lacking. In this Letter, we extend the monodromy relations to higher loops in open string theory. Our construction, based on a contour deformation argument of the open string diagram integrands, leads to new identities that relate planar and nonplanar topologies in string theory. We write one and two-loop monodromy formulas explicitly at any multiplicity. In the field theory limit, at one-loop we obtain identities that reproduce known results. At two loops, we check our formulas by unitarity in the case of the four-point N=4 super-Yang-Mills amplitude.

Analytical study of Yang–Mills theory in the infrared from first principles

Pure Yang–Mills SU(N) theory is studied in the Landau gauge and four dimensional space. While leaving the original Lagrangian unmodified, a double perturbative expansion is devised, based on a massive free-particle propagator. In dimensional regularization, all diverging mass terms cancel exactly in the double expansion, without the need to include mass counterterms that would spoil the symmetry of the Lagrangian. No free parameters are included that were not in the original theory, yielding a fully analytical approach from first principles. The expansion is safe in the infrared and is equivalent to the standard perturbation theory in the UV. At one-loop, explicit analytical expressions are given for the propagators and the running coupling and are found in excellent agreement with the data of lattice simulations. A universal scaling property is predicted for the inverse propagators and shown to be satisfied by the lattice data. Higher loops are found to be negligible in the infrared below 300 MeV where the coupling becomes small and the one-loop approximation is under full control.

Interactions as intertwiners in 4D QFT

In a recent paper we showed that the correlators of free scalar field theory in four dimensions can be constructed from a two dimensional topological field theory based on so(4,2) equivariant maps (intertwiners). The free field result, along with results of Frenkel and Libine on equivariance properties of Feynman integrals, are developed further in this paper. We show that the coefficient of the log term in the 1-loop 4-point conformal integral is a projector in the tensor product of so(4,2) representations. We also show that the 1-loop 4-point integral can be written as a sum of four terms, each associated with the quantum equation of motion for one of the four external legs. The quantum equation of motion is shown to be related to equivariant maps involving indecomposable representations of so(4,2), a phenomenon which illuminates multiplet recombination. The harmonic expansion method for Feynman integrals is a powerful tool for arriving at these results. The generalization to other interactions and higher loops is discussed.

Perturbative quantization of Yang–Mills theory with classical double as gauge algebra

Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.

Next-to-next-to-leading order gravitational spin-squared potential via the effective field theory for spinning objects in the post-Newtonian scheme

The next-to-next-to-leading order spin-squared interaction potential for generic compact binaries is derived for the first time via the effective field theory for gravitating spinning objects in the post-Newtonian scheme. The spin-squared sector is an intricate one, as it requires the consideration of the point particle action beyond minimal coupling, and mainly involves the spin-squared worldline couplings, which are quite complex, compared to the worldline couplings from the minimal coupling part of the action. This sector also involves the linear in spin couplings, as we go up in the nonlinearity of the interaction, and in the loop order. Hence, there is an excessive increase in the number of Feynman diagrams, of which more are higher loop ones. We provide all the Feynman diagrams and their values. The beneficial ``nonrelativistic gravitational'' fields are employed in the computation. This spin-squared correction, which enters at the fourth post-Newtonian order for rapidly rotating compact objects, completes the conservative sector up to the fourth post-Newtonian accuracy. The robustness of the effective field theory for gravitating spinning objects is shown here once again, as demonstrated in a recent series of papers by the authors, which obtained all spin dependent sectors, required up to the fourth post-Newtonian accuracy. The effective field theory of spinning objects allows to directly obtain the equations of motion, and the Hamiltonians, and these will be derived for the potential obtained here in a forthcoming paper.

Loop Stiffness of Grinding Machine Developed for 450 mm Silicon Wafers

Increasing the wafer diameter from φ300 mm to φ450 mm is required to enhance semiconductor devices productivity. A high-stiffness rotary grinding machine equipped with water hydrostatic bearings was developed for a φ450 mm silicon wafer. The grinding machine has an upper structure consisting of a wheel spindle system and a lower structure consisting of a rotary worktable system. The spindle shaft creates both rotary and axial feeding motion. The upper and lower structures are clamped together rigidly by three kinematic couplings. A higher loop stiffness is required for the grinding machine because grinding the larger wafer requires a higher grinding force. This paper investigates the loop stiffness of the developed wafer grinding machine.

Prediction of Geometry of Loop Formed on Terry Fabric Surface Using Mathematical and FEM Modelling

The objective of this study is to understand the loop formation phenomenon of yarn by considering their nonlinear bending behaviour and the effect of loop shape factor on properties of terry fabric. The yarn is modelled as a continuum thin solid beam, and the governing buckling equation is derived using Timoshenko’s elastic theory and the Bernoulli-Euler theorem. Since the formation of loop is effected by large deformation caused by the weight of yarn too, geometric non-linearity is also considered and Runge Kutta method of numerical technique is used to solve the governing equation. Further, finite element modelling technique is also used to see the accuracy of the prediction which is further verified by the actual experimental results. The results of the research prove that the finer yarn produce loops which are having more circularity i.e., higher loop shape factor, as compared to the loops produced from coarse yarn. It is also being proved that the increasing the loop length increases circularity of the loop i.e., higher loop shape factor.

Bootstrapping an NMHV amplitude through three loops

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function $U$, which is closely related to the parity-even part of the ratio function $V$, is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios $(u,v,w)$, as well as on a few two-dimensional planes. Finally, we present an empirical formula for $V$ in terms of elements of the coproduct of the six-gluon MHV remainder function $R_6$ at one higher loop, which works through three loops for $V$ (four loops for $R_6$).