New differential equations for on-shell loop integrals

We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.Comment: 39 pages, 8 figures; v2: typos corrected, definition of harmonic polylogarithms adde

On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM

We study the relationship between the momentum twistor MHV vertex expansion of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of the BCFW recursion relations. We demonstrate explicitly in several examples that the MHV vertex expressions for tree-level amplitudes and loop integrands satisfy the recursion relations. Furthermore, we introduce a rewriting of the MHV expansion in terms of sums over non-crossing partitions and show that this cyclically invariant formula satisfies the recursion relations for all numbers of legs and all loop orders.Comment: 34 pages, 17 figures; v2: Minor improvements to exposition and discussion, updated references, typos fixe

Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills

It has been argued in arXiv:1112.6432 that the planar four-point integrand in N=4 super Yang-Mills theory is uniquely determined by dual conformal invariance together with the absence of a double pole in the integrand of the logarithm in the limit as a loop integration variable becomes collinear with an external momentum. In this paper we reformulate this condition in a simple way in terms of the amplitude itself, rather than its logarithm, and verify that it holds for two- and three-loop MHV integrands for n>4. We investigate the extent to which this collinear constraint and a constraint on the soft behavior of integrands can be used to determine integrands. We find an interesting complementarity whereby the soft constraint becomes stronger while the collinear constraint becomes weaker at larger n. For certain reasonable choices of basis at two and three loops the two constraints in unison appear strong enough to determine MHV integrands uniquely for all n.Comment: 27 pages, 14 figures; v2: very minor change

Some analytic results for two-loop scattering amplitudes

We present analytic results for the finite diagrams contributing to the two-loop eight-point MHV scattering amplitude of planar N=4 SYM. We use a recently proposed representation for the integrand of the amplitude in terms of (momentum) twistors and focus on a restricted kinematics in which the answer depends only on two independent cross-ratios. The theory of motives can be used to vastly simplify the results, which can be expressed as simple combinations of classical polylogarithms.Comment: 18 page

On BCFW shifts of integrands and integrals

In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their `single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choice employed earlier at tree level by Arkani-Hamed and Kaplan. The relation between shifts of integrands and shifts of its integrals is investigated explicitly at one loop. Two particular sources of discrepancy between the integral and integrand are identified related to UV and IR divergences. This is cross-checked with known results for helicity equal amplitudes at one loop. The nature of the on-shell residue at each of the single-cut singularities of the integrand is commented upon. Several natural conjectures and opportunities for further research present themselves.Comment: 43 pages, 6 figures, v2: minor improvement in exposition, typos fixed, bibliography update

SCAMP:standardised, concentrated, additional macronutrients, parenteral nutrition in very preterm infants: a phase IV randomised, controlled exploratory study of macronutrient intake, growth and other aspects of neonatal care

<p>Abstract</p> <p>Background</p> <p>Infants born <29 weeks gestation are at high risk of neurocognitive disability. Early postnatal growth failure, particularly head growth, is an important and potentially reversible risk factor for impaired neurodevelopmental outcome. Inadequate nutrition is a major factor in this postnatal growth failure, optimal protein and calorie (macronutrient) intakes are rarely achieved, especially in the first week. Infants <29 weeks are dependent on parenteral nutrition for the bulk of their nutrient needs for the first 2-3 weeks of life to allow gut adaptation to milk digestion. The prescription, formulation and administration of neonatal parenteral nutrition is critical to achieving optimal protein and calorie intake but has received little scientific evaluation. Current neonatal parenteral nutrition regimens often rely on individualised prescription to manage the labile, unpredictable biochemical and metabolic control characteristic of the early neonatal period. Individualised prescription frequently fails to translate into optimal macronutrient delivery. We have previously shown that a standardised, concentrated neonatal parenteral nutrition regimen can optimise macronutrient intake.</p> <p>Methods</p> <p>We propose a single centre, randomised controlled exploratory trial of two standardised, concentrated neonatal parenteral nutrition regimens comparing a standard macronutrient content (maximum protein 2.8 g/kg/day; lipid 2.8 g/kg/day, dextrose 10%) with a higher macronutrient content (maximum protein 3.8 g/kg/day; lipid 3.8 g/kg/day, dextrose 12%) over the first 28 days of life. 150 infants 24-28 completed weeks gestation and birthweight <1200 g will be recruited. The primary outcome will be head growth velocity in the first 28 days of life. Secondary outcomes will include a) auxological data between birth and 36 weeks corrected gestational age b) actual macronutrient intake in first 28 days c) biomarkers of biochemical and metabolic tolerance d) infection biomarkers and other intravascular line complications e) incidence of major complications of prematurity including mortality f) neurodevelopmental outcome at 2 years corrected gestational age</p> <p>Trial registration</p> <p>Current controlled trials: <a href="http://www.controlled-trials.com/ISRCTN76597892">ISRCTN76597892</a>; EudraCT Number: 2008-008899-14</p

Wilson Loops @ 3-Loops in Special Kinematics

We obtain a compact expression for the octagon MHV amplitude / Wilson loop at 3 loops in planar N=4 SYM and in special 2d kinematics in terms of 7 unfixed coefficients. We do this by making use of the cyclic and parity symmetry of the amplitude/Wilson loop and its behaviour in the soft/collinear limits as well as in the leading term in the expansion away from this limit. We also make a natural and quite general assumption about the functional form of the result, namely that it should consist of weight 6 polylogarithms whose symbol consists of basic cross-ratios only (and not functions thereof). We also describe the uplift of this result to 10 points.Comment: 26 pages. Typos correcte

Massive amplitudes on the Coulomb branch of N=4 SYM

We initiate a systematic study of amplitudes with massive external particles on the Coulomb-branch of N=4 super Yang Mills theory: 1) We propose that (multi-)soft-scalar limits of massless amplitudes at the origin of moduli space can be used to determine Coulomb-branch amplitudes to leading order in the mass. This is demonstrated in numerous examples. 2) We find compact explicit expressions for several towers of tree-level amplitudes, including scattering of two massive W-bosons with any number of positive helicity gluons, valid for all values of the mass. 3) We present the general structure of superamplitudes on the Coulomb branch. For example, the n-point "MHV-band" superamplitude is proportional to a Grassmann polynomial of mixed degree 4 to 12, which is uniquely determined by supersymmetry. We find explicit tree-level superamplitudes for this MHV band and for other simple sectors of the theory. 4) Dual conformal generators are constructed, and we explore the dual conformal properties of the simplest massive amplitudes. Our compact expressions for amplitudes and superamplitudes should be of both theoretical and phenomenological interest; in particular the tree-level results carry over to truncations of the theory with less supersymmetry.Comment: 29 pages, 1 figur

Adrenal Dysfunction in Hemodynamically Unstable Patients in the Emergency Department

Objective: Adrenal failure, a treatable condition, can have catastrophic consequences if unrecognized in critically ill ED patients. The authors' objective was to prospectively study adrenal function in a case series of hemodynamically unstable (high-risk) patients from a large, urban ED over a 12-month period. Methods: In a prospective manner, critically ill adult patients presenting to the ED were enrolled when presenting with a mean arterial blood pressure ≤60 mm Hg requiring vasopressor therapy for more than one hour after receiving fluid resuscitation (central venous pressure of 12-15 mm Hg or a minimum of 40 mL/kg of crystalloid). Patients were excluded if presenting with hemorrhage, trauma, or AIDS, or if steroids were used within the previous six months. An adrenocorticotropic hormone (ACTH) stimulation test was performed and serum cortisol was measured. Treatment for adrenal insufficiency was not instituted. Results: A total of 57 consecutive patients were studied. Of these, eight (14%) had baseline serum cortisol concentrations of <20 Μg/dL (<552 nmol/L), which was considered adrenal insufficiency (AI). Three additional patients (5%) had subnormal 60-minute post-ACTH-stimulation cortisol responses (<30 Μg/dL) and a delta cortisol ≤9 Μg/dL, which is the difference between the baseline and 60-minute levels. This is functional hypoadrenalism (FH). There were no laboratory abnormalities that distinguished patients with AI or FH from those with preserved adrenal function (PAF). Rates of survival to discharge did not differ between the AI group (7 of 8) and PAF patients (21 of 46; p = 0.052). Conclusions: Adrenal dysfunction is common in high-risk ED patients. Overall, it has a frequency of 19% among a homogeneous population of hemodynamically unstable vasopressor-dependent patients. The effect of physiologic glucocorticoid replacement in this setting remains to be determined.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71956/1/j.1553-2712.1999.tb00417.x.pd

Scattering Amplitudes and Toric Geometry

In this paper we provide a first attempt towards a toric geometric interpretation of scattering amplitudes. In recent investigations it has indeed been proposed that the all-loop integrand of planar N=4 SYM can be represented in terms of well defined finite objects called on-shell diagrams drawn on disks. Furthermore it has been shown that the physical information of on-shell diagrams is encoded in the geometry of auxiliary algebraic varieties called the totally non negative Grassmannians. In this new formulation the infinite dimensional symmetry of the theory is manifest and many results, that are quite tricky to obtain in terms of the standard Lagrangian formulation of the theory, are instead manifest. In this paper, elaborating on previous results, we provide another picture of the scattering amplitudes in terms of toric geometry. In particular we describe in detail the toric varieties associated to an on-shell diagram, how the singularities of the amplitudes are encoded in some subspaces of the toric variety, and how this picture maps onto the Grassmannian description. Eventually we discuss the action of cluster transformations on the toric varieties. The hope is to provide an alternative description of the scattering amplitudes that could contribute in the developing of this very interesting field of research.Comment: 58 pages, 25 figures, typos corrected, a reference added, to be published in JHE