Scattering Equations and Feynman Diagrams

We show a direct matching between individual Feynman diagrams and integration measures in the scattering equation formalism of Cachazo, He and Yuan. The connection is most easily explained in terms of triangular graphs associated with planar Feynman diagrams in $\phi^3$-theory. We also discuss the generalization to general scalar field theories with $\phi^p$ interactions, corresponding to polygonal graphs involving vertices of order $p$. Finally, we describe how the same graph-theoretic language can be used to provide the precise link between individual Feynman diagrams and string theory integrands.Comment: 18 pages, 57 figure

Integration Rules for Scattering Equations

As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any M\"obius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.Comment: 30 pages, 29 figure

Integration Rules for Loop Scattering Equations

We formulate new integration rules for one-loop scattering equations analogous to those at tree-level, and test them in a number of non-trivial cases for amplitudes in scalar $\phi^3$-theory. This formalism greatly facilitates the evaluation of amplitudes in the CHY representation at one-loop order, without the need to explicitly sum over the solutions to the loop-level scattering equations.Comment: 22 pages, 17 figure

New Representations of the Perturbative S-Matrix

We propose a new framework to represent the perturbative S-matrix which is well-defined for all quantum field theories of massless particles, constructed from tree-level amplitudes and integrable term-by-term. This representation is derived from the Feynman expansion through a series of partial fraction identities, discarding terms that vanish upon integration. Loop integrands are expressed in terms of "Q-cuts" that involve both off-shell and on-shell loop-momenta, defined with a precise contour prescription that can be evaluated by ordinary methods. This framework implies recent results found in the scattering equation formalism at one-loop, and it has a natural extension to all orders---even non-planar theories without well-defined forward limits or good ultraviolet behavior.Comment: 4+1 pages, 4 figure

String-like dual models for scalar theories

We show that all tree-level amplitudes in $\varphi^p$ scalar field theory can be represented as the $\alpha'\to0$ limit of an $SL(2,R)$-invariant, string-theory-like dual model integral. These dual models are constructed according to constraints that admit families of solutions. We derive these dual models, and give closed formulae for all tree-level amplitudes of any $\varphi^p$ scalar field theory.Comment: 15 pages, 0 figure

Origin and evolution of formation waters, Alberta Basin, Western Canada sedimentary Basin. I. Chemistry

Inorganic chemical analyses and short-chain aliphatic acid content are used to interpret the origin and compositional evolution of formation waters in the Alberta portion of the Western Canada Sedimentary Basin. Forty-three formation water samples were obtained covering a stratigraphic interval from Devonian to Cretaceous. The data show that: (1) there is a subaerially evaporated brine component that shows no apparent contribution of waters derived from evaporite dissolution; and (2) formation waters have maintained characteristics indicative of subaerially evaporated waters, despite subsequent flushing by gravity-driven meteoric waters in the basin.Formation waters are predominantly Na---Cl brines that contain 4-235g/l total dissolved solids (TDS). Short-chain aliphatic acids (SCA) range up to 932 mg/l, with the following abundance: acetate >> propionate > butyrate. Their number varies randomly with subsurface temperature, depth, geological age and salinity. Instead, SCA distributions appear related to proximity to Jurassic and Mississippian source rocks and to zones of active bacterial SO4 reduction.Based on chemical composition, the formation waters can be divided into three groups. Group I waters are from dominantly carbonate reservoirs and Group II from clastics. Groups I and II are differentiated from Group III in that they are composed of a brine end member, formed by evaporation of sea water beyond the point of halite saturation, that has been subsequently diluted 50-80% by a meteoric water end member. Group III waters are from clastic reservoirs and are dilute, meteoric waters that are decoupled from the more saline, stratigraphically lower, waters of Groups I and II.Group I waters have been influenced by clay mineral transformations in shales surrounding the carbonate reservoirs, ankeritization reactions of reservoir dolomites and calcites, and possible decarboxylation reactions. Group II waters indicate significant leaching reactions, particularly of feldspar and clay minerals. Group I and Group II waters both indicate ion exchange reactions were also possible. The waters are near equilibrium with respect to quartz, calcite, dolomite and barite, but are undersaturated with respect to evaporite minerals (halite, anhydrite). Occurrence of feldspar (predominantly albite) and kaolinite seems to control the population of the water cations. Post-Laramide invasion of meteoric waters provided an impetus for many of the diagenetic reactions in both carbonate, but especially in clastic reservoirs. Subsequent hydrochemical isolation of Group I and II waters from further meteoric influences occurred, resulting in pronounced mixing relations and cross-formational fluid flow replacing the once dominant lateral flow.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28507/1/0000304.pd

Origin and evolution of formation waters, Alberta Basin, Western Canada Sedimentary Basin. II. Isotope systematics and water mixing

Isotopic measurements (Sr, O, D) on formation waters from the Alberta Basin have been made, covering a stratigraphic range from Devonian to Upper Cretaceous. These measurements, combined with chemical compositional trends, give evidence for two distinct water regimes. One hydrological regime is composed of waters hosted in Devonian-Lower Cretaceous reservoirs, the other waters from Upper Cretaceous and younger sedimentary rocks. The two regimes are separated by a regional transgressive shale in the Colorado Group, the Second White Speckled Shale Formation.The waters within the Devonian-Lower Cretaceous regime exhibit a large range in 87Sr/86Sr values (0.7076-0.7129), but have similar Sr concentrations, regardless of host lithology. Bulk rock and late-stage diagenetic cements are less radiogenic than present brines. Importantly, brines from Devonian carbonates possess the most radiogenic Sr isotopic signatures of the waters examined. Devonian shales and/or Cambrian shales may be sources of high 87Sr/86Sr ratios in the carbonate-hosted waters. Waters from the Upper Cretaceous clastic units, which have ratios as low as 0.7058, and diagenetic cements from Upper Cretaceous clastic units appear to have precipitated from fluids similar in Sr isotopic value to modern brines. High Sr concentrations in the Cretaceous clastic waters and sedimentary rocks and correspondingly low 87Sr/86Sr ratios suggest that volcanism in Montana during the Cretaceous may have provided a source of sediments to the study area.Cross-formational upward water migration, superimposed on lateral fluid flow, is required to explain the geochemistry and isotopic systematics in the brines from Devonian-Lower Cretaceous reservoirs. Strontium isotope ratios and Sr contents suggest a two component mixing relation for these waters. This system of waters also exhibits [delta]D values characteristic of meteoric values in the Neogene, reflecting post-Laramide flushing of Tertiary waters throughout the basin, with subsequent hydrochemical isolation from more modern waters. In contrast, waters in Upper Cretaceous reservoirs have O and D isotopic compositions similar to those of present day rainfall; these, in conjunction with very dilute Sr concentrations and low Sr ratios, suggest hydrological isolation from the stratigraphically lower system.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28509/1/0000306.pd

An algebraic approach to the scattering equations

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism