<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>ϕ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> amplitudes from the positive tropical Grassmannian: Triangulations of extended diagrams
Bibliographic record
Abstract
The global Schwinger formula, introduced by Cachazo and Early as a single integral over the positive tropical Grassmannian, provides a way to uncover properties of scattering amplitudes which are hard to see in their standard Feynman diagram formulation. In a recent work, Cachazo and one of the authors extended the global Schwinger formula to general <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>ϕ</a:mi> <a:mi>p</a:mi> </a:msup> </a:math> theories. When <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>p</c:mi> <c:mo>=</c:mo> <c:mn>4</c:mn> </c:math> , it was conjectured that the integral decomposes as a sum over cones which are in bijection with noncrossing chord diagrams, and further that these can be obtained by finding the zeroes of a piecewise linear function, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>H</e:mi> <e:mo stretchy="false">(</e:mo> <e:mi>x</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> . In this note we give a proof of this conjecture. We also present a purely combinatorial way of computing <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>ϕ</i:mi> <i:mi>p</i:mi> </i:msup> </i:math> amplitudes by triangulating a trivial extended version of noncrossing ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mi>p</k:mi> <k:mo>−</k:mo> <k:mn>2</k:mn> </k:mrow> </k:math> )-chord diagrams, called extended diagrams, and present a proof of the bijection between triangulated extended diagrams and Feynman diagrams when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mn>4</m:mn> </m:math> . This is reminiscent of recent constructions using Stokes polytopes and accordiohedra. However, the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>ϕ</o:mi> <o:mi>p</o:mi> </o:msup> </o:math> amplitude is now partitioned by a new collection of objects, each of which characterizes a polyhedral cone in the positive tropical Grassmannian in the form of an associahedron or of an intersection of two associahedra. Moreover, we comment on the bijection between extended diagrams and double-ordered biadjoint scalar amplitudes. We also conjecture the form of the general piecewise linear function, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msup> <q:mi>H</q:mi> <q:msup> <q:mi>ϕ</q:mi> <q:mi>p</q:mi> </q:msup> </q:msup> <q:mo stretchy="false">(</q:mo> <q:mi>x</q:mi> <q:mo stretchy="false">)</q:mo> </q:math> , whose zeroes generate the regions in which the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mi>ϕ</u:mi> <u:mi>p</u:mi> </u:msup> </u:math> global Schwinger formula decomposes into. Published by the American Physical Society 2024
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.003 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.025 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".