Jekyll2019-06-28T00:05:28+00:00http://amplitudes.org/feed.xmlAmplitudes.orgComing soon.John Joseph M. CarrascoConverting from Schnetz f-alphabet to MZVs through weight 142019-06-17T00:00:00+00:002019-06-17T00:00:00+00:00http://amplitudes.org/software/fAlphabetToMZVs<h1 id="link">Link:</h1>
<p>Paper and machine readable expressions.</p>
<ul>
<li>Paper: <a href="https://arxiv.org/abs/1906.07116">The Cosmic Galois Group and Extended Steinmann Relations for Planar N=4 SYM Amplitudes</a></li>
<li><a href="https://arxiv.org/src/1906.07116v1/anc">Conversion Code</a></li>
</ul>
<h1 id="authors">Author(s):</h1>
<p>Simon Caron-Huot, Lance J. Dixon, Falko Dulat, Matt von Hippel, Andrew J. McLeod, Georgios Papathanasiou</p>
<h1 id="abstract">Abstract:</h1>
<p>We describe the minimal space of polylogarithmic functions that is required to express the six-particle amplitude in planar N=4 super-Yang-Mills theory through six and seven loops, in the NMHV and MHV sectors respectively. This space respects a set of extended Steinmann relations that restrict the iterated discontinuity structure of the amplitude, as well as a cosmic Galois coaction principle that constrains the functions and the transcendental numbers that can appear in the amplitude at special kinematic points. To put the amplitude into this space, we must divide it by the BDS-like ansatz and by an additional zeta-valued constant ρ. For this normalization, we conjecture that the extended Steinmann relations and the coaction principle hold to all orders in the coupling. We describe an iterative algorithm for constructing the space of hexagon functions that respects both constraints. We highlight further simplifications that begin to occur in this space of functions at weight eight, and distill the implications of imposing the coaction principle to all orders. Finally, we explore the restricted spaces of transcendental functions and constants that appear in special kinematic configurations, which include polylogarithms involving square, cube, fourth and sixth roots of unity</p>
<h1 id="point-of-the-code">Point of the Code:</h1>
<blockquote>
<p>At many of these points, transcendental constants beyond MZVs appear, such as
alternating sums and multiple polylogarithms evaluated at higher roots of unity.
To study the coaction at these points, it is especially useful to work in terms
of an f-alphabet, which makes the coaction structure of these constants manifest.</p>
</blockquote>
<h1 id="see-also">See also:</h1>
<p>O. Schnetz. <a href="https://www.math.fau.de/person/oliver-schnetz/">Computer program hyperlogprocedures</a></p>John Joseph M. Carrascohttp://prettyquestions.comRelated paper and mathematica code.Integrated One and Two Loop Four-Point Super-QCD2019-04-10T00:00:00+00:002019-04-10T00:00:00+00:00http://amplitudes.org/amplitudes/FullN2SQCD2Loops<h1 id="link">Link:</h1>
<p>Paper and Mathematica expressions.</p>
<ul>
<li><a href="https://arxiv.org/abs/1904.05299">The Full-Color Two-Loop Four-Gluon Amplitude in N=2 Super-QCD</a></li>
<li><a href="https://arxiv.org/src/1904.05299v1/anc">Mathematica expressions</a></li>
</ul>
<h1 id="authors">Author(s):</h1>
<p>Claude Duhr, Henrik Johansson, Gregor Kälin, Gustav Mogull, Bram Verbeek</p>
<h1 id="abstract">Abstract:</h1>
<p>We present the fully integrated form of the two-loop four-gluon amplitude in N=2 supersymmetric quantum chromodynamics with gauge group SU(Nc) and with Nf massless supersymmetric quarks (hypermultiplets) in the fundamental representation. Our result maintains full dependence on Nc and Nf, and relies on the existence of a compact integrand representation that exhibits the duality between color and kinematics. Specializing to the N=2 superconformal theory, where Nf=2Nc , we obtain remarkably simple amplitudes that have an analytic structure close to that of N=4 super-Yang-Mills theory, except that now certain lower-weight terms appear. We comment on the corresponding results for other gauge groups.</p>John Joseph M. Carrascohttp://prettyquestions.comPaper and Mathematica expressions.Six-Gluon Amplitudes in Planar N=4 Super-Yang-Mills Theory at Six and Seven Loops2019-03-26T00:00:00+00:002019-03-26T00:00:00+00:00http://amplitudes.org/amplitudes/sixSevenLoop6PointN4Planar<h1 id="link">Link:</h1>
<p>Paper and machine readable expressions.</p>
<ul>
<li>Paper: <a href="https://arxiv.org/abs/1903.10890">Six-Gluon Amplitudes in Planar N=4 Super-Yang-Mills Theory at Six and Seven Loops</a></li>
<li><a href="http://www.slac.stanford.edu/~lance/Cosmic/">Huge Cosmically Normalized Mathematica Files</a></li>
</ul>
<h1 id="authors">Author(s):</h1>
<p>Simon Caron-Huot, Lance Dixon, Falko Dulat, Matt von Hippel, Andrew McLeod and Georgios Papathanasiou</p>
<h1 id="abstract">Abstract:</h1>
<p>We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.</p>John Joseph M. Carrascohttp://prettyquestions.comPaper and mathematica expressions.Four-graviton scattering to three loops in N=8 supergravity2019-02-19T00:00:00+00:002019-02-19T00:00:00+00:00http://amplitudes.org/amplitudes/IntegratedN=8ThreeLoopFourPoint<h1 id="link">Link:</h1>
<p>Paper and Mathematica expressions.</p>
<ul>
<li><a href="https://arxiv.org/abs/1902.07221">Four-graviton scattering to three loops in N=8 supergravity</a></li>
<li><a href="https://arxiv.org/src/1902.07221v1/anc">Mathematica expressions</a></li>
</ul>
<h1 id="authors">Author(s):</h1>
<p>Johannes M. Henn, Bernhard Mistlberger</p>
<h1 id="abstract">Abstract:</h1>
<p>We compute the three-loop scattering amplitude of four gravitons in N=8 supergravity. Our results are analytic formulae for a Laurent expansion of the amplitude in the regulator of dimensional regularisation. The coefficients of this series are closed formulae in terms of well-established harmonic poly-logarithms. Our results display a remarkable degree of simplicity and represent an important stepping stone in the exploration of the structure of scattering amplitudes. In particular, we observe that to this loop order the four graviton amplitude is given by uniform weight 2L functions, where L is the loop order.</p>John Joseph M. Carrascohttp://prettyquestions.comPaper and Mathematica expressions.The two-loop five-point amplitude in N=8 supergravity2019-01-24T00:00:00+00:002019-01-24T00:00:00+00:00http://amplitudes.org/amplitudes/TwoLoopFivePointN8sg<h2 id="two-for-the-price-of-one--virtually-simultaneous-results-from-two-groups">Two for the price of one! Virtually simultaneous results from two groups.</h2>
<h2 id="17-jan-2019">17 Jan 2019</h2>
<h1 id="link">Link:</h1>
<p>Paper and Mathematica expressions.</p>
<ul>
<li><a href="https://arxiv.org/abs/1901.05932">The two-loop five-particle amplitude in N=8 supergravity</a></li>
<li><a href="https://arxiv.org/src/1901.005932/anc">Mathematica expressions</a></li>
</ul>
<h1 id="authors">Author(s):</h1>
<p>D. Chicherin, T. Gehrmann, J. M. Henn, P. Wasser, Y. Zhang, S. Zoia</p>
<h1 id="abstract">Abstract:</h1>
<p>We compute for the first time the two-loop five-particle amplitude in N=8 supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. The amplitude satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate, and extract the finite remainder function. The latter depends on fewer rational factors, and is independent of one of the symbol letters. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as an ancillary file.</p>
<h2 id="24-jan-2019">24 Jan 2019</h2>
<h1 id="link-1">Link:</h1>
<p>Paper and Mathematica expressions.</p>
<ul>
<li><a href="https://arxiv.org/abs/1901.08563/">The two-loop five-point amplitude in N=8 supergravity</a></li>
<li><a href="https://arxiv.org/src/1901.08563/anc">Mathematica expressions</a></li>
</ul>
<h1 id="authors-1">Author(s):</h1>
<p>Samuel Abreu, Lance J. Dixon, Enrico Herrmann, Ben Page, Mao Zeng</p>
<h1 id="abstract-1">Abstract:</h1>
<p>We compute the symbol of the two-loop five-point amplitude in N=8 supergravity. We write an ansatz for the amplitude whose rational prefactors are based on not only 4-dimensional leading singularities, but also d-dimensional ones, as the former are insufficient. Our novel d-dimensional unitarity-based approach to the systematic construction of an amplitude’s rational structures is likely to have broader applications, for example to analogous QCD calculations. We fix parameters in the ansatz by performing numerical integration-by-parts reduction of the known integrand. We find that the two-loop five-point N=8 supergravity amplitude is uniformly transcendental. We then verify the soft and collinear limits of the amplitude. There is considerable similarity with the corresponding amplitude for N=4 super-Yang-Mills theory: all the rational prefactors are double copies of the Yang-Mills ones and the transcendental functions overlap to a large degree. As a byproduct, we find new relations between color-ordered loop amplitudes in N=4 super-Yang-Mills theory.</p>John Joseph M. Carrascohttp://prettyquestions.comPaper and Mathematica expressions.All Summer School Lectures for Amplitudes 20182018-06-15T00:00:00+00:002018-06-15T00:00:00+00:00http://amplitudes.org/lectures/Amp2018SummerSchool<h1 id="link">Link</h1>
<p>All <a href="http://qmap.ucdavis.edu/events/events-past-events/amplitudes-summer-school">2018 Amplitudes summer school lectures</a>,
hosted by Mathematics and Physics (QMAP).</p>
<h1 id="lecturers">Lecturers</h1>
<p>Nima Arkani-Hamed, Zvi Bern, Jacob Bourjaily, Claude Duhr, Eric D'Hoker, Song He, Yutin Huang, Alexander Postnikov.</p>
<h1 id="description">Description</h1>
<p>Scattering amplitudes are the arena where quantum field theory directly confronts experiment, and the precise predictions are crucial for the beyond Standard Model searches at particle colliders such as LHC. At the same time, scattering amplitudes have revealed beautiful and intriguing mathematical structures and patterns which suggest that our fundamental understanding of quantum field theory is far from complete. The study of scattering amplitudes has been a very active area of research over the past few years, with applications ranging from precision calculations for the LHC through new approaches to gauge theories and gravity and ending with exciting connections to pure mathematics.</p>
<h1 id="student-research-presentations">Student Research Presentations</h1>
<p>Cameron Langer,
Matteo Parisi,
Ryota Kojima,
Stefan Stanojevic,
Gongwang Yan,
Hadleigh Frost,
Giulio Salvatori,
Nicholas Early,
Akshay Yelleshpur Srikant,
Julio Parra-Martinez,
Ekta Ekta,
Simone Zoia,
Alex Edison,
Gregor Kälin,
Michael Enciso,
Isobel Nicholson,
Mariana Carrillo Gonzalez,
Sebastian Mizera,
Matthew Heydeman,
Alfredo Guevara,
Stefan Nekovar,
Callum Jones,
Zhewei Yin,
Shruti Paranjape,
James Bonifacio,
Marios Hadjiantonis,
Christian Broennum-Hansen,
Herschel A. Chawdhry,
Taushif Ahmed,
Vasily Sotnikov,
Joe Farrow,
Nat Levine,
Pranjal Nayak,
Sangmin Choi,
Michal Pazderka,
Vsevolod Chestnov,
Sourav Raha,
Haoyu Sun,
Ana Lucia Retore.</p>John Joseph M. Carrascohttp://prettyquestions.comLinks to Video and SlidesA Monte Carlo Approach to the 4D Scattering Equations2018-06-07T00:00:00+00:002018-06-07T00:00:00+00:00http://amplitudes.org/software/2018_MC_Scattering<h1 id="link">Link:</h1>
<p>Paper and mathematica code in source (under arxiv “Other Formats”):
<a href="https://arxiv.org/abs/1806.02732">A Monte Carlo Approach to the 4D Scattering Equations</a>.</p>
<h1 id="author">Author:</h1>
<p>Josef A. Farrow</p>
<h1 id="abstract">Abstract:</h1>
<p>The scattering equation formalism is a general framework for calculation of amplitudes in theories of massless particles. We provide a detailed introduction to the 4D scattering equation framework accessible to non-experts, outline current difficulties solving the equations numerically, and explain how to overcome them with a Monte Carlo algorithm. With this submission we include <code class="highlighter-rouge">treeamps4dJAF</code>, the first publicly available <em>Mathematica</em> package for calculating amplitudes by solving the scattering equations, supporting MHV analytical and Nk−2MHV numerical computations. The package provides a powerful and flexible computational tool for calculating tree-level amplitudes in super Yang Mills theories, Einstein supergravity and conformal supergravity. We tabulate sets of numerical solutions up to 9 points in all MHV sectors and 12 points in the NHMV sector which can be used for fast evaluation of amplitudes.</p>John Joseph M. Carrascohttp://prettyquestions.comJosef Farrow -- Paper and code applying Monte-Carlo to solving 4D CHY Scattering Equations.N=4 SYM and N=8 SG 5-loop integrands2018-04-25T00:00:00+00:002018-04-25T00:00:00+00:00http://amplitudes.org/integrands/uvFiveLoopMaxSG<h1 id="link">Link:</h1>
<p>Paper and Mathematica expressions.</p>
<ul>
<li><a href="https://arxiv.org/abs/1804.09311">Ultraviolet Properties of N = 8 Supergravity at Five Loops</a></li>
<li><a href="https://arxiv.org/src/1804.09311v1/anc">Mathematica expressions</a></li>
</ul>
<h1 id="authors">Author(s):</h1>
<p>Zvi Bern, John Joseph Carrasco, Wei-Ming Chen, Alex Edison, Henrik Johansson, Julio Parra-Martinez, Radu Roiban, Mao Zeng</p>
<h1 id="abstract">Abstract:</h1>
<p>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 Dc=24/5, corresponding to a D8R4 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 Dc=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.</p>John Joseph M. Carrascohttp://prettyquestions.comPaper and Mathematica expressions.Amplitudes related Graph Tools2017-07-11T00:00:00+00:002017-07-11T00:00:00+00:00http://amplitudes.org/software/2017_AmpGraphTools<h1 id="link">Link</h1>
<p>Github hosted amplitudes open graph library <a href="https://github.com/drjjmc/ampGraphTools_mma">ampGraphTools</a>.
Sample implementation of ideas described in TASI 2014 Lectures on <a href="https://arxiv.org/abs/1506.00974">Gauge and Gravity Amplitude Relations</a>.</p>
<h1 id="author">Author</h1>
<p>John Joseph M. Carrasco</p>
<h1 id="example-usage">Example Usage</h1>
<p>Can see examples of usage in live-written Mathematica notebook used to give 4 lectures here <a href="https://figshare.com/articles/Graphical_Methods_for_Sharp_Predictions_Live_Lecture_Notes_-_Amplitudes_2017_Summer_School/5197213">here</a>.</p>
<h1 id="content-of-4-lectures">Content of 4-lectures:</h1>
<p>Introducing graph methods for modern amplitude calculation.</p>
<ul>
<li>
<p>Graphs and Ordered Amplitudes</p>
</li>
<li>
<p>Integrand Verification – two-loops.</p>
</li>
<li>
<p>Integrand Cut Construction – two loops.</p>
</li>
<li>
<p>Introduction to Color-Kinematics</p>
</li>
</ul>
<h1 id="video">Video</h1>
<p>Video for [<a href="https://media.ed.ac.uk/media/Amplitudes+Summer+School+2017+-+ohn+Joseph+CarrascoA+Graphical+Methods+for+Sharp+PredictionA+from+theories+entirely+formal+to+the+utterly+effective+-+lecture+1+/1_aan0792b/60996171">Lecture 1</a>]
[<a href="https://media.ed.ac.uk/media/Amplitudes+Summer+School+-+John+Joseph+CarrascoA+Graphical+Methods+for+Sharp+PredictionA+from+theories+entirely+formal+to+the+utterly+effective+-+lecture+2/1_f4m1wl5h">Lecture 2</a>] [<a href="https://media.ed.ac.uk/media/Amplitudes+Summer+School+2017A+John+Joseph+Carrasco++-+Graphical+Methods+for+Sharp+PredictionA+from+theories+entirely+formal+to+the+utterly+effective+-+lecture+3/1_jqiem0l6">Lecture 3</a>] [<a href="https://media.ed.ac.uk/media/Amplitudes+Summer+School+2017A+John+Joseph+Carrasco+-+Graphical+Methods+for+Sharp+PredictionA+from+theories+entirely+formal+to+the+utterly+effective+-+lecture+4/1_3chpd4tj">Lecture 4</a>]</p>John Joseph M. Carrascohttp://prettyquestions.comJJMC -- Simple implementation of some amplitudes related graph tools.All Summer School Lectures for Amplitudes 20172017-07-07T00:00:00+00:002017-07-07T00:00:00+00:00http://amplitudes.org/lectures/2017AmpSummerSchool<h1 id="link">Link</h1>
<p>All summer school lectures,
hosted by the Higgs Center for Theoretical Physics in Edinburgh,
<a href="https://indico.ph.ed.ac.uk/event/30/">here</a>.</p>
<h1 id="lecturers">Lecturer’s</h1>
<p>Stefan Weinzierl, Marcus Spradlin, John Joseph M. Carrasco, Jaroslav Trnka, and Johannes Henn.</p>
<h1 id="description">Description</h1>
<p>The study of scattering amplitudes has developed substantially over the past few years, with applications ranging from precision calculations for the LHC through new approaches to gauge theories and gravity and ending with pure mathematics. The annual meeting Amplitudes 2017 workshop will be hosted by the Higgs Centre between 10-14 July. This present summer school is held in the week preceding the workshop, and it offers an introduction to some of the central topics to be discussed at the workshop. Five lecturers will deliver sets of lectures and tutorials over the full 5 days of 3-7 July.</p>
<h1 id="student-research-presentations">Student Research Presentations</h1>
<p>Jorrit Bosma, Sebastian Mizera, Gustav Mogull, Chia-Hsien Shen, Andrew McLeod, Theresa Abl, Joe Farrow, and Andres Luna-Godoy.</p>John Joseph M. Carrascohttp://prettyquestions.comLinks to Video and Slides