scholarly journals A note on letters of Yangian invariants

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Song He ◽  
Zhenjie Li
Keyword(s):  
Scattering Amplitudes ◽  
Twistor Space ◽  
Cluster Algebra ◽  
Polynomial Equations ◽  
Algebraic Problem ◽  
The Matrix ◽  
Loop Amplitudes

Abstract Motivated by reformulating Yangian invariants in planar $$ \mathcal{N} $$ N = 4 SYM directly as d log forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the d log’s, which we call “letters”, for any Yangian invariant. These are functions of momentum twistors Z ’s, given by the positive coordinates α’s of parametrizations of the matrix C(α), evaluated on the support of polynomial equations C(α) · Z = 0. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian G(4, n), which is relevant for the symbol alphabet of n-point scattering amplitudes. For n = 6, 7, the collection of letters for all Yangian invariants contains the cluster $$ \mathcal{A} $$ A coordinates of G(4, n). We determine algebraic letters of Yangian invariant associated with any “four-mass” box, which for n = 8 reproduce the 18 multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

scholarly journals Two-loop rational terms in Yang-Mills theories

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Jean-Nicolas Lang ◽  
Stefano Pozzorini ◽  
Hantian Zhang ◽  
Max F. Zoller
Keyword(s):  
Scattering Amplitudes ◽  
Gauge Theories ◽  
A Posteriori ◽  
Four Dimensions ◽  
Yang Mills ◽  
General Properties ◽  
Finite Set ◽  
Numerical Tools ◽  
Loop Amplitudes

Abstract Scattering amplitudes in D dimensions involve particular terms that originate from the interplay of UV poles with the (D − 4)-dimensional parts of loop numerators. Such contributions can be controlled through a finite set of process-independent rational counterterms, which make it possible to compute loop amplitudes with numerical tools that construct the loop numerators in four dimensions. Building on a recent study [1] of the general properties of two-loop rational counterterms, in this paper we investigate their dependence on the choice of renormalisation scheme. We identify a nontrivial form of scheme dependence, which originates from the interplay of mass and field renormalisation with the (D−4)-dimensional parts of loop numerators, and we show that it can be controlled through a new kind of one-loop counterterms. This guarantees that the two-loop rational counterterms for a given renormalisable theory can be derived once and for all in terms of generic renormalisation constants, which can be adapted a posteriori to any scheme. Using this approach, we present the first calculation of the full set of two-loop rational counterterms in Yang-Mills theories. The results are applicable to SU(N) and U(1) gauge theories coupled to nf fermions with arbitrary masses.

Enveloping Algebras of Semi-Simple Lie Algebras

1950 ◽  
Vol 2 ◽  
pp. 257-266 ◽  
Author(s):  
N. Jacobson
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Basis ◽  
Multiplication Table ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Enveloping Algebras ◽  
Starting Point ◽  
The Matrix

In a recent paper we studied systems of equations of the form(1) (2) where as usual [a,b] = ab — ba and ϕ(λ) is a polynomial. Equations of this type have arisen in quantum mechanics. In our paper we gave a method of determining the matrix solutions of such equations. The starting point of our discussion was the observation that if the elements xi satisfy (1) then the elements xi, [xj,xk] satisfy the multiplication table of a certain basis of the Lie algebra of skew symmetric (n + 1) ⨯ (n + 1) matrices. We proved that if (2) is imposed as an added condition, then the algebra generated by the has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.

AN ALGEBRAIC PROBLEM ARISING IN BIOMATHEMATICS

2016 ◽  
Vol 78 (6-6) ◽  
Author(s):  
Sisilia Sylviani ◽  
Ema Carnia ◽  
A. K. Supriatna
Keyword(s):  
Population Growth ◽  
Matrix Model ◽  
Special Form ◽  
Matrix Multiplication ◽  
Diagonal Matrix ◽  
Permutation Matrix ◽  
Block Diagonal Matrix ◽  
Algebraic Problem ◽  
The Matrix ◽  
Block Diagonal

This paper discusses a matrix model that describes the dynamics of a population with m live stages and lives in n patch seen from algebra viewpoint. The matrix D describes population growth in a patch or location. The matrix D is defined as a matrix obtained from matrix multiplication of a permutation matrix with a block diagonal matrix that its diagonal blocks is matrices with non-negative entries and transpose of a permutation matrix [4]. It will be shown that the permutation matrix contained in D has a special form.

YANG-MILLS AMPLITUDES FROM STRING THEORY IN TWISTOR SPACE

2005 ◽  
Vol 20 (15) ◽  
pp. 3416-3419 ◽  
Author(s):  
MARCUS SPRADLIN
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Twistor Space ◽  
Mathematical Structure ◽  
Feynman Diagrams ◽  
Tree Level ◽  
Algebraic Equations ◽  
Yang Mills

Tree-level gluon scattering amplitudes in Yang-Mills theory frequently display simple mathematical structure which is completely obscure in the calculation of Feynman diagrams. We describe a novel way of calculating these amplitudes, motivated by a conjectured relation to twistor space, in which the problem of summing Feynman diagrams is replaced by the problem of solving a certain set of algebraic equations.

scholarly journals Twistor Space Structure Of One-Loop Amplitudes In Gauge Theory

2004 ◽  
Vol 2004 (10) ◽  
pp. 074-074 ◽  
Author(s):  
Freddy Cachazo ◽  
Peter Svrcek ◽  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Twistor Space ◽  
Space Structure ◽  
Loop Amplitudes

scholarly journals Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Niklas Henke ◽  
Georgios Papathanasiou
Keyword(s):  
Scattering Amplitudes ◽  
Finite Subset ◽  
Tropical Geometry ◽  
Cluster Algebra ◽  
Cluster Algebras ◽  
Square Root ◽  
Mathematical Foundation ◽  
Yang Mills ◽  
Square Roots ◽  
Related Approach

Abstract We further exploit the relation between tropical Grassmannians and Gr(4, n) cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory at higher multiplicity n ≥ 8. As a mathematical foundation that provides access to square-root symbol letters in principle for any n, we analyse infinite mutation sequences in cluster algebras with general coefficients. First specialising our analysis to the eight-particle amplitude, and comparing it with a recent, closely related approach based on scattering diagrams, we find that the only additional letters the latter provides are the two square roots associated to the four-mass box. In combination with a tropical rule for selecting a finite subset of variables of the infinite Gr(4, 9) cluster algebra, we then apply our results to obtain a collection of 3, 078 rational and 2, 349 square-root letters expected to appear in the nine-particle amplitude. In particular these contain the alphabet found in an explicit 2-loop NMHV symbol calculation at this multiplicity.

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