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Program computes fine structure energy levels of ruby as a function of magnetic field. Included in program is matrix formulation, each row of which contains a magnetic field and four corresponding energy levels.
The asymptotic kinematic limits of S-matrices are dominated by large logarithms which, roughly speaking, fall into two categories: those which are controlled by a renormalization group (RG) scale, which we may think of as logs involving ratios of invariant mass scales, and those which are functions of ratios of rapidities, so called “rapidity logs”. It has been pointed out by Caron-Huot and Wilhlem [1] that RG anomalous dimension can be extracted from the phase of the S-matrix, which can greatly simplify calculations via unitarity methods. In this paper we generalize the results of [1] to show that the phase can be used to reconstruct rapidity anomalous dimensions, by performing a special type of complex boost. The methodology introduced allows one to calculate without the need for a rapidity regulator which can lead to significant simplifications. We demonstrate the use of this method to derive the rapidity anomalous dimensions in the Sudakov form factor and the two parton soft function at two loops order.
Abstract Theα ′ -expansion of string theory provides a rich set of higher-dimension operators, indexed byζvalues, which can be used to study color-kinematics duality and the double copy. These two powerful properties, actually first noticed in tree-level string amplitudes, simplify the construction of both gauge and gravity amplitudes. However, their applicability and limitations are not fully understood. We attempt to construct a set of color-kinematics dual numerators at one loop and four points for insertions of operator combinations corresponding to the lowest fourζ 2 -free operator insertions from the open superstring:ζ 3 ,ζ 5 ,$$ {\zeta}_3^2 $$ ζ 3 2 , andζ 7 . We succeed in finding a representation for the first three in terms of box, triangle, and bubble numerators. In the case ofζ 7 we find an obstruction to a fully color-dual representation related to the bubble-on-external-leg type diagrams. We discuss two paths around this obstruction, both of which signal an incompatability between color-kinematics duality and manifesting certain desired properties. Using the constructed color-dual numerators, we find two different Bern-Carrasco-Johansson double copies that produce candidate closed-string-insertion numerators. Both approaches to the double copy match the kinematics of the cuts, with relative normalization set by either summing over both double copies including degeneracy or by including an explicit prefactor on the double-copy numerator definitions.
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This write-up follows the presentation at the symposium, with emphasis on topics and ideas discussed there. It is purposefully informal, not a review of the field, and neither does it include a complete list of references. However, I hope that readers might find somwe comments useful or amusing, and may appreciate the challenges and reasons for excitement about recent progress and future opportunities in flavor physics.
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Lattice quantum chromodynamics (QCD) calculations share a defining challenge by requiring a small finite range of spatial separation z between quark/gluon bilinears for controllable power corrections in the perturbative QCD factorization, and a large hadron boost p z for a successful determination of collinear parton distribution functions (PDFs). However, these two requirements make the determination of PDFs from lattice data very challenging. We present the application of generative machine learning algorithms to estimate the polarized and unpolarized gluon correlation functions utilizing short-distance data and extending the correlation up to z p z ≲ 14 , surpassing the current capabilities of lattice QCD calculations. We train physics-informed machine learning algorithms to learn from the short-distance correlation at z ≲ 0.36 fm and take the limit, p z → ∞ , thereby minimizing possible contamination from the higher-twist effects for a successful reconstruction of the polarized gluon PDF. We also expose the bias and problems with underestimating uncertainties associated with the use of model-dependent and overly constrained functional forms, such as x α ( 1 − x ) β and its variants to extract PDFs from the lattice data. We propose the use of generative machine learning algorithms to mitigate these issues and present our determination of the polarized and unpolarized gluon PDFs in the nucleon. Published by the American Physical Society 2025
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Univac computer programs for energy levels and transition probabilities for ruby and emeralds