Single-valued harmonic polylogarithms and the multi-Regge limit

A bstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, ( w , w ∗...

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Vydáno v:	The journal of high energy physics Ročník 2012; číslo 10
Hlavní autoři:	Dixon, Lance J., Duhr, Claude, Pennington, Jeffrey
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer-Verlag 01.10.2012 Springer Nature B.V
Témata:	Approximation Classical and Quantum Gravitation Complex variables Constants Eigenvalues Elementary Particles Gluons Harmonic functions High energy physics Kinematics Mathematical analysis Mellin transforms Phenomenology-HEP, Theory-HEP,HEPTH Physics Physics and Astronomy Quantum Field Theories Quantum Field Theory Quantum Physics Relativity Theory String Theory Yang-Mills theory Scattering Amplitudes Extended Supersymmetry Supersymmetric gauge theory
ISSN:	1029-8479, 1029-8479
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Abstract	A bstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, ( w , w ∗ ). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables ( ν , n ) which are related to ( w , w ∗ ) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n ) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N 3 LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.
AbstractList	We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w, w∗). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν, n) which are related to (w, w∗) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N3LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops. A bstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, ( w , w ∗ ). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables ( ν , n ) which are related to ( w , w ∗ ) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n ) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N 3 LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops. We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, ( w , w ∗ ). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables ( ν , n ) which are related to ( w , w ∗ ) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n ) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N 3 LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.
ArticleNumber	74
Author	Duhr, Claude Dixon, Lance J. Pennington, Jeffrey
Author_xml	– sequence: 1 givenname: Lance J. surname: Dixon fullname: Dixon, Lance J. organization: SLAC National Accelerator Laboratory, Stanford University – sequence: 2 givenname: Claude surname: Duhr fullname: Duhr, Claude organization: Institut für Theoretische Physik, ETH Zürich – sequence: 3 givenname: Jeffrey surname: Pennington fullname: Pennington, Jeffrey email: jpennin@stanford.edu organization: SLAC National Accelerator Laboratory, Stanford University
BackLink	https://www.osti.gov/servlets/purl/1053433$$D View this record in Osti.gov
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Snippet	A bstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the... We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the... We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar super Yang-Mills theory are the single-valued...
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SubjectTerms	Approximation Classical and Quantum Gravitation Complex variables Constants Eigenvalues Elementary Particles Gluons Harmonic functions High energy physics Kinematics Mathematical analysis Mellin transforms Phenomenology-HEP, Theory-HEP,HEPTH Physics Physics and Astronomy Quantum Field Theories Quantum Field Theory Quantum Physics Relativity Theory String Theory Yang-Mills theory
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Title	Single-valued harmonic polylogarithms and the multi-Regge limit
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