Polylogarithms, Multiple Zeta Values and Superstring Amplitudes (1304.7267v2)

Abstract: A formalism is provided to calculate tree amplitudes in open superstring theory for any multiplicity at any order in the inverse string tension. We point out that the underlying world-sheet disk integrals share substantial properties with color-ordered tree amplitudes in Yang-Mills field theories. In particular, we closely relate world-sheet integrands of open-string tree amplitudes to the Kawai-Lewellen-Tye representation of supergravity amplitudes. This correspondence helps to reduce the singular parts of world-sheet disk integrals -including their string corrections- to lower-point results. The remaining regular parts are systematically addressed by polylogarithm manipulations.

• The paper elucidates a robust link between superstring disk integrals and Yang-Mills amplitudes by detailing precise pole analysis and decomposition techniques.
• The paper introduces a systematic polylogarithm computation framework based on Goncharov's algebra to simplify complex transcendental integrals at tree level.
• The paper demonstrates computational feasibility by achieving high-order α'-expansions and employing motivic multiple zeta values to streamline amplitude calculations.

Analyzing Polylogarithms, Multiple Zeta Values, and Superstring Amplitudes

The paper under analysis provides an advanced formalism for calculating tree-level amplitudes in open superstring theory, capable of handling any multiplicity and order in the inverse string tension $\alpha'$. This work connects the intricate properties of world-sheet disk integrals in superstring theory with tree amplitudes in Yang-Mills (YM) field theories, particularly highlighting resemblances to the Kawai-Lewellen-Tye (KLT) representation of supergravity amplitudes. By leveraging this correspondence, the authors propose strategies to simplify singular components of world-sheet disk integrals and extend the remaining calculations using polylogarithms.

Key Contributions

1. Relationship with YM Field Theories: The paper elucidates substantial parallels between the disk integrals of superstring scattering amplitudes and color-ordered tree amplitudes in YM theories. This association serves as a crucial tool for decomposing complex string-theoretical integrals into more tractable field-theoretic components.
2. Pole Analysis and Reduction Techniques: A detailed prescription is provided for identifying the pole structure within N-point scattering processes, allowing for the reduction of complex integrals into simpler, lower-point results. This strategy employs momentum-corrected factorization properties intrinsic to the correspondences with YM field theory.
3. Polylogarithm Computation Framework: The authors introduce a systematic approach for calculating the regulated parts of integrals using multiple polylogarithms. By employing Goncharov's algebra of polylogarithms, the paper demonstrates how intricate transcendental functions at loop levels in quantum field theories appear at the tree level in superstring theories.
4. Computational Feasibility for High Orders: The framework is designed to be applicable for computing the $\alpha'$-expansion for any multiplicity and order systematically. Exemplifying this capability, the authors achieve computations up to substantial weights (e.g., order $\alpha'^{21}$ for N=5).
5. Hopf Algebra and Motivic Structures: Advancing the algebraic underpinnings, the paper leverages the motivic multiple zeta values endowed with a Hopf algebra structure to streamline the decomposition of string amplitudes, providing a commensurate formulation of KLT-like relations in superstring theory.

Theoretical and Practical Implications

The paper underlines a refined comprehension of scattering amplitudes within superstring theory and its overlap with other pivotal theoretical frameworks in physics, such as gauge and gravity theories. Practically, these insights could lead to enhanced computational methodologies applicable to high-energy theoretical physics, potentially facilitating more efficient algorithmic approaches concerning transcendental integrals in related fields.

Future Directions

The developed techniques present numerous avenues for further inquiry. The paper's use of motivic structures suggests potential exploration into broader algebraic systems and their applications in theoretical physics. Additionally, the connections between tree-level string amplitudes and field-theory amplitudes can be further deepened, possibly extending results to loop-level corrections or other amplitude calculations in quantum field theories.

In conclusion, the paper provides a robust bridge between superstring theories and YM field theories through sophisticated algebraic methodologies and integral computations, reinforcing the mathematical structures underpinning theoretical physics.