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Master Integrals Evaluation: Methods and Applications in Feynman Diagram Computation

Explore the method of differential equations, analytic evaluation, and reduction of master integrals in the study of Feynman diagrams. Discover techniques like Glue-and-cut, Mellin-Barnes, Gegenbauer polynomial, and unitarity method. Learn how to reduce complex diagrams to a combination of master integrals and solve coupled differential equations satisfied by them. Dive into examples of evaluating 3-loop diagrams and converting integrals to harmonic polylogs. Uncover relationships among harmonic polylogs and reduction to a basis set. Find more details in academic sources like arXiv:1211.5204 [hep-th].

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Master Integrals Evaluation: Methods and Applications in Feynman Diagram Computation

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  1. Evaluation of Master Integrals: Method of differential equation Y. Sumino (Tohoku Univ.)

  2. Diagram Computation: Method of Differential Eq. Analytic evaluation of Feynman diagrams: Many methods but no general one • Glue-and-cut • Mellin-Barnes • Differential eq. • Gegenbauer polynomial • Unitarity method . . . .

  3. can be reduced to a combination of master integrals Master integrals can be chosen finite as . Derivative of master integrals w.r.t. an external kinematical variable A combination of master integrals System of linear coupled diff. eq. satisfied by finite master integrals (D=4).

  4. ☆ Example: evaluation of a 3-loop diagram Some of the lines of original diag. are pinched.

  5. ☆ Example: evaluation of a 3-loop diagram Some of the lines of original diag. are pinched.

  6. ☆ Example: evaluation of a 3-loop diagram Solution: ( ) Boundary cond. at and fix the right sol. ; , etc. : sol. to homogeneous eq. Using this method recursively, a diagram can be expressed in an iterated (nested) integral form.

  7. Iterated (nested)integrals: In many cases these can be converted to (generalized) harmonic polylogs (HPLs) by appropriate variable transformations. , etc. Many relations hold among HPLs Reduction to a small set of basis HPLs See e.g. hep-ph/0507152, arXiv:1211.5204 [hep-th]

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