Y. Sumino (Tohoku Univ.) Evaluation of Master Integrals:

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Presentation transcript:

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

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 .

Master integrals can be chosen finite as 𝐷→4 (𝜀→0) . can be reduced to a combination of master integrals Master integrals can be chosen finite as 𝐷→4 (𝜀→0) . 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).

☆ Example: evaluation of a 3-loop diagram 𝑚=0 𝑚=1 Some of the lines of original diag. are pinched.

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

( ) ☆ Example: evaluation of a 3-loop diagram Solution: ( ) Boundary cond. at 𝑧→0 and 𝑧→∞ fix the right sol. ; , etc. : sol. to homogeneous eq. Using this method recursively, a diagram can be expressed in an iterated (nested) integral form.

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]