1. Y. Sumino (Tohoku Univ.) 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. 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).

4. ☆ Example: evaluation of a 3-loop diagram
𝑚=0
𝑚=1
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 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.

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/ ,
arXiv: [hep-th]