Corrections to the material

[mmikhasenko]

• slide 1: so cool figure :-)

• slide 4: this is all first sheet, no second shown here.

• The expression for rho does not have discontinuity on right.

• slide 5: little * next to green stars are not explained

• slide 6: Im CM = rho, other way around.
  Violation of analyticity: mention pseudothreshold, and s=0
  idea -> solution.

[mmikhasenko]

the 3d plots, it might be helpful to indicate where one can pass, where not

e.g. by the color of the line

[mmikhasenko]

as a minor improvement of the plot

you can color the line segment of the same color that has to be equal to each other.
+i0 solid
-i0 dashed

[Zeyna777]

Hi Misha,
thank you for the feedback! Will update it asap. I have some questions (just for clarfication):

Question 1:
„The expression for rho does not have discontinuity on right“

I know that the branch cut of a complex squareroot goes to the left (from bp to -inf.). I just thought the most common choice in particle physics would be to the right. But technically you could rotate it in an abitrary direction in the complex plane.

Question 2 (dumb I know but just out of curiosity):

“Im CM = rho, other way around“

How this changes the logical statment? If CM=-i rho then i Cm= rho. Maybe I am thinking the wrong way.

[mmikhasenko]

Q1: the mathematical sqrt sign is well defined using Arg function in the range -pi, pi. I'm not aware of any programming language, implementation that would mean anything different for sqrt.

See the principal sqrt in
https://en.m.wikipedia.org/wiki/Square_root

The principal square root function is thus defined using the non-positive real axis as a branch cut.

Q2: neither is true:

CM != irho,
rho != -iCM

However, Im CM = rho, and importantly only above the threshold.

The functions differ by their real parts everywhere except a few points, and by their imaginary part below the threshold.