Discrepancy in threshold behaviour of scalar three-point function C0[m^2, M^2, s, m^2, 0, M^2] ​ using PaXEvaluate and series expansions

[balmaduch]

Hi everyone,
I'm studying the scalar three-point function
C0[m^2, M^2, s, m^2, 0, M^2]
evaluated at different values of s, both at the threshold s = (M + m)^2 and for generic values with s >= (M + m)^2.

Evaluating it at the threshold using PaXEvaluate and PaXSeries yields a finite result apart from the expected IR divergence. Code used:

Assuming[m > 0 && M > m && s >= (m + M)^2, 
 PaXEvaluate[
   C0[m^2, M^2, s, m^2, 0, M^2], 
   PaXAnalytic -> True, 
   PaXImplicitPrefactor -> (2 π)^(+2 Epsilon), 
   PaXSeries -> {{s, (M + m)^2, 0}}, 
   PaXC0Expand -> True
 ] /. {1/Epsilon -> 1/Epsilon + EulerGamma - Log[4 π]}
]

However, starting from the generic expression for $s \geq (M + m)^2$ and taking the limit $s \to (M + m)^2$ by introducing a small positive parameter Ev, produces a different result that includes an extra term diverging at threshold:

Assuming[m > 0 && M > m && s >= (m + M)^2 && Ev > 0, 
 PaXEvaluate[
   C0[m^2, M^2, s, m^2, 0, M^2] /. s -> (M + m)^2 + Ev^2, 
   PaXAnalytic -> True, 
   PaXImplicitPrefactor -> (2 π)^(+2 Epsilon), 
   PaXC0Expand -> True
 ] /. {1/Epsilon -> 1/Epsilon + EulerGamma - Log[4 π]}
 // Series[#, {Ev, 0, 0}] &// Normal // FullSimplify]

This additional divergent term does not appear when evaluating directly at the threshold.

I would appreciate any insight on why these two approaches yield different results. Could this discrepancy be due to how the series expansion or limit is handled, a typo, or am I missing something fundamental?

Thanks in advance!

[vsht]

Hi,

since Package-X is not being developed anymore, I doubt that its easy to answer the question what the code is doing here.

In my (admittedly limited) understanding, when the naive Taylor expansion is not possible due to an obvious singularity,
Package-X will try to construct an alternative ("analytic" in the package terminology) expansion, which, for the cases I needed myself, was identical to extracting the hard region in the asymptotic expansion sense. However, the results may differ depending on the number of scales and the type of singularities involved.

In your case the integral obviously has a threshold singularity since the general result for C0[m^2, M^2, s, m^2, 0, M^2], taken
from https://qcdloop.fnal.gov/tridiv6.pdf blows up when you insert s -> (M + m)^2

I don't think that it is particularly surprising, given that the most generic result for C0 from the literature is not applicable to a lot of singular cases that need to be treated separately.