This appendix gives the details of the calculation of integration of UVDP parameters for the vertex correction insertion diagram (*b*).

Note that the integration of *k* _{2} naturally separate \(\mathcal{M}^{(b)}\) into four pieces with different powers of factor \(\frac{1}{u+\frac{1}{(1+v_{1})(1+v_{2})}}\), each of which we denote as \(\mathcal{M}^{(b)}_{i}\), *i*=0,1,2,3. In each piece, there are different degree of divergences, quadratically or logarithmically divergent and regular. In the following, we will use a second subscript 2,0,*R* to represent these parts. Let us compute these parts one by one.

Firstly,

\(\mathcal{M}^{(b)}_{0}\) contains only the single quadratically divergent term

\(\mathcal{M}^{(b)}_{02}\), whose result can be given directly as

where, as noted before, we have set

\(\mu_{s}^{2}\to0\) in the end since it can be regarded as just the IR regularization in the LORE and does not affect the UV structure of Feynman integrals. In the following, we will always apply this method to simplify our results.

Also,

\(\mathcal{M}^{(b)}_{1}\) merely includes one term,

\(\mathcal{M}^{(b)}_{12}\), but the calculation is a little involved since it can be divided further to several pieces

\(\mathcal{M}^{(b)}_{122}\),

\(\mathcal{M}^{(b)}_{120}\), and

\(\mathcal{M}^{(b)}_{12R}\) according to the degree of divergences each of them contain:

Each term above can be calculated straightforwardly:

Since it is easy to prove that

\(\mathcal{M}^{(b)}_{12R}\) is finite and thus irrelevant to our present discussion of UV divergence structure, we omit its calculation here and simply write down the result for

\(\mathcal{M}^{(b)}_{12} \) as:

where ∼ means that they are equal up to the divergence part.

By observation,

\(\mathcal{M}^{(b)}_{2}\) contains quadratically and logarithmically divergent parts,

\(\mathcal{M}^{(b)}_{22}\) and

\(\mathcal{M}^{(b)}_{20}\). The computation of

\(\mathcal{M}^{(b)}_{22}\) is much like that of

\(\mathcal{M}^{(b)}_{12}\) and the result is shown below:

We can prove that the term

\(\mathcal{M}^{(b)}_{22R}\) is finite. So we find that the divergence behavior of the integration

\(\mathcal{M}^{(b)}_{22}\) is

$$ \mathcal{M}^{(b)}_{22}\sim \frac{8ie^4\varGamma(4)}{(16\pi^2)^2}g^{\mu\nu} \biggl[\frac{1}{6}M_c^2+\frac{19}{36\cdot6}p^2\biggl(\ln\frac{M_c^2}{-p^2}- \gamma_\omega\biggr)\biggr]. $$

(C9)

\(\mathcal{M}^{(b)}_{20}\) contains many different Lorentz structures, so it is useful to further refine it into three parts according to their Lorentz structure:

\(\mathcal{M}^{(b)}_{201}\) represents terms proportional to

*p* ^{ μ } *p* ^{ ν },

\(\mathcal{M}^{(b)}_{202}\) proportional to

*g* ^{ μν } *p* ^{2} and

\(\mathcal{M}^{b}_{203}\) proportional to

\(g^{\mu\nu}\mu_{u}^{2}\).

where the last subscript of

\(\mathcal{M}^{(b)}_{2010(R)}\) represents the divergence degree of the part.

As before, we can prove that the part

\(\mathcal{M}^{(b)}_{201R}\) is finite, so the divergent part for

\(\mathcal{M}^{(b)}_{201}\) is

By the very similar way, we can obtain the result of

\(\mathcal{M}^{(b)}_{202} \).

By integrating the expression before

\(\frac{1}{16\pi^{2}}(\ln\frac{M_{c}^{2}}{-p^{2}}-\gamma_{\omega})\), we obtain

\(\mathcal{M}^{(b)}_{2020}\) The part of

\(\mathcal{M}^{(b)}_{202R}\) can also be proved to be finite, so the divergence part of

\(\mathcal{M}^{(b)}_{202}\) is:

Much like

\(\mathcal{M}^{(b)}_{201}\) and

\(\mathcal{M}^{(b)}_{202}\) in structure, it is expected that the

\(\mathcal{M}^{(b)}_{203}\) can be calculated similarly. However, a very important new feature will appear: the overlapping divergence structure hidden in our current LORE procedure.

where

\(\mathcal{M}^{(b)}_{2030}\) represents the part proportional to

\((\ln \frac{M_{c}^{2}}{-p^{2}}-\gamma_{\omega})\) while

\(\mathcal{M}^{(b)}_{203R}\) for the rest regular terms. The integration of

\(\mathcal{M}^{(b)}_{2030}\) is straightforward.

Note that the above parameter integration before

\((\ln\frac{M_{c}^{2}}{-p^{2}}-\gamma_{\omega})\) is also divergent, which is the signal of overlapping divergences. It is clearer when we use the analogy between the Feynman diagrams and the electric circuits, which tells us that the integration of

*v* _{1} and

*v* _{2} should reproduce the divergences coming from the subdiagrams of left and right vertex corrections in Fig.

4 since

\(\frac{1}{1+v_{1}}\) (

\(\frac{1}{1+v_{2}}\)) times the effective propagator

\(k_{1}^{2}+x_{1}(1-x_{1})p^{2}\)([(

*k* _{1}+

*k* _{2}+(

*x* _{2}−

*x* _{1})

*p*)]

^{2}+

*x* _{2}(1−

*x* _{2})

*p* ^{2}) for the left (right) circle in Eq. (

26). When

*v* _{1}(

*v* _{2}) tends to infinity, the left effective propagator

\(\frac{1}{1+v_{1}}[k_{1}^{2}+x_{1}(1-x_{1})p^{2}] \) (the right counterpart

\(\frac{1}{1+v_{2}}\{[(k_{1}+k_{2}+(x_{2}-x_{1})p)]^{2}+x_{2}(1-x_{2})p^{2}\}\)) approaches zero, which means that the left (right) half circle collapse to a point and that sub-circuit is short-cut in the electric circuit language. This singular behavior will become manifest as the divergence in the final integration of UVDP parameter

*v* _{1}(

*v* _{2}), just as the ones shown in Eq. (

C17) above. Thus, the result in Eq. (

C17) can be understood as the subdivergences coming from left and right vertex correction times the overall one

\((\ln\frac{M_{c}^{2}}{-p^{2}}-\gamma_{\omega})\), which is the definition of overlapping divergence [

1,

6].

This overlapping divergence take us some further difficulties that the integration of

\(\mathcal{M}^{(b)}_{203R}\) turn out to be divergent, which gives us further contributions to our UV-divergence structure.

Because of the complication in the logarithmic function, it is difficult to get a closed analytical expression for this kinds of integration. However, if we only focus the divergence behavior of the integral, then we can use the method introduced in the calculation of

\(\mathcal{M}^{(a_{1})}_{2R}\) to greatly simplify the integral and to obtain the asymptotical results. To disentangle the overlapping divergences, we need first to know in what parameter space region the divergence happen. From our experience when working

\(\mathcal{M}^{(b)}_{2030}\), the divergences take place when

*v* _{1}→∞ and

*v* _{2}→∞.

Now we consider the region where

*v* _{1}→∞ and

*v* _{2}→0. Fist we choose a very large number, say

*V*, and set the integration region is only confined in

*v* _{1}≫

*V*. In such a region,

\(\frac{1}{1+v_{1}}\) are small quantities, so we can expand the expression according to

\(\frac{1}{1+v_{1}}\). The leading term of

\(\mathcal{M}^{(b)}_{203R}\) is

where in the last line we extended the lower bound of the integration range to 0 as before for convenience.

Since in the other asymptotic region

*v* _{2}→∞,

*v* _{1}→0, we can obtain a similar expression except for the exchange of

*v* _{1}↔

*v* _{2} and

*x* _{2}↔

*x* _{1}. Thus, we can expect to get the same asymptotic result. Therefore, the divergence behavior of

\(\mathcal{M}^{(b)}_{203R}\) is

\(\mathcal{M}^{(b)}_{3}\) can be naturally divided into three parts

\(\mathcal{M}^{(b)}_{32}\),

\(\mathcal{M}^{(b)}_{30}\) and

\(\mathcal{M}^{(b)}_{3R}\) according to the divergence degree followed by integration of loop momentum

*k* _{2}.

\(\mathcal{M}^{(b)}_{32}\) contains only one term, so the calculation is straightforward:

Since we can prove that

\(\mathcal{M}^{(b)}_{32R}\) is finite, the divergence structure we are concerned with now is given by the addition of

\(\mathcal{M}^{(b)}_{322}\) and

\(\mathcal{M}^{(b)}_{320}\):

Like

\(\mathcal{M}^{(b)}_{20}\),

\(\mathcal{M}^{(b)}_{30}\) contains three parts differentiating by their Lorentz structures:

\(\mathcal{M}^{(b)}_{301}\) represents the part proportional to

*p* ^{ μ } *p* ^{ ν },

\(\mathcal{M}^{(b)}_{301}\) for

*g* ^{ μν } *p* ^{2}, and

\(\mathcal{M}^{(b)}_{303}\) for

\(g^{\mu\nu}\mu_{u}^{2}\). Let us first calculate

\(\mathcal{M}^{(b)}_{301}\).

Actual calculation of

\(\mathcal{M}^{(b)}_{3010}\) shows that the integral before

\((\ln\frac{M_{c}^{2}}{p^{2}}-\gamma_{\omega})\) is logarithmically divergent. So like the part

\(\mathcal{M}^{(b)}_{2030}\),

\(\mathcal{M}^{(b)}_{3010}\) also involves overlapping divergences.

For the more challenging part

\(\mathcal{M}^{(b)}_{301R}\), we follow the approach already used in the derivation of

\(\mathcal{M}^{(b)}_{203R}\) in order to obtain only the asymptotic results. We first focus on the region

*v* _{1}→∞, while

*v* _{2}→0, the expression for

\(\mathcal{M}^{(b)}_{301R}\) can be simplified to

where the integration over

*v* _{1} is confined in the region from

*V*≫1 to infinity. It is easy to see that the simplified integral is essentially the same as that for

\(\mathcal{M}^{(b)}_{203R}\), (

C19), so we can straightforwardly write down the result for

\(\mathcal{M}^{(b)}_{301R}\) where we have already doubled the result for

\(\mathcal{M}^{(b)}_{301Rv_{1}}\) due to another asymptotic limit region

*v* _{2}→∞,

*v* _{1}→0, which gives exactly the same result.

A very similar calculation as

\(\mathcal{M}^{(b)}_{301}\) can give us the results for

\(\mathcal{M}^{(b)}_{302}\),

Note that the coefficient before

\((\ln\frac{M_{c}^{2}}{-p^{2}}-\gamma_{\omega})\) is divergent, which means that

\(\mathcal{M}^{(b)}_{302}\) also contains overlapping divergences. In order to deal with the resultant divergent integral

\(\mathcal{M}^{(b)}_{302R}\), following the procedure used by

\(\mathcal{M}^{(b)}_{203R}\) and

\(\mathcal{M}^{(b)}_{301R}\), we need first to simplify the integral in the region

*v* _{1}→∞,

*v* _{2}→0

In order to obtain

\(\mathcal{M}^{(b)}_{302R}\), the above result needs to be doubled to take into account another contribution from the region

*v* _{2}→∞,

*v* _{1}→0, which is given below:

For the accomplishment of the computation of

\(\mathcal{M}^{(b)}_{30}\), we have to calculate

\(\mathcal{M}^{(b)}_{303}\):

where

\(\mathcal{M}^{(b)}_{3030}\) represents the part proportional to the logarithmic divergence coming from

\(I^{R}_{0}\) while

\(\mathcal{M}^{(b)}_{303R}\) the rest parts. It is a direct exercise to obtain

\(\mathcal{M}^{(b)}_{3030}\) and the result is

$$ \everymath{\displaystyle }\begin{array}[b]{@{}l} \mathcal{M}^{(b)}_{3030} = -\frac{8ie^4\varGamma(4)}{(16\pi^2)^2}\cdot \frac{25}{18\cdot36}g^{\mu\nu}p^2 \biggl(\ln\frac{M_c^2}{-p^2}-\gamma_\omega\biggr), \\ \noalign {\vspace {-8pt}} \end{array} $$

(C34)

while part

\(\mathcal{M}^{(b)}_{303R}\) can be proven finite. So the divergence structure of

\(\mathcal{M}^{(b)}_{303}\) is given by

$$ \everymath{\displaystyle }\begin{array}[b]{@{}l} \mathcal{M}^{(b)}_{303} \sim -\frac{8ie^4\varGamma(4)}{(16\pi^2)^2} \cdot \frac{25}{18\cdot36}g^{\mu\nu}p^2 \biggl(\ln\frac{M_c^2}{-p^2}-\gamma_\omega\biggr). \\ \noalign {\vspace {-8pt}} \end{array} $$

(C35)

For the part

\(\mathcal{M}^{(b)}_{3R}\), we can prove that most terms are finite for the integration of UVDP or Feynman parameters, except for the following one:

By putting the divergence parts of the terms Eqs. (

C1), (

C5), (

C9), (

C12), (

C15), (

C17), (

C20), (

C24), (

C26), (

C28), (

C30), (

C32), (

C35), (

C36) together, we finally arrive at the divergence behavior of the diagram

\(\mathcal{M}^{(b)}\).