Consistency of loop regularization method and divergence structure of QFTs Beyond one-loop order

Abstract

We study the problem how to deal with tensor-type two-loop integrals in the Loop Regularization (LORE) scheme. We use the two-loop photon vacuum polarization in the massless Quantum Electrodynamics (QED) as the example to present the general procedure. In the processes, we find a new divergence structure: the regulated result for each two-loop diagram contains a gauge-violating quadratic harmful divergent term even combined with their corresponding counterterm insertion diagrams. Only when we sum up over all the relevant diagrams do these quadratic harmful divergences cancel, recovering the gauge invariance and locality.

Appendices

Appendix A: Useful formula for the regularization ILIs in LORE

In the following, we will list some of the most useful regularized ILIs for reference:

(A1)

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

with

(A13)

where

(A14)

(A15)

(A16)

(A17)

(A18)

The explicit form of y _−2(α−2)(x) is already given by Eq. (9).

Appendix B: Details of calculation of self-energy insertion diagram

In this appendix, we shall present the details of the calculation of the UVDP parameter integrals for the self-energy correction insertion diagram (a ₁) shown in Fig. 2.

With the simplification by setting $\mu_{s}^{2}\rightarrow0$ and $y_{i}(\frac{\mu^{2}}{M_{c}^{2}}) \rightarrow0$ in Eq. (21), we have the following expression for $\mathcal{M}^{(a_{1})}_{2}$

(B1)

Here $\mathcal{M}^{(a_{1})}_{22}$, $\mathcal{M}^{(a_{1})}_{20}$, and $\mathcal{M}^{(a_{1})}_{2R}$ represent quadratical divergent, logarithmical divergent, and regular part in the $\mathcal{M}^{(a_{1})}_{2}$

(B2)

where in the third line we use the trick that $q_{u}^{2}\equiv-q_{o}^{2} u$, which can effectively transform a UVDP parameter integral into a 1-folded irreducible loop integral (ILI), which is the fundamental object to be regulated in the framework of LORE. In the following, we will frequently use this trick without any explanation.

(B3)

The integral

(B4)

is difficult since the complicated form in the logarithm. However, if we only focus on its divergence behavior, we can make a lot of simplification. As discussed in Sect. 2 in our previous paper [1], the analogy between circuit diagram and Fig. 2 tells us that the divergence appear only in the region v ₁→∞. Therefore, we can set v ₁ in the region v ₁>V, which V is a very large number V≫1. In this asymptotic region, we have $\frac{v_{1}}{1+v_{1}}\rightarrow1$ and the above expression simplifies to

(B5)

where $\mathcal{M}^{(a_{1})}_{2Ru}$, $\mathcal{M}^{(a_{1})}_{2Rx}$, and $\mathcal{M}^{(a_{1})}_{2Rv_{1}}$ represent the three parts in the second line, respectively, the results of which are given as follows:

(B6)

(B7)

(B8)

Since we are only interested in the divergence behavior, the last finite term can be neglected due to the complexity of its calculation. Below we only give the regulated result for the first term with LORE method and leave the details of derivation to Appendix D.

(B9)

Putting Eqs. (B6), (B7), and (B9) together, we arrive at the result for $\mathcal{M}^{(a_{1})}_{2R}$

(B10)

With the experience of the calculation of $\mathcal{M}^{(a_{1})}_{2}$, the computation of logarithmic part $\mathcal{M}^{(a_{1})}_{0}$ is straightforward:

(B11)

From the definition of $\mathcal{M}^{(a_{1})}_{00(R)}$, it is easy to see that they are essentially the same as counterparts in the quadratic part $\mathcal{M}^{(a_{1})}_{20(R)}$. Thus, we can write down the final expression for $\mathcal{M}^{(a_{1})}_{00(R)}$ immediately following $\mathcal{M}^{(a_{1})}_{20(R)}$.

(B12)

(B13)

By summing Eqs. (B2), (B3), (B10), (B12), (B13), we finally obtain our result for diagram (a ₁)

(B14)

Appendix C: Details of calculation of vertex correction insertion diagram

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

(C1)

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:

(C2)

Each term above can be calculated straightforwardly:

(C3)

(C4)

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:

(C5)

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:

(C6)

(C7)

(C8)

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 $g^{\mu\nu}\mu_{u}^{2}$.

(C10)

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

(C11)

As before, we can prove that the part $\mathcal{M}^{(b)}_{201R}$ is finite, so the divergent part for $\mathcal{M}^{(b)}_{201}$ is

(C12)

By the very similar way, we can obtain the result of $\mathcal{M}^{(b)}_{202}$.

(C13)

By integrating the expression before $\frac{1}{16\pi^{2}}(\ln\frac{M_{c}^{2}}{-p^{2}}-\gamma_{\omega})$, we obtain $\mathcal{M}^{(b)}_{2020}$

(C14)

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

(C15)

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.

(C16)

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.

(C17)

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.

(C18)

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

(C19)

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

(C20)

$\mathcal{M}^{(b)}_{3}$ 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:

(C21)

(C22)

(C23)

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

(C24)

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

(C25)

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.

(C26)

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

(C27)

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 $\mathcal{M}^{(b)}_{301R}$

(C28)

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

(C29)

(C30)

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

(C31)

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:

(C32)

For the accomplishment of the computation of $\mathcal{M}^{(b)}_{30}$, we have to calculate $\mathcal{M}^{(b)}_{303}$:

(C33)

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:

(C36)

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

(C37)

Appendix D: Derivation of the integration $\protect\int^{\infty}_{0} \frac{d v_{2}}{1+v_{2}}\ln(1+v_{2})$

Recall that in the above derivation of the two-loop massless QED photon vacuum polarization diagrams we encountered a new parameter integral,

(D1)

which does not appear at one-loop level. Thus, for completion of our calculation, in this section we shall derive its regulated result with the LORE method.

With the same philosophy as before, we would like to transform this UVDP parameter integral to a momentum-like one. In order to do so, we just need to multiply a free energy scale $-q_{o}^{2}$ in the numerator and denominator simultaneously,

(D2)

where we have defined $q^{2}=-q_{o}^{2} v_{2}$ and separated the integral into two parts I ₁ and I ₂. The integration I ₂ can be easily worked out with the LORE method since it is exactly the one encountered in the one-loop calculations. The result of I ₂ is

(D3)

where we define $\mu_{q}^{2}\equiv\mu_{s}^{2}-q_{o}^{2}$.

The integration I ₁ is really new and requires us to calculate more carefully. In the following, we shall give the detailed derivation of this integration in the LORE method. As usually done in the LORE method, we need to apply a series of regulators to the integral:

(D4)

where we defined $\hat{\mathcal{M}}^{2}_{l}=-q_{o}^{2}+\mu_{s}^{2}+l M_{R}^{2}=\mu_{q}^{2}+ l M_{R}^{2}$. With the help of the following equality:

(D5)

we can easily work out the above integration:

(D6)

where we have introduced the two series:

$$ M_c^2 (N) \equiv \sum^N_{l=1}c^N_l (l\ln l)M_R^2, \quad \mbox{and}\quad \gamma_\omega(N) \equiv \sum^N_{l=1} c^N_l \ln l + \ln\Biggl[\sum^N_{l^\prime}c^N_{l^\prime} \bigl(l^\prime \ln l^\prime\bigr)\Biggr], $$

(D7)

both of which in the limit of N→∞ can be shown [3] to have finite limits

(D8)

We also define a new function

(D9)

which can be seen that when the UV scale $M_{c}^{2}$ tends to infinity the function $y_{0}^{\prime}(\frac{\mu_{q}^{2}}{M_{c}^{2}})$ vanishes since the expansion of $\ln(1+\frac{\mu_{q}^{2}}{l M_{R}^{2}})$ is of the order of $O(\frac{\mu_{q}^{2}}{M_{R}^{2}})$. The constant α _ω is defined as

$$ \alpha_\omega \equiv \lim_{N\to\infty}\Biggl[\sum^N_{l=1} c_l^N (\ln l)^2 +\Biggl(\sum^N_{l=1}c_l^N \ln l\Biggr)^2\Biggr]. $$

(D10)

It can be shown from our numerical calculation that α _ω is finite and is α _ω =1.62931….

By combining the expression I ₁ and I ₂, we can give our final regularized result

(D11)

When we set the IR scale $\mu_{s}^{2}$ to 0 which can be viewed as the IR cutoff in the LORE method, then the regulated integral can be simplified to

(D12)

If we further take the limit M _c →∞, the last term $y_{0}^{\prime}(\frac{-q_{o}^{2}}{M_{c}^{2}})\to0$ also, which gives the result we use in our previous calculations.