Any other integral in this thesis can be obtained with the computational
elements illustrated here.

###  B.3 Integration in the loop momentum

After Feynman parametrization, all integrals assume the form of the rational
functions

$\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{\left(1,k_{\mu},k_{\mu\nu},k_{\mu\nu\rho},...\right)}{\left(k^{2}+Q\right)^{\alpha}}.$
(B.39)

To solve the integral, we start with the form

$I\left(k,M,n\right)=\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{1}{\left(k^{2}-M^{2}\right)^{\alpha}}=\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{1}{\left(k^{2}-2k\cdot
q+Q\right)^{\alpha}},$ (B.40)

where $2\alpha>n$ and $M^{2}=q^{2}-Q.$ The auxiliary variable $q$ helps to
develop the tensor integrals. The integration measure
$\mathrm{d}^{n}k=\mathrm{d}^{n-1}k\mathrm{d}k_{0}$. The square the momentum
loop $k^{2}=k_{0}^{2}-\mathbf{k}^{2};$ and
$\mathbf{k}^{2}=\sum_{i=1}^{n-1}k_{i}^{2}.$ The integral (B.40) only

$\displaystyle I\left(Q,n\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{1}{\left(k^{2}-M^{2}\right)^{\alpha}}=\int\frac{\mathrm{d}^{n-1}k}{\left(2\pi\right)^{n}}\left[\int_{-\infty}^{+\infty}\mathrm{d}k_{0}f\left(k_{0}\right)\right]$
(B.41) $\displaystyle f\left(k_{0}\right)$ $\displaystyle=$
$\displaystyle\left[k_{0}^{2}-(\sqrt{\mathbf{k}^{2}+M^{2}}-i\varepsilon)^{2}\right]^{-\alpha},$
(B.42) $\displaystyle f\left(k_{0}\right)$ $\displaystyle\sim$
$\displaystyle\frac{1}{k_{0}^{2\alpha}},\text{ as
}k_{0}^{2}\rightarrow\infty.$ (B.43)

The poles and prescription coming from Feynman propagators

$\displaystyle k_{0}^{2}$ $\displaystyle=$
$\displaystyle\sqrt{\mathbf{k}^{2}+M^{2}}-i\varepsilon$ (B.44) $\displaystyle
k_{0}^{2}$ $\displaystyle=$
$\displaystyle-\sqrt{\mathbf{k}^{2}+M^{2}}+i\varepsilon.$ (B.45)

To compute the integral, we extend the integration for $k_{0}\in\mathbb{C}$
and consider the following contour $C=C_{1}+C_{2}+C_{3}+C_{4}$ in the figure
below

Figure B.1: Contour of integration

Then take the integral over that contour

$F_{C}(\mathbf{k}^{2},M^{2})=\int_{C}\mathrm{d}k_{0}f\left(k_{0}\right)=\left(\int_{C_{1}}+\int_{C_{2}}+\int_{C_{3}}+\int_{C_{4}}\right)\mathrm{d}k_{0}f\left(k_{0}\right)=0$
(B.46)

since there are no poles inside the closed path of integration. We write the
integral as $F_{C}=F_{1}+F_{2}+F_{3}+F_{4}$; the semi-circle contributions
vanish in the limit
$\lim_{R\rightarrow\infty}\left(F_{C_{3}}+F_{C_{4}}\right)=0.$ The reminder
contribution gives the desired relation

$\lim_{R\rightarrow\infty}F_{C_{1}}=-\lim_{R\rightarrow\infty}F_{C_{2}}\rightarrow\int_{-\infty}^{\infty}\mathrm{d}k_{0}f\left(k_{0}\right)=-\int_{+i\infty}^{-i\infty}\mathrm{d}k_{0}f\left(k_{0}\right).$
(B.47)

Changing the integration variable in the last integral over the imaginary axis
by adopting $k_{0}=ik_{0}^{\prime},$ we may write

$I\left(Q,n\right)=\int\frac{\mathrm{d}^{n-1}k}{\left(2\pi\right)^{n-1}}\int_{-\infty}^{\infty}\mathrm{d}k_{0}\frac{1}{\left(k_{0}^{2}-\mathbf{k}^{2}-M^{2}\right)^{\alpha}}=\int\frac{\mathrm{d}^{n-1}k}{\left(2\pi\right)^{n-1}}\int_{-\infty}^{\infty}\mathrm{d}k_{0}^{\prime}\frac{i}{\left(-k_{0}^{\prime
2}-\mathbf{k}^{2}-M^{2}\right)^{\alpha}}$ (B.48)

and effectively we have an euclidean signature $(k^{\prime 2}:=k_{0}^{\prime
2}+\sum_{i=1}^{n-1}k_{i}^{2})$ to perform the integral

$I\left(Q,n\right)=i\left(-1\right)^{\alpha}\int\frac{\mathrm{d}^{n}k^{\prime}}{\left(2\pi\right)^{n}}\frac{1}{\left(k^{\prime
2}+M^{2}\right)^{\alpha}}.$ (B.49)

Now we introduce spherical coordinates to these variables and split the radius
and solid angle integrations

$I\left(Q,n\right)=\frac{i\left(-1\right)^{\alpha}}{\left(2\pi\right)^{n}}\int_{S^{n-1}}\mathrm{d}\Omega\int_{0}^{\infty}\mathrm{d}rr^{n-1}\frac{1}{\left(r^{2}+M^{2}\right)^{\alpha}}.$
(B.50)

The solid angle furnish

$\frac{1}{\left(2\pi\right)^{n}}\int_{S^{n-1}}\mathrm{d}\Omega=\frac{2}{\left(4\pi\right)^{n/2}\Gamma\left(\frac{n}{2}\right)}$
(B.51)

and simple manipulations bring the form

$I\left(Q,n\right)=\frac{2i\left(-1\right)^{\alpha}}{\left(4\pi\right)^{n/2}\Gamma\left(\frac{n}{2}\right)}\frac{1}{2M^{2\left(\alpha-n/2\right)}}\int_{0}^{\infty}\mathrm{d}\left(r^{\prime
2}\right)\left(r^{\prime 2}\right)^{\left(n-2\right)/2}\left(r^{\prime
2}+1\right)^{-\alpha}.$ (B.52)

Another variables change $r^{\prime
2}=\left(1-y\right)/y\rightarrow\mathrm{d}\left(r^{\prime
2}\right)=-\mathrm{d}y1/y^{2},$ the Beta function is

$B\left(\alpha-\frac{n}{2},\frac{n}{2}\right)=\int_{0}^{1}\mathrm{d}yy^{\left(\alpha-n/2\right)-1}\left(1-y\right)^{n/2-1}.$

We have

$\displaystyle I\left(Q,n\right)$ $\displaystyle=$
$\displaystyle\frac{i\left(-1\right)^{\alpha}}{\left(4\pi\right)^{n/2}\Gamma\left(\frac{n}{2}\right)M^{2\left(\alpha-n/2\right)}}\int_{0}^{1}\mathrm{d}yy^{\left(\alpha-n/2\right)-1}\left(1-y\right)^{n/2-1}$
(B.53) $\displaystyle=$
$\displaystyle\frac{i\left(-1\right)^{\alpha}}{\left(4\pi\right)^{n/2}}\frac{\Gamma\left(\alpha-\frac{n}{2}\right)}{\Gamma\left(\alpha\right)M^{2\left(\alpha-n/2\right)}},$
(B.54)

thus, from $M^{2}=q^{2}-Q$ follows

$I\left(Q,n\right)=\frac{i\left(-1\right)^{\alpha}}{\left(4\pi\right)^{n/2}}\frac{\Gamma\left(\alpha-\frac{n}{2}\right)}{\Gamma\left(\alpha\right)\left(q^{2}-Q\right)^{\alpha-n/2}}.$
(B.55)

Now taking derivatives concerning the variable q on both sides and shifting
the parameters $\alpha\rightarrow\alpha-1$ in the form

$I\left(Q,n\right)=\frac{i}{\left(-4\pi\right)^{n/2}}\frac{\Gamma\left(\alpha-n/2\right)}{\Gamma\left(\alpha\right)\left(Q-q^{2}\right)^{\left(\alpha-n/2\right)}},$
(B.56)

the explicit derivative is

$\frac{\partial I}{\partial
q^{\mu_{1}}}=\frac{i}{\left(-4\pi\right)^{n/2}}\frac{2q_{\mu_{1}}\Gamma\left(\alpha-n/2+1\right)}{\Gamma\left(\alpha\right)\left(Q-q^{2}\right)^{\alpha+1-n/2}}=\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{2\alpha
k_{\mu_{1}}}{\left(k^{2}-2k\cdot q+Q\right)^{\alpha+1}},$ (B.57)

follows the relation

$\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{k_{\mu_{1}}}{\left(k^{2}-2k\cdot
q+Q\right)^{\alpha}}=\frac{i}{\left(-4\pi\right)^{n/2}}q_{\mu_{1}}\frac{\Gamma\left(\alpha-n/2\right)}{\Gamma\left(\alpha\right)\left(Q-q^{2}\right)^{\alpha-n/2}}.$
(B.58)

Recursively

$\int\frac{\mathrm{d}^{n}k}{\left(2\pi\right)^{n}}\frac{k_{\mu_{2}}k_{\mu_{1}}}{\left(k^{2}+Q\right)^{\alpha}}=\frac{i}{\left(-4\pi\right)^{n/2}}\frac{1}{2}g_{\mu_{12}}\frac{\Gamma\left(\alpha-n/2-1\right)}{\Gamma\left(\alpha\right)Q^{\alpha-n/2-1}}.$
(B.59)

From the formulae presented, it is possible to obtain a general result,
adopting $n=2\omega$, which reads

$\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}\frac{k_{\mu_{1}}\cdots
k_{\mu_{2l+1}}}{\left(k^{2}+Q\right)^{\alpha}}=0$ (B.60)
$\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}\frac{k_{\mu_{1}}\cdots
k_{\mu_{2l}}}{\left(k^{2}+Q\right)^{\alpha}}=\frac{i}{\left(4\pi\right)^{\omega}}\frac{1}{2^{l}}g_{(\mu_{1}\mu_{2}}\cdots
g_{\mu_{2l-1},\mu_{2l})}\frac{\Gamma\left(\alpha-\omega-l\right)}{\Gamma\left(\alpha\right)Q^{\alpha-\omega-l}}$
(B.61)

It is interesting to note that these results imply in the properties:

$\displaystyle\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}f\left(k^{2}\right)k_{\mu}$
$\displaystyle=$ $\displaystyle 0$ (B.62)
$\displaystyle\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}k_{\mu}k_{\nu}f\left(k^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{g_{\mu\nu}}{2\omega}\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}k^{2}f\left(k^{2}\right)$
(B.63)
$\displaystyle\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}k_{\mu\nu\alpha\beta}f\left(k^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{\left(g_{\mu\nu\alpha\beta}+g_{\mu\alpha\nu\beta}+g_{\mu\beta\nu\alpha}\right)}{4\left(\omega+1\right)}\int\frac{\mathrm{d}^{2\omega}k}{\left(2\pi\right)^{2\omega}}k^{4}f\left(k^{2}\right).$
(B.64)

##  Appendix C The One point Integrals in Two Dimensions

After performing the Dirac traces present in the definitions we established
for the perturbative amplitudes in (7.33) and (7.34), their integrals
naturally decompose in Feynman integrals that we define in the equations
(3.52) and (3.53). The calculations follow the IReg method by applying the
separation identity (3.4) on the divergent integrals. The finite part is
integrated and projected in definitions (3.27) and (3.28). The residual
divergent part is projected onto divergent objects of the set, expressed in
(3.5) and their relations in the session (3.1).

We start with integrals that have only one propagator. These have only
divergent structures. The finite parts after separating the labels, cancel out
when they are integrated.

Integral $J_{1}:$ by power-counting this integral has a superficial degree of
divergence is logarithmic

$\bar{J}_{1}\left(k_{i}\right)=I_{\log}$ (C.1)

From the next integral, it is necessary to specify the $k_{1}$ and $k_{2}$
labels of the integral.

Integral $J_{1\mu_{1}}:$ superficial degree of divergence is linear

$\displaystyle 2\bar{J}_{1\mu_{1}}\left(k_{1}\right)$ $\displaystyle=$
$\displaystyle-\left(P-q\right)^{\nu_{1}}\Delta_{2\mu_{1}\nu_{1}}$ (C.2)
$\displaystyle 2\bar{J}_{1\mu_{1}}\left(k_{2}\right)$ $\displaystyle=$
$\displaystyle-\left(P+q\right)^{\nu_{1}}\Delta_{2\mu_{1}\nu_{1}}$ (C.3)

Integral $J_{1\mu_{12}}:$ superficial degree of divergence is quadratic

$\displaystyle\bar{J}_{1\mu_{12}}\left(k_{1}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\Delta_{1\mu_{12}}+g_{\mu_{12}}I_{\mathrm{quad}}\right)-\frac{1}{8}\left(P-q\right)^{2}\Delta_{2\mu_{12}}$
$\displaystyle+\frac{1}{8}\left(P-q\right)^{\nu_{1}}\left[\left(P-q\right)^{\nu_{2}}W_{3\mu_{12}\nu_{12}}-2\left(P-q\right)_{(\mu_{1}}\Delta_{2\mu_{2})\nu_{1}}\right]$
$\displaystyle\bar{J}_{1\mu_{12}}\left(k_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\Delta_{1\mu_{12}}+g_{\mu_{12}}I_{\mathrm{quad}}\right)-\frac{1}{8}\left(P+q\right)^{2}\Delta_{2\mu_{12}}$
$\displaystyle+\frac{1}{8}\left(P+q\right)^{\nu_{1}}\left[\left(P+q\right)^{\nu_{2}}W_{3\mu_{12}\nu_{12}}-2\left(P+q\right)_{(\mu_{1}}\Delta_{2\mu_{2})\nu_{1}}\right]$

Integral $J_{1\mu_{123}}:$ superficial degree of divergence is cubic and are
the integrals with the highest power-counting

$\displaystyle J_{1\mu_{123}}\left(k_{1}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left(P-q\right)^{\nu_{1}}W_{2\mu_{123}\nu_{1}}+\frac{1}{4}\left(P-q\right)_{(\mu_{1}}\Delta_{1\mu_{23})}$
$\displaystyle-\frac{1}{48}\left(P-q\right)^{\nu_{1}}\left(P-q\right)^{\nu_{2}}\left(P-q\right)^{\nu_{3}}W_{4\mu_{123}\nu_{123}}$
$\displaystyle+\frac{1}{16}\left(P-q\right)^{\nu_{1}}\left(P-q\right)^{\nu_{2}}\left(P-q\right)_{(\mu_{1}}W_{3\mu_{23})\nu_{12}}$
$\displaystyle+\frac{1}{16}\left(P-q\right)^{2}\left[\left(P+p\right)^{\nu_{1}}W_{3\mu_{123}\nu_{1}}-\left(P-q\right)_{(\mu_{1}}\Delta_{2\mu_{23})}\right]$
$\displaystyle-\frac{1}{8}\left(P-q\right)^{\nu_{1}}\left(P-q\right)_{(\mu_{1}}\left(P-q\right)_{\mu_{2}}\Delta_{2\mu_{3})\nu_{1}}$
$\displaystyle J_{1\mu_{123}}\left(k_{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\left(P+q\right)^{\nu_{1}}W_{2\mu_{123}\nu_{1}}+\frac{1}{4}\left(P+q\right)_{(\mu_{1}}\Delta_{1\mu_{23})}$
$\displaystyle-\frac{1}{48}\left(P+q\right)^{\nu_{1}}\left(P+q\right)^{\nu_{2}}\left(P+q\right)^{\nu_{3}}W_{4\mu_{123}\nu_{123}}$
$\displaystyle+\frac{1}{16}\left(P+q\right)^{\nu_{1}}\left(P+q\right)^{\nu_{2}}\left(P+q\right)_{(\mu_{1}}W_{3\mu_{23})\nu_{12}}$
$\displaystyle+\frac{1}{16}\left(P+q\right)^{2}\left[\left(P+q\right)^{\nu_{1}}W_{3\mu_{123}\nu_{1}}-\left(P+q\right)_{(\mu_{1}}\Delta_{2\mu_{23})}\right]$
$\displaystyle-\frac{1}{8}\left(P+q\right)^{\nu_{1}}\left(P+q\right)_{(\mu_{1}}\left(P+q\right)_{\mu_{2}}\Delta_{2\mu_{3})\nu_{1}}.$

For instance, we calculated the $J_{1}^{\mu_{1}\mu_{2}}\left(k_{i}\right)$.
The complete expression:

$\displaystyle J_{1}^{\mu_{1}\mu_{2}}\left(k_{i}\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{k^{\mu_{1}}k^{\mu_{2}}}{D_{i}}$
$\displaystyle+\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left(k_{i}^{\mu_{1}}k^{\mu_{2}}+k_{i}^{\mu_{2}}k^{\mu_{1}}\right)\frac{1}{D_{i}}$
$\displaystyle+k_{i}^{\mu_{1}}k_{i}^{\mu_{2}}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{1}{D_{i}}.$

Using the expansions for two first integral above, we have

$\displaystyle\left[\frac{k^{\mu_{1}}k^{\mu_{2}}}{D_{i}}\right]_{\mathrm{even}}$
$\displaystyle=$
$\displaystyle\frac{k^{\mu_{1}}k^{\mu_{2}}}{D_{\lambda}}-k_{i}^{2}\frac{k^{\mu_{1}}k^{\mu_{2}}}{D_{\lambda}^{2}}+4k_{i\nu_{12}}\frac{k^{\mu_{1}}k^{\mu_{2}}k^{\nu_{1}}k^{\nu_{2}}}{D_{\lambda}^{3}}$
(C.9) $\displaystyle\left[\frac{k^{\mu_{1}}}{D_{i}}\right]_{\mathrm{even}}$
$\displaystyle=$
$\displaystyle-2k_{i\nu_{1}}\frac{k^{\mu_{1}}k^{\nu_{1}}}{D_{\lambda}^{2}}.$
(C.10)

So the expanded integral is given by

$\displaystyle J_{1}^{\mu_{1}\mu_{2}}\left(k_{i}\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left(\frac{k^{\mu_{1}}k^{\mu_{2}}}{D_{\lambda}}-k_{i}^{2}\frac{k^{\mu_{1}}k^{\mu_{2}}}{D_{\lambda}^{2}}+4k_{i\nu_{1}}k_{i\nu_{2}}\frac{k^{\mu_{1}}k^{\mu_{2}}k^{\nu_{1}}k^{\nu_{2}}}{D_{\lambda}^{3}}\right)$
$\displaystyle-2k_{i\nu_{1}}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left(k_{i}^{\mu_{1}}\frac{k^{\mu_{2}}k^{\nu_{1}}}{D_{\lambda}^{2}}+k_{i}^{\mu_{2}}\frac{k^{\mu_{1}}k^{\nu_{1}}}{D_{\lambda}^{2}}\right)$
$\displaystyle+k_{i}^{\mu_{1}}k_{i}^{\mu_{2}}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{1}{D_{i}}.$

Identifying the divergent objects in Section (3.1)

$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{8k^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}}{D_{\lambda}^{3}}$
$\displaystyle=$ $\displaystyle
W_{3}^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}+g^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}I_{\text{{log}}}$
$\displaystyle W_{3}^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}$ $\displaystyle=$
$\displaystyle\square_{3}^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}+\frac{1}{2}g^{(\mu_{1}\nu_{1}}\Delta_{2}^{\mu_{2}\nu_{2})}$
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{2k^{\mu_{1}\mu_{2}}}{D_{\lambda}^{2}}$
$\displaystyle=$
$\displaystyle\Delta_{2}^{\mu_{1}\mu_{2}}+g^{\mu_{1}\mu_{2}}I_{\text{{log}}}$
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{2k^{\mu_{1}\mu_{2}}}{D_{\lambda}}$
$\displaystyle=$
$\displaystyle\Delta_{1}^{\mu_{1}\mu_{2}}+g^{\mu_{1}\mu_{2}}I_{\text{{quad}}}.$

Substituting in (C), we can see the scalars $I_{\log}$ cancel and remains the
final expression

$\displaystyle J_{1}^{\mu_{1}\mu_{2}}\left(k_{i}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\Delta_{1}^{\mu_{1}\mu_{2}}+g^{\mu_{1}\mu_{2}}I_{\mathrm{quad}}\right]+\frac{1}{2}k_{i\nu_{12}}W_{3}^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}$
$\displaystyle-
k_{i}^{\mu_{1}}k_{i\nu_{1}}\Delta_{2}^{\mu_{2}\nu_{1}}-k_{i}^{\mu_{2}}k_{i\nu_{1}}\Delta_{2}^{\mu_{1}\nu_{1}}-\frac{1}{2}k_{i}^{2}\Delta_{2}^{\mu_{1}\mu_{2}}.$

The expression above can be written as (C) and C replacing the routing $k_{i}$
by $2k_{1}=\left(P-q\right)$ or $2k_{2}=\left(P+q\right).$

##  Appendix D Function
$Z_{k}^{\left(-1\right)}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)$

As we saw throughout the text, it is sometimes interesting to consider
explicit forms of these functions due to their importance in discussing some
important aspects of amplitudes. So we consider the following function

$Z_{k}^{\left(-1\right)}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)\equiv\int_{0}^{1}d\mathrm{z}\frac{z^{k}}{Q\left(q^{2},m_{1}^{2},m_{2}^{2}\right)},$

where $Q=q^{2}z\left(1-z\right)+\left(m_{1}^{2}-m_{2}^{2}\right)z-m_{1}^{2}$
is the polynomial form of denominator. Since all the functions
$Z_{k}^{\left(-1\right)}$ can be put in terms of the functions
$Z_{0}^{\left(-1\right)},$ we will consider in this appendix the calculation
explicitly only of the function, defined by

$Z_{0}^{\left(-1\right)}\left(q^{2},m_{1}^{2},m_{2}^{2}\right)=\int_{0}^{1}\frac{1}{Q}.$
(D.1)

One way to integrate is to write the polynomial present in the denominator
through its roots. We do

$Q=-q^{2}\left[z^{2}-\frac{1}{q^{2}}\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)z+\frac{m_{1}^{2}}{q^{2}}\right]=-q^{2}\left(z-\alpha\right)\left(z-\beta\right).$
(D.2)

Where the roots of the polynomial are $\alpha$ and $\beta$ given by

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)+\sqrt{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)^{2}-4m_{1}^{2}q^{2}}}{2q^{2}};$
(D.3) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{q^{2}+m_{1}^{2}-m_{2}^{2}-\sqrt{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)^{2}-4m_{1}^{2}q^{2}}}{2q^{2}},$
(D.4)

where $\alpha$ and $\beta$ satisfy the following relations:

$\displaystyle\alpha+\beta$ $\displaystyle=$
$\displaystyle\frac{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)}{q^{2}};\qquad\alpha\beta=\frac{m_{1}^{2}}{q^{2}}$
(D.5) $\displaystyle\alpha-\beta$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)^{2}-4m_{1}^{2}q^{2}}}{q^{2}}.$
(D.6)

Rewriting Eq. (D.1) as

$Z_{0}^{\left(-1\right)}=-\frac{1}{q^{2}}\int_{0}^{1}\frac{1}{\left(z-\alpha\right)\left(z-\beta\right)}=-\frac{1}{q^{2}}\frac{1}{\alpha-\beta}\int_{0}^{1}d\mathrm{z}\left[\frac{1}{\left(z-\alpha\right)}-\frac{1}{\left(z-\beta\right)}\right].$
(D.7)

Using the passage

$\int_{0}^{1}d\mathrm{z}\frac{1}{\left(z-\alpha\right)}=\ln\left(1-\alpha\right)-\ln\left(-\alpha\right)=\ln\left(\frac{\alpha-1}{\alpha}\right),$
(D.8)

we will have

$Z_{0}^{\left(-1\right)}=-\frac{1}{q^{2}}\frac{1}{\alpha-\beta}\left\\{\ln\left(\frac{\alpha-1}{\alpha}\right)-\ln\left(\frac{\beta-1}{\beta}\right)\right\\}=\frac{1}{q^{2}}\frac{1}{\alpha-\beta}\left[\ln\left(\frac{\alpha-1}{\alpha}\right)\left(\frac{\beta}{\beta-1}\right)\right].$
(D.9)

From that, we can write the explicit form for the function
$Z_{0}^{\left(-1\right)},$

$Z_{0}^{\left(-1\right)}=\frac{1}{\sqrt{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)^{2}-4m_{1}^{2}q^{2}}}\ln\left[\frac{\left(m_{1}^{2}+m_{2}^{2}-q^{2}\right)+\sqrt{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)^{2}-4m_{1}^{2}q^{2}}}{\left(m_{1}^{2}+m_{2}^{2}-q^{2}\right)-\sqrt{\left(q^{2}+m_{1}^{2}-m_{2}^{2}\right)^{2}-4m_{1}^{2}q^{2}}}\right]$
(D.10)

In the kinematical limit, where $q^{2}\ll 1,$ we have the result

$Z_{0}^{\left(-1\right)}=\frac{1}{\left(m_{1}^{2}-m_{2}^{2}\right)}\ln\left[\frac{\left(m_{1}^{2}-m_{2}^{2}\right)+\left(m_{1}^{2}-m_{2}^{2}\right)}{\left(m_{1}^{2}-m_{2}^{2}\right)-\left(m_{1}^{2}-m_{2}^{2}\right)}\right].$
(D.11)

##  Appendix E Subamplitudes

We cast vector subamplitudes in this appendix. They are ordered following the
amplitudes that originate them ($AVV$, $VAV$, $VVA$, and $AAA$) and then
grouped according to the version. That emphasizes patterns attributed to each
version and additional terms depending on the squared mass.

First version:

$\displaystyle\left(t^{VPP}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[-K_{1}^{\nu_{1}}S_{23}+K_{2}^{\nu_{1}}S_{13}-K_{3}^{\nu_{1}}S_{12}\right]\frac{1}{D_{123}}$
(E.1) $\displaystyle\left(t^{ASP}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[-K^{\nu_{1}}S_{23}+K_{2}^{\nu_{1}}\left(S_{13}+2m^{2}\right)-K_{3}^{\nu_{1}}\left(S_{12}+2m^{2}\right)\right]\frac{1}{D_{123}}$
(E.2) $\displaystyle\left(t^{APS}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[K_{1}^{\nu_{1}}\left(S_{23}+2m^{2}\right)-K_{2}^{\nu_{1}}\left(S_{13}+2m^{2}\right)+K_{3}^{\nu_{1}}S_{12}\right]\frac{1}{D_{123}}$
(E.3) $\displaystyle\left(t^{VSS}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[K_{1}^{\nu_{1}}\left(S_{23}+2m^{2}\right)-K_{2}^{\nu_{1}}S_{13}+K_{3}^{\nu_{1}}\left(S_{12}+2m^{2}\right)\right]\frac{1}{D_{123}}$
(E.4) $\displaystyle(T^{VPP})^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{31}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+(p_{21}^{\nu_{1}}-p_{32}^{\nu_{1}})I_{\log}\right]-4\left(p_{21}\cdot
p_{32}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[(p_{31}^{\nu_{1}}p_{21}^{2}-p_{21}^{\nu_{1}}p_{31}^{2})J_{3}+p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)\right]$
$\displaystyle\left(T^{ASP}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{31}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+\left(p_{21}^{\nu_{1}}-p_{32}^{\nu_{1}}\right)I_{\log}\right]-4\left(p_{21}\cdot
p_{32}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}-p_{21}^{\nu_{1}}p_{31}^{2}-4m^{2}p_{32}^{\nu_{1}}\right)J_{3}+p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)\right]$
$\displaystyle-\left(T^{APS}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{31}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+(p_{21}^{\nu_{1}}-p_{32}^{\nu_{1}})I_{\log}\right]-4\left(p_{21}\cdot
p_{32}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}-p_{21}^{\nu_{1}}p_{31}^{2}+4m^{2}p_{21}^{\nu_{1}}\right)J_{3}+p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)\right]$
$\displaystyle-\left(T^{VSS}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{31}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+(p_{21}^{\nu_{1}}-p_{32}^{\nu_{1}})I_{\log}\right]-4\left(p_{21}\cdot
p_{32}+4m^{2}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}-p_{21}^{\nu_{1}}p_{31}^{2}-4m^{2}p_{31}^{\nu_{1}}\right)J_{3}+p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)\right]$

Second version:

$\displaystyle\left(t^{SAP}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[K_{1}^{\nu_{1}}S_{23}+K_{2}^{\nu_{1}}\left(S_{13}+2m^{2}\right)-K_{3}^{\nu_{1}}\left(S_{12}+2m^{2}\right)\right]\frac{1}{D_{123}}$
(E.9) $\displaystyle\left(t^{PVP}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[-K_{1}^{\nu_{1}}S_{23}-K_{2}^{\nu_{1}}S_{13}+K_{3}^{\nu_{1}}S_{12}\right]\frac{1}{D_{123}}$
(E.10) $\displaystyle\left(t^{PAS}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle-\left[K_{1}^{\nu_{1}}\left(S_{23}+2m^{2}\right)+K_{2}^{\nu_{1}}S_{13}-K_{3}^{\nu_{1}}\left(S_{12}+2m^{2}\right)\right]\frac{1}{D_{123}}$
(E.11) $\displaystyle\left(t^{SVS}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[K_{1}^{\nu_{1}}\left(S_{23}+2m^{2}\right)+K_{2}^{\nu_{1}}\left(S_{13}+2m^{2}\right)-K_{3}^{\nu_{1}}S_{12}\right]\frac{1}{D_{123}}$
(E.12) $\displaystyle-\left(T^{SAP}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle
2\left[P_{21}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+\left(p_{32}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{32}\cdot
p_{31}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{21}^{\nu_{1}}p_{31}^{2}-p_{31}^{\nu_{1}}p_{21}^{2}+4m^{2}p_{32}^{\nu_{1}}\right)J_{3}+p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)+p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$
$\displaystyle\left(T^{PVP}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{21}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+\left(p_{32}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{32}\cdot
p_{31}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{21}^{\nu_{1}}p_{31}^{2}-p_{31}^{\nu_{1}}p_{21}^{2}\right)J_{3}+p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)+p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$
$\displaystyle\left(T^{PAS}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{21}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+\left(p_{32}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{32}\cdot
p_{31}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{21}^{\nu_{1}}p_{31}^{2}-p_{31}^{\nu_{1}}p_{21}^{2}+4m^{2}p_{31}^{\nu_{1}}\right)J_{3}+p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)+p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$
$\displaystyle-\left(T^{SVS}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{21}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}+\left(p_{32}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{32}\cdot
p_{31}-4m^{2}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{21}^{\nu_{1}}p_{31}^{2}-p_{31}^{\nu_{1}}p_{21}^{2}-4m^{2}p_{21}^{\nu_{1}}\right)J_{3}+p_{32}^{\nu_{1}}J_{2}\left(p_{32}\right)+p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$

Third version:

$\displaystyle\left(t^{SPA}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[K_{1}^{\nu_{1}}\left(S_{23}+2m^{2}\right)-K_{2}^{\nu_{1}}\left(S_{13}+2m^{2}\right)-K_{3}^{\nu_{1}}S_{12}\right]\frac{1}{D_{123}}$
(E.17) $\displaystyle\left(t^{PSA}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[-K_{1}^{\nu_{1}}\left(S_{23}+2m^{2}\right)+K_{2}^{\nu_{1}}S_{13}+K_{3}^{\nu_{1}}\left(S_{12}+2m^{2}\right)\right]\frac{1}{D_{123}}$
(E.18) $\displaystyle\left(t^{PPV}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle-\left[-K_{1}^{\nu_{1}}S_{23}+K_{2}^{\nu_{1}}S_{13}+K_{3}^{\nu_{1}}S_{12}\right]\frac{1}{D_{123}}$
(E.19) $\displaystyle\left(t^{SSV}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle\left[-K_{1}^{\nu_{1}}S_{23}+K_{2}^{\nu_{1}}\left(S_{13}+2m^{2}\right)+K_{3}^{\nu_{1}}\left(S_{12}+2m^{2}\right)\right]\frac{1}{D_{123}}$
(E.20) $\displaystyle\left(T^{SPA}\right)^{\nu_{1}}$ $\displaystyle=$
$\displaystyle
2\left[P_{32}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}-\left(p_{21}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{21}\cdot
p_{31}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}+p_{21}^{\nu_{1}}p_{31}^{2}-4m^{2}p_{21}^{\nu_{1}}\right)J_{3}-p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$
$\displaystyle-\left(T^{PSA}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{32}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}-\left(p_{21}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{21}\cdot
p_{31}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}+p_{21}^{\nu_{1}}p_{31}^{2}-4m^{2}p_{31}^{\nu_{1}}\right)J_{3}-p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$
$\displaystyle\left(T^{PPV}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{32}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}-\left(p_{21}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{21}\cdot
p_{31}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}+p_{21}^{\nu_{1}}p_{31}^{2}\right)J_{3}-p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$
$\displaystyle-\left(T^{SSV}\right)^{\nu_{1}}$ $\displaystyle=$ $\displaystyle
2\left[P_{32}^{\nu_{2}}\Delta_{3\nu_{2}}^{\nu_{1}}-\left(p_{21}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)I_{\log}\right]+4\left(p_{21}\cdot
p_{31}-4m^{2}\right)J_{3}^{\nu_{1}}$
$\displaystyle+2\left[\left(p_{31}^{\nu_{1}}p_{21}^{2}+p_{21}^{\nu_{1}}p_{31}^{2}-4m^{2}\left(p_{21}^{\nu_{1}}+p_{31}^{\nu_{1}}\right)\right)J_{3}-p_{21}^{\nu_{1}}J_{2}\left(p_{21}\right)-p_{31}^{\nu_{1}}J_{2}\left(p_{31}\right)\right]$

##  Appendix F Surface Terms

The surface terms used in this work appear in a totally symmetrical way in the
indices, for the first time treated from the point of view of the IReg
strategy. The meaning of the notation used is

$g_{(\mu_{12}}g_{\mu_{34})}=g_{\mu_{12}}g_{\mu_{34}}+g_{\mu_{13}}g_{\mu_{24}}+g_{\mu_{14}}g_{\mu_{23}}.$
(F.1)

For instance, in the case of permutations involving six indices as the product
of the metrics by the logarithmically divergent object $\Delta_{2\mu\nu},$ we
have forty-five terms given by,

$\displaystyle g_{(\mu_{12}}g_{\mu_{34}}$ (F.2) $\displaystyle=$
$\displaystyle\Delta_{2\mu_{12}}g_{(\mu_{34}}g_{\mu_{56})}+\Delta_{2\mu_{13}}g_{(\mu_{24}}g_{\mu_{56})}+\Delta_{2\mu_{14}}g_{(\mu_{23}}g_{\mu_{56})}+\Delta_{2\mu_{15}}g_{(\mu_{23}}g_{\mu_{46})}+\Delta_{2\mu_{16}}g_{(\mu_{23}}g_{\mu_{45})}$
$\displaystyle+\Delta_{2\mu_{23}}g_{(\mu_{14}}g_{\mu_{56})}+\Delta_{2\mu_{24}}g_{(\mu_{13}}g_{\mu_{56})}+\Delta_{2\mu_{25}}g_{(\mu_{13}}g_{\mu_{46})}+\Delta_{2\mu_{26}}g_{(\mu_{13}}g_{\mu_{45})}$
$\displaystyle+\Delta_{2\mu_{34}}g_{(\mu_{12}}g_{\mu_{56})}+\Delta_{2\mu_{35}}g_{(\mu_{12}}g_{\mu_{46})}+\Delta_{2\mu_{36}}g_{(\mu_{12}}g_{\mu_{45})}$
$\displaystyle+\Delta_{2\mu_{45}}g_{(\mu_{12}}g_{\mu_{36})}+\Delta_{2\mu_{46}}g_{(\mu_{12}}g_{\mu_{35})}$
$\displaystyle+\Delta_{2\mu_{56}}g_{(\mu_{12}}g_{\mu_{34})}.$

This can be written succinctly as

$g_{(\mu_{12}}g_{\mu_{34}}\Delta_{2\mu_{56})}=\sum_{i_{2}>i_{1}=1}^{5}\Delta_{2i_{1}i_{2}}g_{(i_{3}i_{4}}g_{i_{5}i_{6})}\text{
with }i_{n}\not=i_{m}$ (F.3)

where $i_{n}$ denotes $\mu_{i_{n}}$. For the box terms we may also write

$g_{(\mu_{1}\mu_{2}}\square_{3\mu_{3456})}=\sum_{i_{2}>i_{1}=1}^{5}g_{\mu_{i_{1}}\mu_{i_{2}}}\square_{3\mu_{i_{3}}\mu_{i_{4}}\mu_{i_{5}}\mu_{i_{6}}}.$
(F.4)

###  F.1 Uniqueness Factor: Combination of the violating terms

As we saw, surface terms violate several symmetry relations. However, if the
relations are satisfied, relations between surface terms emerge for their
traces and the finite part. Through the strategy (3), we saw that all the
divergent objects were organized into standardized objects as to their tensor
degree and power counting. We have

$\displaystyle\Xi_{\nu_{23}}^{\left(a\right)}$ $\displaystyle=$
$\displaystyle\left[2\square_{3\rho\nu_{23}}^{\rho}-2\Delta_{2\nu_{23}}-g_{\nu_{23}}\Delta_{2\rho}^{\rho}\right]$
(F.5) $\displaystyle\Xi_{\alpha_{12}\nu_{23}}^{\left(b\right)}$
$\displaystyle=$
$\displaystyle\left[3\Sigma_{4\rho\alpha_{12}\nu_{23}}^{\rho}-8\square_{3\alpha_{12}\nu_{23}}-g_{\alpha_{12}\nu_{23}}\Delta_{2\rho}^{\rho}\right]$
(F.6) $\displaystyle\Xi_{\alpha_{1}\alpha_{2}}^{\mathrm{quad}}$
$\displaystyle=$
$\displaystyle\left[W_{2\rho\alpha_{1}\alpha_{2}}^{\rho}-2\Delta_{1\alpha_{12}}+2g_{\alpha_{12}}I_{\text{{quad}}}-2m^{2}\left(\Delta_{2\alpha_{12}}+g_{\alpha_{12}}I_{\log}\right)\right].$
(F.7)

In this way, this organization allows us to write the $U$-factor as

$\displaystyle U_{\alpha_{1}\alpha_{2}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\theta_{\alpha_{1}\alpha_{2}}\left(2\Delta_{2\rho}^{\rho}+i/\pi\right)+\frac{1}{9}\left(3P^{\nu_{2}}P^{\nu_{3}}+q^{\nu_{23}}\right)\Xi_{\alpha_{1}\alpha_{2}\nu_{23}}^{b}$
$\displaystyle+\frac{1}{18}\left(3P^{\nu_{2}}P^{\nu_{3}}+q^{\nu_{23}}\right)g_{(\alpha_{1}\alpha_{2}}\Xi_{\nu_{2}\nu_{3})}^{a}$
$\displaystyle-\frac{1}{2}\left(P^{2}+q^{2}\right)\Xi_{\alpha_{12}}^{a}-P^{\nu_{1}}P_{(\alpha_{2}}\Xi_{\alpha_{1})\nu_{1}}^{a}+4\Xi^{\mathrm{quad}}$

The uniqueness factor that arises in the basic permutations

$U_{\alpha_{2}\nu_{1}}=\left(4\Upsilon_{\nu_{1}\alpha_{2}}+2q_{\nu_{1}}\Upsilon_{\alpha_{2}}+2q_{\alpha_{2}}\Upsilon_{\nu_{1}}+q_{\alpha_{2}}q_{\nu_{1}}\Upsilon\right),$
(F.9)

its explicit expression reads

$\displaystyle U_{\alpha_{2}\nu_{1}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\theta_{\nu_{1}\alpha_{2}}\left(2\Delta_{2\rho}^{\rho}+i/\pi\right)$
$\displaystyle+\frac{1}{9}\left(3P^{\nu_{2}}P^{\nu_{3}}+q^{\nu_{23}}\right)\left[3\Sigma_{4\rho\nu_{1}\alpha_{2}\nu_{23}}^{\rho}-8\square_{3\nu_{1}\alpha_{2}\nu_{23}}-g_{\nu_{1}\alpha_{2}\nu_{23}}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle+\frac{1}{18}\left(3P^{\nu_{2}}P^{\nu_{3}}+q^{\nu_{23}}\right)\left[g_{(\alpha_{1}\alpha_{2}}\left(2\square_{3\rho\nu_{23})}^{\rho}-2\Delta_{2\nu_{23})}-g_{\nu_{23})}\Delta_{2\rho}^{\rho}\right)\right]$
$\displaystyle-\frac{1}{2}\left(P^{2}+q^{2}\right)\left[2\left(\square_{3\rho\nu_{1}\alpha_{2}}^{\rho}-\Delta_{2\nu_{1}\alpha_{2}}\right)-g_{\nu_{1}\alpha_{2}}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle-
P_{\alpha_{2}}P^{\nu_{2}}\left[2\left(\square_{3\rho\nu_{1}\nu_{2}}^{\rho}-\Delta_{2\nu_{1}\nu_{2}}\right)-g_{\nu_{1}\nu_{2}}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle-
P_{\nu_{1}}P^{\nu_{2}}\left[2\left(\square_{3\rho\alpha_{2}\nu_{2}}^{\rho}-\Delta_{2\alpha_{2}\nu_{2}}\right)-g_{\alpha_{2}\nu_{2}}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle+4\left[W_{2\rho\nu_{1}\alpha_{2}}^{\rho}-2\Delta_{1\nu_{1}\alpha_{2}}+2g_{\nu_{1}\alpha_{2}}I_{\text{{quad}}}-2m^{2}\left(\Delta_{2\nu_{1}\alpha_{2}}+g_{\nu_{1}\alpha_{2}}I_{\log}\right)\right].$

In the massless limit and independent of unique or vanishing surface terms

$U_{\alpha_{2}\nu_{1}}=-\frac{1}{3}\theta_{\nu_{1}\alpha_{2}}\left(2\Delta_{2\rho}^{\rho}+\frac{i}{\pi}\right)=-\frac{1}{3}\theta_{\nu_{1}\alpha_{2}}\Upsilon$
(F.11)

$\displaystyle U_{\alpha_{1}\alpha_{2}}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\theta_{\alpha_{1}\alpha_{2}}\Upsilon+\frac{1}{9}\left(3P^{\nu_{12}}+q^{\nu_{12}}\right)\Xi_{\alpha_{1}\alpha_{2}\nu_{12}}^{\left(b\right)}-P^{\nu_{1}}P_{(\alpha_{2}}\Xi_{\alpha_{1})\nu_{1}}^{\left(a\right)}$
$\displaystyle+\frac{1}{18}\left(3P^{\nu_{12}}+q^{\nu_{12}}\right)g_{(\alpha_{1}\alpha_{2}}\Xi_{1\nu_{12})}^{\left(a\right)}-\frac{1}{2}\left(P^{2}+q^{2}\right)\Xi_{\alpha_{12}}^{\left(a\right)}+4\Xi^{\mathrm{quad}},$

where the definitions

$\displaystyle\Xi_{\alpha_{12}}^{\mathrm{quad}}$ $\displaystyle=$
$\displaystyle\square_{2\rho\alpha_{12}}^{\rho}+\frac{1}{2}g_{\alpha_{12}}\Delta_{1\rho}^{\rho}+\Delta_{1\alpha_{12}}+2g_{\alpha_{1}\alpha_{2}}I_{\text{{quad}}}-2m^{2}\left(\Delta_{2\alpha_{12}}+g_{\alpha_{12}}I_{\log}\right)$
(F.13) $\displaystyle=$
$\displaystyle\square_{2\rho\alpha_{12}}^{\rho}+\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{g_{\alpha_{12}}k^{2}}{D_{\lambda}}+\frac{2k_{\alpha_{12}}}{D_{\lambda}}\right]-2m^{2}\left(\Delta_{2\alpha_{12}}+g_{\alpha_{12}}I_{\log}\right)$
(F.14)
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{2k_{\alpha_{12}}}{D_{\lambda}^{2}}$
$\displaystyle=$ $\displaystyle\Delta_{2\alpha_{12}}+g_{\alpha_{12}}I_{\log}$
(F.15)
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{2k_{\alpha_{12}}}{D_{\lambda}}$
$\displaystyle=$
$\displaystyle\Delta_{1\alpha_{12}}+g_{\alpha_{1}\alpha_{2}}I_{\text{{quad}}}$
(F.16)
$\frac{1}{2}g_{\alpha_{12}}\Delta_{1\rho}^{\rho}+\Delta_{1\alpha_{12}}+2g_{\alpha_{1}\alpha_{2}}I_{\text{{quad}}}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left(g_{\alpha_{12}}k^{2}+2k_{\alpha_{12}}\right)\frac{1}{D_{\lambda}}$
(F.17)
$\square_{2\rho\alpha_{12}}^{\rho}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[4k^{2}k_{\alpha_{12}}-6k_{\alpha_{12}}D_{\lambda}-g_{\alpha_{12}}k^{2}D_{\lambda}\right]\frac{1}{D_{\lambda}^{2}}.$
(F.18)

Where the quadratic form can be made null as

$\Xi_{\alpha_{12}}^{\mathrm{quad}}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[4\left(k^{2}-m^{2}\right)-4D_{\lambda}\right]\frac{k_{\alpha_{12}}}{D_{\lambda}^{2}}=0.$
(F.19)

###  F.2 Bilinears reductions and the accessible values to the uniqueness
factor

Observing the expressions

$\displaystyle\Xi_{\nu_{23}}^{\left(a\right)}$ $\displaystyle=$ $\displaystyle
2\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left\\{\left[\frac{8k^{2}k_{\nu_{23}}}{D_{\lambda}^{3}}-\frac{g_{\nu_{23}}k^{2}+6k_{\nu_{23}}}{D_{\lambda}^{2}}\right]\right.$
$\displaystyle-\left.\left[\frac{2k_{\nu_{23}}}{D_{\lambda}^{2}}-\frac{g_{\nu_{23}}}{D_{\lambda}}\right]-g_{\nu_{23}}\left[\frac{k^{2}}{D_{\lambda}^{2}}-\frac{1}{D_{\lambda}}\right]\right\\}.$

If it is linear and bilinears are reduced, follow the solid resu

$\Xi_{\nu_{23}}^{\left(a\right)}=4m^{2}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{4k_{\nu_{23}}}{D_{\lambda}^{3}}-g_{\nu_{23}}\frac{1}{D_{\lambda}^{2}}\right]=-4m^{2}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{\partial}{\partial
k^{\nu_{3}}}\frac{k_{\nu_{2}}}{D_{\lambda}^{2}}\equiv 0.$ (F.21)

The last passage involves defining a surface term that appears in 4D. Here it
is finite and indisputably zero.

As the higher rank term, they appear in the violations of RAGFs and unicity of
odd amplitudes

$\Xi_{\mu_{1234}}^{\left(b\right)}=\left[3\Sigma_{4\rho\mu_{1234}}^{\rho}-8\square_{3\mu_{1234}}-g_{\mu_{1234}}\Delta_{2\rho}^{\rho}\right],$
(F.22)

we will have for the first term

$3g^{\mu_{12}}\Sigma_{4\mu_{123456}}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{144k^{2}k_{\mu_{3456}}}{D_{\lambda}^{4}}-\frac{8\left[10k_{\mu_{3456}}+k^{2}g_{(\mu_{34}}k_{\mu_{56})}\right]}{D_{\lambda}^{3}}\right],$
(F.23)

using the formula
$g^{\mu_{12}}g_{(\mu_{12}}k_{\mu_{3456})}=10k_{\mu_{3456}}+k^{2}g_{(\mu_{34}}k_{\mu_{56})}$
and the definitions

$\displaystyle 8\square_{3\mu_{3456}}$ $\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{64k_{\mu_{3456}}}{D_{\lambda}^{3}}-\frac{8g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{2}}\right]$
(F.24) $\displaystyle g_{(\mu_{12}}g_{\mu_{34})}\Delta_{2\rho}^{\rho}$
$\displaystyle=$ $\displaystyle
g_{(\mu_{12}}g_{\mu_{34})}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{2k^{2}}{D_{\lambda}^{2}}-\frac{2}{D_{\lambda}}\right],$
(F.25)

it is obtained the result

$\displaystyle\left[3g^{\mu_{12}}\Sigma_{4\mu_{123456}}-8\square_{3\mu_{3456}}-g_{(\mu_{12}}g_{\mu_{34})}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle=$
$\displaystyle\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left\\{\left[\frac{144k_{\mu_{3456}}}{D_{\lambda}^{3}}-\frac{8g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{2}}-\frac{2g_{(\mu_{12}}g_{\mu_{34})}}{D_{\lambda}}\right]\frac{k^{2}}{D_{\lambda}}\right.$
$\displaystyle\hskip
60.0pt\left.-\left[\frac{144k_{\mu_{3456}}}{D_{\lambda}^{3}}-\frac{8g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{2}}-\frac{2g_{(\mu_{12}}g_{\mu_{34})}}{D_{\lambda}}\right]\right\\}.$

Reducing bilinears by adding and subtracting the mass makes obtaining the
identity

$\frac{k^{2}}{k^{2}-m^{2}}=1+\frac{m^{2}}{k^{2}-m^{2}}.$ (F.27)

We reach at

$\displaystyle\left[3g^{\mu_{12}}\Sigma_{4\mu_{123456}}-8\square_{3\mu_{3456}}-g_{(\mu_{12}}g_{\mu_{34})}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle=$ $\displaystyle
m^{2}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left\\{\frac{144k_{\mu_{3456}}}{D_{\lambda}^{4}}-\frac{8g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{3}}-\frac{2g_{(\mu_{12}}g_{\mu_{34})}}{D_{\lambda}^{2}}\right\\}.$

Mass terms do not vanish identically; what remains are precisely convergent
surface terms

$\displaystyle\left[3g^{\mu_{12}}\Sigma_{4\mu_{123456}}-8\square_{3\mu_{3456}}-g_{(\mu_{12}}g_{\mu_{34})}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle=$ $\displaystyle
m^{2}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left\\{12\left[\frac{12k_{\mu_{3456}}}{D_{\lambda}^{4}}-\frac{g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{3}}\right]+4\frac{g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{3}}-\frac{g_{(\mu_{12}}g_{\mu_{34})}}{D_{\lambda}^{2}}-\frac{g_{(\mu_{12}}g_{\mu_{34})}}{D_{\lambda}^{2}}\right\\},$

these terms own integrands that are typical of four dimensions. Integrating in
2D they are precisely zero

$\Delta_{3;\mu_{ij}}=-\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{\partial}{\partial
k^{\mu_{i}}}\frac{k_{\mu_{j}}}{D_{\lambda}^{2}}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{4k_{\mu_{ij}}}{D_{\lambda}^{3}}-\frac{g_{\mu_{ij}}}{D_{\lambda}^{2}}\right]\equiv
0$ (F.30)
$\square_{4;\mu_{3456}}=-\frac{1}{2}\sum_{i=1}^{4}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\frac{\partial}{\partial
k^{\mu_{i}}}\frac{k_{\mu_{1}\cdots\hat{\mu}_{i}\cdots\mu_{4}}}{D_{\lambda}^{3}}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left[\frac{12k_{\mu_{3456}}}{D_{\lambda}^{4}}-\frac{g_{(\mu_{34}}k_{\mu_{56})}}{D_{\lambda}^{3}}\right]\equiv
0$ (F.31)

thereby

$\left[3g^{\mu_{12}}\Sigma_{4\mu_{123456}}-8\square_{3\mu_{3456}}-g_{(\mu_{12}}g_{\mu_{34})}\Delta_{2\rho}^{\rho}\right]=m^{2}\left[12\square_{4\mu_{3456}}+g_{(\mu_{34}}\Delta_{3\mu_{56})}\right]=0$
(F.32)

if the total derivative character of the expression is desired, we can also
write in the form

$\displaystyle\left[3g^{\mu_{12}}\Sigma_{4\mu_{123456}}-8\square_{3\mu_{3456}}-g_{(\mu_{12}}g_{\mu_{34})}\Delta_{2\rho}^{\rho}\right]$
$\displaystyle=$ $\displaystyle
m^{2}\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left\\{-6\sum_{i=1}^{4}\frac{\partial}{\partial
k^{\mu_{i}}}\frac{k_{\mu_{1}\cdots\hat{\mu}_{i}\cdots\mu_{4}}}{D_{\lambda}^{3}}-g_{(\mu_{3}\mu_{4}}\frac{\partial}{\partial
k^{\mu_{5}}}\frac{k_{\mu_{6})}}{D_{\lambda}^{2}}\right\\}.$

Quadratic term in the Uniqueness factor: We assume bilinear reduction this
term cancels identically independent from the definition of the quadratic
scalar

$U_{\alpha_{12}}^{\mathrm{quad}}=\int\frac{\mathrm{d}^{2}k}{\left(2\pi\right)^{2}}\left\\{\frac{16k^{2}k_{\alpha_{12}}}{D_{\lambda}^{2}}-\frac{16k_{\alpha_{12}}}{D_{\lambda}}\left[1-m^{2}\frac{1}{D_{\lambda}}\right]\right\\}$
(F.34)

in other words $U_{\alpha_{12}}^{\mathrm{quad}}=0$.

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Spacecraft as it flew over that storm (Parisi et al., 2021). This may mean
that storms in Saturn’s atmosphere could potentially generate at least some of
the waves in the rings, although further work is needed to ascertain whether
it is likely that suitably intense and deep storms exist in Saturn.

Alternatively, these gravitational waves could be generated by global
oscillations with suitably strong $m=3$ components and pattern speeds close to
the planet’s bulk rotation rate. Giant planets are expected to have a complex
spectrum of inertial and/or gravito-inertial modes with suitable pattern
speeds that are currently being investigated in part because of their
potential affects on giant planet’s tidal response and dissipation (Barker et
al., 2016; Fuller et al., 2016; Ogilvie, 2020; Lin and Ogilvie,
2021).888Rossby waves with $m=3$ have also been observed in Saturn’s
stratosphere (Guerletet al., 2018), but those structures do not involve enough
material to generate sufficiently strong gravitational perturbations. These
structures may even exhibit some amount of time-variability similar to the
observed gravitational anomalies (Barker and Lithwick, 2013; Barker, 2016).
However, in order for these sorts of global oscillations to produce detectable
ring waves, they would have to generate perturbations in the planet’s
gravitational potential comparable to those associated with the fundamental
normal modes or mixed fundamental and gravitational normal modes responsible
for several other waves in the C ring (Hedman and Nicholson, 2014; Fuller,
2014; French et al., 2016; Hedman and Nicholson, 2016; Mankovich and Fuller,
2021; Dewberry et al., 2021). While the relative amplitudes of the various
modes inside giant planets is currently an active area of research (Wu, 2005;
Markham and Stevenson, 2018; Wu and Lithwick, 2019; Markham et al., 2020; Lin
and Ogilvie, 2021; Dewberry and Lai, 2021), we are not aware of any explicit
predictions for the likely range of gravitational signals generated by
oscillations with the pattern speeds of the waves considered here. We
therefore expect that additional work is needed to ascertain whether any of
these oscillations could produce these waves.

Another way to determine whether these gravitational anomalies are due to
compact storms or global oscillations is by comparing the strength of
gravitational perturbations with different $m$-values. Global oscillations can
produce signals with a single azimuthal wavenumber, but compact mass anomalies
carried around by Saturn’s winds produce gravitational perturbations similar
to orbiting satellites and so should generate a series of waves with
predictable locations and relative amplitudes (El Moutamid et al., 2016).
While there are hints of wave signals near some of these predicted locations,
it is not yet practical to use these potential signals to help constrain the
sources of the relevant gravitational perturbations because the vast majority
of these waves generated by compact storms should occur in the A and B rings.
These rings have much higher surface mass densities than the C ring, which
makes identifying these waves more difficult because it reduces their
amplitudes and enhances offsets in their pattern speeds. Furthermore, the B
ring is nearly opaque, while the A ring contains many satellite-driven density
waves, both of which complicate isolating the appropriate wave signals.
Comparing any potential wave signals with the C-ring structures considered
here is therefore a non-trivial task that will likely be a productive avenue
for future work.

###  5.2 Pattern speeds

Turning to the pattern speeds of the signals, it is important to first note
that correlating the overall distribution of gravitational signatures with
aspects of Saturn’s visible atmosphere is challenging because the
detectability of gravitational anomalies depends on local ring properties,
with the plateaux apparently being better able to preserve weaker signals over
a longer period of time than other parts of the rings. Nevertheless, given
that the strongest signals both inside and outside the plateaux correspond to
potential perturbations around 100 cm2/s2, Figure 29 most likely provides a
reasonably complete survey of those particular gravitational anomalies for the
time period between 2010 and 1995, and a more patchy history of weaker
signatures.

At first, the overall similarity of the distribution of gravitational anomaly
pattern speeds to the rotation rates of Saturn’s winds at first looks like it
would strongly support the idea that these gravitational signals come from
compact storms. However, closer inspection reveals there is surprisingly
little correlation between these strong anomalies in the planet’s
gravitational field and the most dramatic storms observed in its clouds.

While Saturn’s visible cloud structures are usually relatively subdued
compared to Jupiter’s, roughly once each Saturn year there is a major event
that produces large atmospheric disturbances known as Great White Spots, two
of which have occurred in the last 40 years (Sánchez-Lavega et al., 2018). One
appeared in Saturn’s equatorial jet in 1990, while the other appeared at mid-
northern latitudes in late 2010 (Sánchez-Lavega et al., 2018). Given that
these are the most obvious events in Saturn’s atmosphere over the past 40
years, one might naturally expect that these would produce strong
gravitational signatures, but this does not seem to be the case.

There is no obvious signature of the 1990 event in the rings. This discrepancy
could potentially be explained by the fact that the wave generated by this
event would have occurred outside of any plateaux. In these regions, even
relatively strong waves seem to fade away when they are more than 20 years
old, likely due to enhanced dissipation rates. The signal from the 1990 event
therefore might have damped away before Cassini was able to observe it.

The rotation rates of various components of the 2010 giant storm were between
808∘/day and 810∘/day (Sánchez-Lavega et al., 2018), which is close to two of
our strongest waves. However, attributing either of these gravitational field
anomalies to the 2010 event is difficult because this region contains multiple
patterns that had existed since at least 1980, and while the intensity of the
pattern at 810.5∘/day decreased a few years before 2010, and the intensity of
the pattern around 808∘/day might have begun to rise around that time, there
is not any obvious dramatic change in the strength or locations of these long-
lived anomalies that can be attributed to the 2010 event. More detailed
modeling of these sorts of outbursts could reveal that the 2010 event was a
brief surface manifestation of a deep storm that produced the gravitational
asymmetry, but at the moment any connection between the 2010 event and these
long-lived gravitational anomalies appears to be indirect.

Besides the lack of obvious correlations with major atmospheric outbursts, it
is also worth noting that some of the gravitational anomaly pattern speed fall
outside the range of rotation rates in Saturn’s visible winds. In particular
there are two waves that are generated by perturbations with pattern speeds of
around 804.7∘/day (see Section 4.5). These are substantially slower than the
lowest rotation rates associated with Saturn’s observed winds (García-Melendo
et al., 2011), so if they are to be attributed to discrete storms, they would
need to be in a deep atmospheric layer with winds that differ substantially
from the visible cloud tops.

In this context, it is interesting to note that the anomalies responsible for
the observed waves can be divided into two broad classes based on their
appearance in the maps shown in Section 4. One class consists of relatively
transient features lasting less than a decade (like the ones around 833∘/day)
and the other containing more persistent anomalies that appear to last for
several decades (like the ones at 807.6∘/day and 810∘/day). Of course, such a
classification is more challenging to apply to weaker features, but we can
note that all the signals with pattern speeds above 815∘/day appear to be
transient. By contrast, for pattern speeds below 815∘/day, clearly transient
signals are only found between 807.6∘/day-$808.5^{\circ}$/day (see Figure 23).
The rest of the signals are either certainly long-lived (e.g. the ones at
807.5∘/day and 810∘/day) or are potentially long-lived features found outside
the plateaux (e.g. the signals around $805^{\circ}$/day and 813∘/day, although
difficulties in properly modeling the wave dissipation in these regions makes
this identification less certain).

Interestingly, the distribution of transient features between 808∘/day and
842∘/day matches the range of rotation rates seen in Saturn’s visible winds.
By contrast, the more persistent patterns all have pattern speeds
substantially less than the planet’s bulk rotation rate of 818∘/day (Mankovich
et al., 2019). This could indicate that the transient features are due to
relatively shallow atmospheric phenomena that are carried around the planet at
roughly the same rate as the visible winds, while the more persistent features
are from a deeper layer whose dynamics and rotation state are distinct from
the visible clouds. Such a finding might be consistent with recent
measurements of the static gravity field that indicate the planet contains a
sub-co-rotating region at moderate depths (Iess et al., 2019), but much more
work is needed to rigorously evaluate this idea. Alternatively, the longer-
lived structures could represent global oscillations with pattern speeds that
are slightly slower than the planet’s bulk rotation rate, such as inertial and
gravito-inertial waves (Barker et al., 2016; Ogilvie, 2020; Lin and Ogilvie,
2021; Saio et al., 2021).

We can also note that the variations in both the amplitudes and pattern speeds
of these gravitational anomalies provide yet another means to better constrain
their origins. On Jupiter, both the sizes and drift rates of long-lived storms
like the Great Red Spot have been observed to change over timescales of years
to decades (Simon et al., 2018; Barrado-Izagirre et al., 2021; Wong et al.,
2021; Morales-Juberías et al., 2022), and the variations in these storms’
drift rates around the planet are even comparable to those found in Saturn’s
gravitational anomalies. Detailed examinations of the evolution of structures
in Saturn’s atmosphere over the course of the Cassini Mission and beyond could
therefore potentially reveal features with a history that matches some of the
signals identified in this work.

###  5.3 The $m=-1$ wave and a potential connection with solar tides

The previous discussion has focused on the $m=3$ patterns, but we shouldn’t
forget that the persistent $m=3$ feature with a pattern speed of around
810∘/day appears to be associated with an $m=-1$ anomaly rotating three times
faster. Since this is the only $m=3$ wave with a $m=-1$ companion, this
anomaly is likely to be a special case. A full understanding of these waves
will likely require detailed dynamical models of the planet’s interior that
are beyond the scope of this paper. However, comparing the terms in the
gravitational potential required to produce these two waves suggests that
solar tides could be playing an important role.

To illustrate why tidal processes could be relevant to this system, let us
focus on a scenario where the $m=3$ waves are generated by discrete storms
carried around by Saturn’s winds. Recall that each wave is generated by a
specific term in the gravitational expansion with the form
$\cos(|m|(\lambda-\Omega_{p}t))$. Hence the $m=3$ wave requires a term in the
gravitational potential that goes like $\cos(3\lambda-3\Omega_{3}t)$ with
$\Omega_{3}\simeq 810^{\circ}$/day, while the $m=-1$ wave requires a term that
goes like $\cos(\lambda-\Omega_{1}t)$ with $\Omega_{1}\simeq
2430^{\circ}$/day. The first term arises naturally if we consider a compact
mass $M_{A}$ moving on a circular trajectory in Saturn’s equatorial plane at a
radius $r_{A}$ and a rate $\Omega_{A}$ because in this case, the distance
between the mass anomaly and a point in the ring at radius $r$ and inertial
longitude $\lambda$ is:

$d=\sqrt{r_{A}^{2}+r^{2}-2rr_{A}\cos(\lambda-\Omega_{A}t)}.$ (55)

The corresponding perturbation to the gravitational potential
$\Phi^{\prime}=GM_{A}/d$ can therefore be expanded as a power law series in
$\cos(\lambda-\Omega_{A}t)$ and then re-written in the following form:

$\Phi^{\prime}=\frac{GM_{A}}{r}\sum_{m=1}^{\infty}f_{m}(r_{A}/r)\cos(m(\lambda-\Omega_{A}t)).$
(56)

where $f_{m}(r_{A}/r)$ are functions of the ratio $r_{A}/r$ that can be
expressed in terms of Laplace coefficients. For a mass anomaly carried around
by Saturn’s winds, $\Omega_{A}$ can easily be around 810∘/day, and so the
$m=3$ term in this expansion goes like $\cos(3\lambda-3\Omega_{A}t)$, as
desired. Note that if we instead made the anomaly inside the planet a dipole
(cf. Parisi et al., 2020, 2021), then $M_{A}$ would be replaced by the dipole
moment of the anomaly divided by $r_{A}$, and the $f_{m}$ functions would be
different, but the overall form of the series would be the same. Either of
these models therefore naturally produces the term in the potential needed to
drive the $m=3$ wave.

However, the $m=1$ term in this expansion will go like
$\cos(\lambda-\Omega_{A}t)$ and so the pattern speed of the corresponding wave
will be a factor of three too small. The $m=-1$ wave therefore cannot easily
be explained as the result of a constant mass anomaly moving at a steady rate
comparable to the planet’s rotation. Instead, we would need a mass anomaly
moving around the planet at three times the planet’s spin rate, which
currently seems unlikely.

Now consider the possibility that as the anomaly moves around the planet, its
effective mass $M_{A}$ (or, equivalently, its dipole moment) can vary with
longitude and time. These variations could correspond to oscillations in the
storm’s vertical position or extent in the planet’s atmosphere. In principle,
such variations could be arbitrarily complex, but for the sake of simplicity,
let us say that they have the following form:

$M_{A}=M_{0}\left[1+\mu_{x}\cos(m_{x}(\lambda-\Omega_{x}t))\right]$ (57)

for some values of $m_{x}$ and $\Omega_{x}$ (note that this corresponds to the
effective anomaly mass varying with longitude in a frame rotating at the rate
$\Omega_{x}$). In this case, the $m=3$ term in Equation 56 becomes:

$\scriptsize\begin{split}\Phi^{\prime}_{3}=\frac{GM_{0}}{r}f_{3}(r_{A}/r)\cos(3\lambda-3\Omega_{A}t))\\\
+\frac{GM_{0}\mu_{x}}{2r}f_{3}(r_{A}/r)\cos((3+m_{x})\lambda-(3\Omega_{A}+m_{x}\Omega_{x})t)\\\
+\frac{GM_{0}\mu_{x}}{2r}f_{3}(r_{A}/r)\cos((3-m_{x})\lambda-(3\Omega_{A}-m_{x}\Omega_{x})t)\end{split}$
(58)

The first term in this expression has the correct form to produce the $m=3$
wave with a pattern speed of $\Omega_{A}$, while the other two terms will
produce waves with different $m$-numbers and pattern speeds. More
specifically, if we let $m_{x}=2$, then the last term will go like
$\cos(\lambda-(3\Omega_{A}-2\Omega_{x})t)$, and so produce a $|m|=1$ wave with
a pattern speed of $3\Omega_{A}-2\Omega_{x}$. Recall that the pattern speed of
the $m=-1$ wave is close to three times Saturn’s rotation rate, so this term
in the potential will naturally generate the desired wave provided that
$\Omega_{x}$ is sufficiently close to zero. Furthermore, slow changes in
$\Omega_{A}$ and $M_{0}$ will produce the observed coordinated changes in the
pattern speeds and amplitudes of the two waves, while slow changes in
$\mu_{x}$ allow the relative amplitudes of the two waves to change over time.

At first, it might seem that $m_{x}=2$ and $2\Omega_{x}<<3\Omega_{A}$ are
rather arbitrary choices, but in fact these particular parameter values
correspond to tidal distortions in the structure of the planet. In order to
produce an $m=3$ wave with pattern speed $\Omega_{3}$ and an $m=-1$ wave with
pattern speed $\Omega_{1}$, we need the magnitude of the mass anomaly to go
like:

$\footnotesize
M_{A}=M_{0}\left[1+\mu_{x}\cos\left(2\lambda-[3\Omega_{3}-\Omega_{1}]t\right)\right]$
(59)

This behavior is reminiscent of tidal phenomena, which have $m=2$ structures
that rotate slowly relative to inertial space compared to the planet’s spin
rate. We therefore may posit that as the anomaly moves around the planet, its
amplitude is modified by some aspect of the planet’s internal structure that
is affected by a tidal force.

We can even ascertain which sort of tidal potential is relevant by evaluating
the pattern speed of the tidal term $\Omega_{x}=(3\Omega_{3}-\Omega_{1})/2$.
From visual inspection of the maps shown in Figures 20 and 21, we can see that
this difference is well less than a degree per day, which immediately
eliminates tides due to Saturn’s major moons, which all move around the planet
at speeds of at least several degrees per day. This leaves the Sun as the most
likely source of this term, which from Saturn’s perspective moves around the
planet at only 0.03∘/day. To see if this rate is consistent with the observed
values of $\Omega_{3}$ and $\Omega_{1}$, we found the pattern speeds
corresponding to the peak potential perturbation for the $m=3$ signals between
809 and 811∘/day, and the $m=-1$ signals between 2428∘/day and 2432∘/day for
all years between 2004 and 1996 in all three epochs (when all three epochs
have decent signal-to-noise and consistent values for
$(3\Omega_{3}-\Omega_{1})/2$). If we assume a surface mass density of 1.3
g/cm2 for both waves, then we find that $(3\Omega_{3}-\Omega_{1})/2$ is around
0.085∘/day, but this drops to around 0.050∘/day if we assume a surface mass
density of around 0.7 g/cm2 for the $m=-1$ wave, consistent with the
observations described above. Besides such systematic uncertainties, the
finite width of the signals in Figures 20 and 21 suggest that there are
additional uncertainties in the individual pattern speeds that are of order
0.05∘/day. A rigorous examination of these uncertainties is beyond the scope
of this report, but even this rough assessment indicates that the observed
values of $\Omega_{x}=(3\Omega_{3}-\Omega_{1})/2$ appear to be reasonably
consistent with that expected for solar tides.

We may therefore suggest that the atmospheric structure responsible for the
$810^{\circ}$/day wave has been passing through a region of Saturn’s
atmosphere with a particularly strong response to solar tides that somehow
modulate the magnitude of the gravitational anomaly associated with that
structure. The primary challenge for this idea is that the comparable
amplitudes of the potential perturbations associated with the $m=3$ and $m=-1$
waves imply that the modulation term $\mu_{x}$ is of order unity. This means
that the perturbation responsible for the $m=3$ wave must undergo substantial
variations in its effective strength as it moves around the planet and through
the tidal bulge. This is not likely to be the case for a simple mass
concentration, since it would require the anomaly to gain and lose a
substantial amount of mass twice each rotation.

Large changes in the potential perturbation strength could be possible for
more complex structures in the planet. For example, consider the dipolar model
that has recently been used to model the gravitational perturbations
associated with Jupiter’s Great Red Spot (Parisi et al., 2020, 2021). This
model treats strong storms as a mass excess at one level in the atmosphere
some distance above a comparable mass deficit. If we imagine the vertical
positions of these two layers changed by different amounts in response to the
relevant tidal potential, then the separation between them would change as
they move around the planet, which would naturally cause the dipole moment of
the anomaly to vary in the appropriate way. The comparable amplitudes of the
two ring waves would then require that the variations in the separation
between the two mass anomalies be comparable to their average separation.
Similar scenarios could potentially also work with more global oscillations
involving mass concentrations and deficits at different levels in the
atmosphere. In either case, the relevant structure would need to span an
atmospheric layer that has a strong gradient in its tidal response, which
would most likely involve some sort of resonant interaction with the solar
tide similar to those that have been considered in Earth’s atmosphere (Zahnle
and Walker, 1987).

There are two different ways one might attempt to test and further explore
this idea. On the one hand, this model does predict that there should be a
third term in the gravitational potential of comparable magnitude to the one
responsible for the $m=-1$ wave. This term is generated by the middle term in
Equation 58, which would produce a $m=5$ perturbation in the gravitational
potential with a pattern speed of around 486∘/day, and so generate a $m=5$
wave at around 136,400 km from Saturn’s center in the outer A ring. Initial
searches of this region found no evidence for such a wave. However, this is a
ring region with a much higher surface mass density and many satellite-
generated waves, which may obscure this particular wave signal. The tidal term
could also potentially interact with other terms in the expansion in Equation
56, producing additional waves that could exist in other ring regions.
However, more work is needed to ascertain whether the expected signals from
these gravitational perturbations are likely to be detectable in the currently
available data.

A potentially more productive approach for testing this idea would involve
modeling of the two observable signals and seeing whether they can be
generated by realistic atmospheric structures. Recall that only one $m=3$ wave
has a corresponding $m=-1$ wave, and the amplitude of the $m=-1$ term appears
to have been declining over the past few decades even as the amplitude of the
$m=3$ term has been increasing. According to the above model, this implies
that the tidal resonance that generates strong gradients in anomaly
separations occurs in a particular part of the atmosphere that is rotating at
810∘/day. It would therefore be worthwhile to determine what properties a
layer in Saturn’s atmosphere would need to have (e.g. mean density, scale
height, rotation rate and relevant resonance frequencies) in order to produce
a tidal resonance at that particular rotation rate and ascertain whether those
layers could also support a suitably strong gravitational anomaly. Such
studies are likely to provide insights into where these atmospheric structures
can be found and what these structures might be.

Acknowledgements: The authors thank C. Mankovich J. Fuller, M. Marley, R.
French, M. Tiscareno, M. Parisi, A. Simon and K. Soderlund for helpful
discussions. The authors also wish to thank the reviewers for their helpful
comments. This work was supported by NASA Cassini Data Analysis Program Grant
NNX178AF85G.

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## Appendix A: Expected relationships between wave amplitudes and perturbation
strengths

While there is not yet a complete theory for density waves generated by time-
variable periodic forces, the theory of linear density waves generated by
purely periodic perturbing forces provides useful insights into the
relationship between wave amplitudes and gravitational perturbation strengths.
For these waves, the asymptotic form of the wave amplitude $A$ in an optical
depth profile999Note this is different from the wavelet amplitude signal
$A_{w}$, which depends on both radius and wavenumber is typically written in
terms of distance from the resonance radius using the following expression
(Nicholson et al., 1990; Marley and Porco, 1993; Tiscareno and Harris, 2018,
with the approximation that the local mean motion is
$\sqrt{GM_{P}/r_{L}^{3}}$):

$A(r)=\frac{\Phi^{\prime}_{m}}{\pi
G\sigma_{0}r_{L}}\sqrt{\frac{3|m-1|M_{P}}{2\pi\sigma_{0}r_{L}^{2}}}\frac{(r-r_{L})}{r_{L}}e^{-(r-r_{L})^{3}/r_{D}^{3}}$
(60)

where $\sigma_{0}$ is the local ring surface mass density, $r_{L}$ is the
resonant radius, $r_{D}$ is the characteristic damping length of the wave and
$\Phi^{\prime}_{m}$ is the effective gravitational potential perturbation
acting on the ring. Note that this potential is often expressed in terms of a
normalized torque from the resonance $\mathcal{T}/\sigma_{o}$ using the
relationship:

$\Phi^{\prime}_{m}=\sqrt{\frac{3|m-1|n_{L}^{2}}{m\pi^{2}}\frac{\mathcal{T}}{\sigma_{0}}},$
(61)

For a first-order resonance with a satellite of mass $M_{s}$ with semi-major
axis $a_{s}$ the perturbing potential is given by the expression (Nicholson et
al., 1990)

$\Phi^{\prime}_{m}=\sqrt{\pi}\frac{GM_{s}}{a_{s}}\left[\alpha\frac{db^{m}_{1/2}}{d\alpha}+2mb^{m}_{1/2}\right],$
(62)

where $\alpha=r_{L}/a_{s}$ and $b^{m}_{1/2}(\alpha)$ is a Laplace coefficient
(Shu, 1984; Murray and Dermott, 1999). Expressions of a similar form can be
obtained for higher-order satellite resonances (Tiscareno and Harris, 2018).
Meanwhile, for a standard planetary normal mode of degree $\ell$ and azimuthal
wavenumber $m$ (Marley and Porco, 1993):

$\Phi^{\prime}_{m}=(2m+\ell+1)\Phi^{\prime}_{\ell m}$ (63)

where $\Phi^{\prime}_{\ell m}$ is the relevant component of the planet’s
gravitational potential due to that normal mode.

Equation 60 can be re-cast in terms of elapsed time $\delta
t=(r-r_{L})/v_{g}$, and making the further approximation that
$\kappa_{L}\simeq\sqrt{GM_{P}/r_{L}^{3}}$. This yields the following
expression:

$A(r)={\Phi^{\prime}_{m}}\sqrt{\frac{3|m-1|}{2\pi G\sigma_{0}r^{3}_{L}}}\delta
te^{-\delta t^{3}/t_{D}^{3}}$ (64)

where $t_{D}$ is a characteristic damping time that depends on the ring’s
kinematic viscosity $\nu$ as follows (Shu, 1984; Tiscareno et al., 2007):

$t_{D}=\left(\frac{8r_{L}^{2}\kappa_{L}^{2}}{7|m-1|^{2}\nu
n_{L}^{4}}\right)^{1/3}$ (65)

The amplitude of the wave is therefore proportional to $\Phi^{\prime}_{m}$ for
all elapsed times or wavenumbers, with a constant of proportionality that is a
function of $\delta t$ and depends upon $\sigma_{0}$, $t_{D}$ and $|m-1|$.
Note that for both the $m=3$ and $m=-1$ waves considered here $|m-1|=2$, so
the amplitude evolution of both types of wave fragments should be comparable
to each other in regions with similar $\sigma_{0}$ and $\nu$. This is why the
satellite-driven waves with $m=3$ can be used to estimate the conversion
factors for all the waves of interest to this study.