We will choose the quantity $\tilde{B}_{(kl)(mn)}$ momentarily. Because the
B-type couplings satisfy Eq. (142), the sum of all unique $B$ objects
satisfies

$\displaystyle
B_{(kl)(mn)}+B_{(km)(ln)}+B_{(kn)(lm)}=\vec{\mu}^{\,2}\,a_{klmn}+\tilde{B}_{(kl)(mn)}+\tilde{B}_{(km)(ln)}+\tilde{B}_{(kn)(lm)}$
(147)

where $\vec{\mu}^{\,2}\equiv\mu_{k}^{2}+\mu_{l}^{2}+\mu_{m}^{2}+\mu_{n}^{2}$.
That is, we can ensure the convenient property

$\displaystyle B_{(kl)(mn)}+B_{(km)(ln)}+B_{(kn)(lm)}\dot{=}0$ (148)

as long as we choose $\tilde{B}_{(kl)(mn)}$ such that

$\displaystyle\tilde{B}_{(kl)(mn)}+\tilde{B}_{(km)(ln)}+\tilde{B}_{(kn)(lm)}=-\vec{\mu}^{\,2}\,a_{klmn}$
(149)

One immediate choice (and the choice we take now) is to set each $\tilde{B}$
equal to one third of $-\vec{\mu}^{\,2}\,a_{klmn}$

$\displaystyle\tilde{B}_{(kl)(mn)}\dot{=}-\dfrac{1}{3}\,a_{klmn}$ (150)

This yields (as a replacement rule for $b_{k^{\prime}l^{\prime}mn}$ and
definition of $B_{(kl)(mn)}$)

$\displaystyle\boxed{b_{k^{\prime}l^{\prime}mn}=B_{(kl)(mn)}+\dfrac{1}{6}\bigg{[}2(\mu_{k}^{2}+\mu_{l}^{2})-(\mu_{m}^{2}+\mu_{n}^{2})\bigg{]}a_{klmn}}$
(151)

where $B$ is symmetric within each pair and between pairs:

$\displaystyle B_{(kl)(mn)}=B_{(mn)(kl)}=B_{(mn)(lk)}$ (152)

and satisfies the additional constraint

$\displaystyle\boxed{B_{(kl)(mn)}+B_{(km)(ln)}+B_{(kn)(lm)}=0}$ (153)

such that only two among $\\{B_{(kl)(mn)},B_{(km)(ln)},B_{(kn)(lm)}\\}$ are
linearly independent. Note that $B_{(kl)(mn)}$ has the same symmetry
properties as $\sum_{j}\mu_{j}^{2p}\,a_{klj}\,a_{mnj}$. Because Eq. (151)
reduces B-type couplings to A-type couplings as much is as possible, we refer
to it as the quartic B-to-A rule. This and Eq. (143) comprise the desired
B-to-A formulas.

The above rules are sufficient as-is for reducing the sum
$\sum_{j}\mu_{j}^{2}\,a_{klj}\,a_{mnj}$ and yielding the first non-trivial sum
rule. Using the cubic coupling equation Eq. (141) with completeness yields,

$\displaystyle b_{k^{\prime}l^{\prime}mn}$
$\displaystyle=\dfrac{1}{2}\left[\mu_{k}^{2}+\mu_{l}^{2}\right]a_{klmn}-\dfrac{1}{2}\sum_{j=0}\mu_{j}^{2}\,a_{klj}\,a_{mnj}$
(154)

Meanwhile, the LHS can be simplified via Eq. (151). Solving for the
undetermined sum then gives us,

$\displaystyle\boxed{\sum_{j=0}\mu_{j}^{2}\,a_{klj}\,a_{mnj}=-2\,B_{(kl)(mn)}+\dfrac{1}{3}\,\vec{\mu}^{\,2}\,a_{klmn}}$
(155)

where $\vec{\mu}^{\,2}\equiv\mu_{k}^{2}+\mu_{l}^{2}+\mu_{m}^{2}+\mu_{n}^{2}$.
We next turn our attention to $\sum_{j}\mu_{j}^{4}\,a_{klj}\,a_{mnj}$ and then
$\sum_{j}\mu_{j}^{6}\,a_{klj}\,a_{mnj}$.

###  B.4 The $\mu_{j}^{4}$ Sum Rule

The $\sum_{j}\mu_{j}^{4}\,a_{klj}\,a_{mnj}$ relation is relatively
straightforward. As defined in Eq. (132), we can rewrite
$c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$ in terms of B-type cubic
couplings, to which we can then apply the B-to-A formulas:

$\displaystyle c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
$\displaystyle=\sum_{j=0}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}n^{\prime}j}$
(156)
$\displaystyle=\dfrac{1}{4}\sum_{j}\left[\mu_{k}^{2}+\mu_{l}^{2}-\mu_{j}^{2}\right]\left[\mu_{m}^{2}+\mu_{n}^{2}-\mu_{j}^{2}\right]a_{klj}a_{mnj}$
(157)
$\displaystyle=\dfrac{1}{4}(\mu_{k}^{2}+\mu_{l}^{2})(\mu_{m}^{2}+\mu_{n}^{2})a_{klmn}-\dfrac{1}{4}(\vec{\mu}^{\,2})\sum_{j}\mu_{j}^{2}a_{klj}a_{mnj}+\dfrac{1}{4}\sum_{j}\mu_{j}^{4}a_{klj}a_{mnj}$
(158)

such that, using Eq. (155) and solving for the undetermined sum
$\sum_{j}\mu_{j}^{4}\,a_{klj}\,a_{mnj}$,

$\displaystyle\boxed{\sum_{j}\mu_{j}^{4}a_{klj}a_{mnj}=4\,c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}-2\,(\vec{\mu}^{\,2})\,B_{(kl)(mn)}+\left[\dfrac{1}{3}(\vec{\mu}^{\,2})^{2}-(\mu_{k}^{2}+\mu_{l}^{2})(\mu_{m}^{2}+\mu_{n}^{2})\right]a_{klmn}}$
(159)

as desired. Deriving the $\sum_{j}\mu_{j}^{6}\,a_{klj}\,a_{mnj}$ relation
requires significantly more work.

###  B.5 The $\mu_{j}^{6}$ Sum Rule

####  B.5.1 Elastic

As a warm-up to the inelastic case, let us first derive the $\mu_{j}^{6}$ sum
rule (and review the other sum rules) as they appear in the elastic case, i.e.
when $k=l=m=n$. This will provide the general flow of the argument which is
made more complicated in the inelastic case. Definitions for $x$, $a$, $b$,
etc. are included in subsection B.1.

Using the spin-2 completeness relation and differential equation alone, we
have previously derived many elastic coupling relations Chivukula:2019rij ;
Chivukula:2019zkt ; Chivukula:2020hvi ; Foren:2020egq . For example, there are
the elastic B-to-A formulas

$\displaystyle
b_{n^{\prime}n^{\prime}j}=\dfrac{1}{2}\big{[}2\mu_{n}^{2}-\mu_{j}^{2}\big{]}a_{nnj}\hskip
35.0ptb_{j^{\prime}n^{\prime}n}=\dfrac{1}{2}\mu_{j}^{2}a_{nnj}\hskip
35.0ptb_{n^{\prime}n^{\prime}nn}=\dfrac{1}{3}\mu_{n}^{2}a_{nnnn}$ (160)

which allow us to rewrite any spin-2 exclusive B-type couplings in terms of
A-type couplings. Using these in combination with completeness, we find

$\displaystyle a_{nnnn}$ $\displaystyle=\sum_{j}a_{nnj}^{2}$ $\displaystyle
b_{n^{\prime}n^{\prime}nn}$
$\displaystyle=\sum_{j}b_{n^{\prime}n^{\prime}j}a_{nnj}=\mu_{n}^{2}\sum_{j}a_{nnj}^{2}-\dfrac{1}{2}\sum_{j}\mu_{j}^{2}a_{nnj}^{2}$
$\displaystyle c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}$
$\displaystyle\equiv\sum_{j}b_{n^{\prime}n^{\prime}j}^{2}=\mu_{n}^{4}\sum_{j}a_{nnj}^{2}-\mu_{n}^{2}\sum_{j}\mu_{j}^{2}a_{nnj}^{2}+\dfrac{1}{4}\sum_{j}\mu_{j}^{4}a_{nnj}^{2}$

which imply various expressions for sums of the form
$\sum_{j}\mu_{j}^{2p}a_{nnj}^{2}$:

$\displaystyle\sum_{j=0}^{+\infty}a_{nnj}^{2}$ $\displaystyle=a_{nnnn}$ (161)
$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{2}a_{nnj}^{2}$
$\displaystyle=\dfrac{4}{3}\mu_{n}^{2}a_{nnnn}$ (162)
$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{4}a_{nnj}^{2}$
$\displaystyle=4c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{4}{3}\mu_{n}^{2}a_{nnnn}$
(163)

These relations allow us to quickly rewrite various sums between B-type
couplings, including

$\displaystyle\sum_{j=0}^{+\infty}b_{n^{\prime}n^{\prime}j}b_{j^{\prime}n^{\prime}n}$
$\displaystyle=\dfrac{1}{3}\mu_{n}^{4}a_{nnnn}-c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}~{}.$
(164)

As discussed in the main text, a combination of the GW model sum-rules
insuring cancellation of the ${\cal O}(s^{3})$ and ${\cal O}(s^{2})$
contributions to amplitude may be written

$\displaystyle\text{Stabilized RS1:}\hskip
15.0pt\sum_{j=0}^{+\infty}\big{[}5\mu_{n}^{2}-\mu_{j}^{2}\big{]}\mu_{j}^{4}a_{nnj}^{2}=\dfrac{16}{3}\mu_{n}^{6}a_{nnnn}+9\sum_{i=0}^{+\infty}\mu_{(i)}^{2}a_{n^{\prime}n^{\prime}(i)}^{2}~{},$
(25)

We can use our existing relations to rewrite this expression as:

$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{6}a_{nnj}^{2}$
$\displaystyle=20\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{4}{3}\mu_{n}^{6}a_{nnnn}+9\sum_{i=0}^{+\infty}\mu_{(i)}^{2}a_{n^{\prime}n^{\prime}(i)}^{2}$
(165)

It is this variant we seek now to prove.

To begin, note that the B-to-A formulas relate the sum
$\sum_{j}\mu_{j}^{6}a_{nnj}^{2}$ to the sum
$\sum_{j}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$ like so:

$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{6}a_{nnj}^{2}=16\mu_{j}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+4\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$
(166)

Thus, if we determine a means of rewriting
$\sum_{j}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$ in terms of
$c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}$ and $a_{nnnn}$, then we will
have a means of doing the same for the desired sum.

We will arrive at the desired form by considering two integrals of total
derivatives, each of which vanishes because $(\partial_{\varphi}\psi_{n})$
vanishes at the orbifold fixed points $\varphi\in\\{0,\pi\\}$.

Integral 1: To begin, consider the following trivial integral:

$\displaystyle\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\partial_{\varphi}\bigg{\\{}\bigg{[}(\partial_{\varphi}\psi_{j})-6(\partial_{\varphi}A)\psi_{j}\bigg{]}\varepsilon^{-6}(\partial_{\varphi}\psi_{n})^{2}\bigg{\\}}=0$
(167)

By evaluating the net derivative and using the spin-2 mode differential
equation to simplify second derivatives of spin-2 wavefunctions,242424It is
useful to repackage each $(\partial_{\varphi}\psi)$ instead as
$\varepsilon^{+4}(\mathcal{D}\psi)$ where
$\mathcal{D}\equiv\varepsilon^{-4}\partial_{\varphi}$ because then the spin-2
mode equation may be utilized more directly in the form
$\partial_{\varphi}\mathcal{D}\psi_{n}=-\mu_{n}^{2}\varepsilon^{-2}\psi_{n}$.
we attain

$\displaystyle 0$
$\displaystyle=-12\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}A)^{2}\varepsilon^{-6}(\partial_{\varphi}\psi_{n})^{2}\psi_{j}\bigg{\\}}+12(kr_{c})\mu_{n}^{2}x^{(-4)}_{n^{\prime}nj}-2\mu_{n}^{2}b_{j^{\prime}n^{\prime}n}$
$\displaystyle\hskip
35.0pt-6\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-6}(\partial_{\varphi}\psi_{n})^{2}\psi_{j}\bigg{\\}}-\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}$
(168)

We can then construct an instance of
$\sum_{j}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$ within this by multiplying
it by $b_{n^{\prime}n^{\prime}j}$ and summing over $j$. This yields

$\displaystyle 0$
$\displaystyle=-12\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}A)^{2}\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}+12(kr_{c})\mu_{n}^{2}x^{(-6)}_{n^{\prime}n^{\prime}n^{\prime}n}-2\mu_{n}^{2}\sum_{j=0}^{+\infty}b_{n^{\prime}n^{\prime}j}b_{j^{\prime}n^{\prime}n}$
$\displaystyle\hskip
35.0pt-6\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}-\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$
(169)

Integral 2: Next consider the following trivial integral:

$\displaystyle\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\partial_{\varphi}\bigg{\\{}\bigg{[}\dfrac{3}{2}(\partial_{\varphi}A)\varepsilon^{-2}(\partial_{\varphi}\psi_{n})-\mu_{n}^{2}\psi_{n}\bigg{]}\varepsilon^{-6}(\partial_{\varphi}\psi_{n})^{3}\bigg{\\}}=0$
(170)

Evaluating this derivative in the same way as we did with the first integral,
we find

$\displaystyle 0$
$\displaystyle=12\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}A)^{2}\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}-12(kr_{c})\mu_{n}^{2}x^{(-6)}_{n^{\prime}n^{\prime}n^{\prime}n}-\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+3\mu_{n}^{4}b_{n^{\prime}n^{\prime}nn}$
$\displaystyle\hskip
35.0pt+\dfrac{3}{2}\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}$
(171)

Combining: Summing Eqs. (169) & (171) and then solving for
$\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$ immediately
yields:

$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}$
$\displaystyle=-\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+3\mu_{n}^{4}b_{n^{\prime}n^{\prime}nn}-2\mu_{n}^{2}\sum_{j=0}^{+\infty}b_{n^{\prime}n^{\prime}j}b_{j^{\prime}n^{\prime}n}$
$\displaystyle\hskip
35.0pt-\dfrac{9}{2}\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}$
(172)

We already know how to rewrite $b_{n^{\prime}n^{\prime}nn}$ and
$\sum_{j}b_{n^{\prime}n^{\prime}j}b_{j^{\prime}n^{\prime}n}$ in terms of
$c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}$ and $a_{nnnn}$; namely, Eqs.
(160) & (164). Using these, Eq. (172) becomes,

$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{n^{\prime}n^{\prime}j}^{2}=\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{1}{3}\mu_{n}^{6}a_{nnnn}-\dfrac{9}{2}\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}$
(173)

Finally, applying this result to Eq. (166) gives us an expression for the
desired “$\mu_{j}^{6}$” sum:

$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{6}a_{nnj}^{2}=20\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{4}{3}\mu_{n}^{6}a_{nnnn}-18\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{}.$
(174)

Using the “$\mu^{4}_{j}$” sum in Eq. (163) we obtain the result quoted in the
text

$\displaystyle\sum_{j=0}^{+\infty}\bigg{[}5\mu_{n}^{2}-\mu_{j}^{2}\bigg{]}\mu_{j}^{4}a^{2}_{nnj}$
$\displaystyle=\dfrac{16}{3}\mu_{n}^{6}a_{nnnn}+18\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)\varepsilon^{-8}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{}.$
(27)

This is all that is required for the elastic case discussed in the main text.
The inelastic case covered in the next subsubsection is logically similar, but
involves longer expressions and significantly more algebra.

####  B.5.2 Inelastic

Before beginning the derivation of the inelastic $\mu_{j}^{6}$ sum rule, it is
advantageous to define a well-organized polynomial basis with which we can
write our results succinctly. In particular, because the sums we wish to
simplify ($\sum_{j}\mu^{2p}\,a_{klj}\,a_{mnj}$) and relevant quartic degrees
of freedom all have (at least) the symmetries of $B_{(kl)(mn)}$, it is useful
to define a symmetrization operation that forms quantities with symmetries
identical to $B_{(kl)(mn)}$:

$\displaystyle\langle f_{klmn}\rangle$
$\displaystyle\equiv\bigg{\\{}\bigg{[}f_{klmn}+(k\leftrightarrow
l)\bigg{]}+(m\leftrightarrow n)\bigg{\\}}+(kl\leftrightarrow mn)$ (175)
$\displaystyle=f_{klmn}+f_{lkmn}+f_{klnm}+f_{lknm}+f_{mnkl}+f_{mnlk}+f_{nmkl}+f_{nmlk}$
(176)

This allows us to quickly construct a finite basis for polynomials of
$\mu^{2}\in\\{\mu_{k}^{2},\mu_{l}^{2},\mu_{m}^{2},\mu_{n}^{2}\\}$ having the
aforementioned symmetry structures. For a single power of $\mu^{2}$, there is
only one basis element:

$\displaystyle\alpha_{(kl)(mn)}^{(1,1)}\equiv\langle\mu_{k}^{2}\rangle=2\vec{\mu}^{\,2}\equiv
2(\mu_{k}^{2}+\mu_{l}^{2}+\mu_{m}^{2}+\mu_{n}^{2})$ (177)

For two powers of $\mu^{2}$, there are three:

$\displaystyle\alpha_{(kl)(mn)}^{(2,1)}\equiv\langle\mu_{k}^{4}\rangle\hskip
35.0pt\alpha_{(kl)(mn)}^{(2,2)}\equiv\langle\mu_{l}^{2}\mu_{k}^{2}\rangle\hskip
35.0pt\alpha_{(kl)(mn)}^{(2,3)}\equiv\langle\mu_{m}^{2}\mu_{k}^{2}\rangle$
(178)

and for three powers of $\mu^{2}$, there are four:

$\displaystyle\alpha_{(kl)(mn)}^{(3,1)}\equiv\langle\mu_{k}^{6}\rangle\hskip
18.0pt$ $\displaystyle\hskip
18.0pt\alpha_{(kl)(mn)}^{(3,2)}\equiv\langle\mu_{l}^{2}\mu_{k}^{4}\rangle$
$\displaystyle\alpha_{(kl)(mn)}^{(3,3)}\equiv\langle\mu_{m}^{2}\mu_{k}^{4}\rangle\hskip
18.0pt$ $\displaystyle\hskip
18.0pt\alpha_{(kl)(mn)}^{(3,4)}\equiv\langle\mu_{m}^{2}\mu_{l}^{2}\mu_{k}^{2}\rangle$
(179)

With these, we can generically construct any polynomial of the squared masses
(up to cubic degree) having the aforementioned symmetry properties:

$\displaystyle M_{(kl)(mn)}^{(1)}(c_{1})$
$\displaystyle=2c_{1}\,\vec{\mu}^{\,2}$ (180) $\displaystyle
M_{(kl)(mn)}^{(2)}(c_{1},c_{2},c_{3})$
$\displaystyle=\sum_{i=1}^{3}c_{i}\,\alpha_{(kl)(mn)}^{(2,i)}$ (181)
$\displaystyle M_{(kl)(mn)}^{(3)}(c_{1},c_{2},c_{3},c_{4})$
$\displaystyle=\sum_{i=1}^{4}c_{i}\,\alpha_{(kl)(mn)}^{(3,i)}$ (182)

Note that these symbols are intentionally linear in their $c_{i}$ arguments.
In this language, Eq. (159) may be rewritten as

$\displaystyle\boxed{\sum_{j}\mu_{j}^{4}\,a_{klj}\,a_{mnj}=4\,c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}-2\,\vec{\mu}^{\,2}\,B_{(kl)(mn)}+\dfrac{1}{6}\,M^{(2)}_{(kl)(mn)}(1,1,-1)\,a_{klmn}}$
(183)

We now proceed to the $\sum_{j}\mu_{j}^{6}\,a_{klj}\,a_{mnj}$ rule.

As in the previous subsection, we begin our derivation by applying the B-to-A
formulas to a sum of cubic B-type couplings, and then apply existing sum
rules:

$\displaystyle\sum_{j}\mu_{j}^{2}\,b_{k^{\prime}l^{\prime}j}\,b_{m^{\prime}n^{\prime}j}$
$\displaystyle=\dfrac{1}{4}\sum_{j}\mu_{j}^{6}\,a_{klj}\,a_{mnj}-\vec{\mu}^{\,2}\,c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
$\displaystyle-\dfrac{1}{12}\,\vec{\mu}^{\,2}\,(\mu_{k}^{2}+\mu_{l}^{2}-\mu_{m}^{2}-\mu_{n}^{2})^{2}\,a_{klmn}+\dfrac{1}{2}\,\bigg{[}(\vec{\mu}^{\,2})^{2}-(\mu_{k}^{2}+\mu_{l}^{2})\,(\mu_{m}^{2}+\mu_{n}^{2})\bigg{]}B_{(kl)(mn)}$
(184)

On the RHS, only the desired sum $\sum_{j}\mu_{j}^{6}a_{klj}a_{mnj}$ remains
undetermined. However, unlike the previous subsection, we do not yet have a
simplification of the LHS of this expression. To find such a simplification,
we concoct a vanishing combination of two integrals (namely,
$(\text{I1})_{(kl)j}$ and $(\text{I2})_{k(lmn)}$ of Eqs. (185) and (191)),
each of which vanishes independently because their integrands are total
derivatives.

Integral 1: The first integral yields a vanishing combination of cubic
quantities, and is defined as

$\displaystyle(\text{I1})_{(kl)j}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\partial_{\varphi}\bigg{\\{}\Big{[}\tfrac{1}{2}(\partial_{\varphi}\psi_{j})-3(\partial_{\varphi}A)\psi_{j}\Big{]}\varepsilon^{-6}(\partial_{\varphi}\psi_{k})(\partial_{\varphi}\psi_{l})\bigg{\\}}$
(185) $\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\partial_{\varphi}\bigg{\\{}\tfrac{1}{2}\varepsilon^{+6}\mathcal{D}_{j}\mathcal{D}_{k}\mathcal{D}_{l}-3A^{\prime}\,\varepsilon^{+2}\,\psi_{j}\mathcal{D}_{k}\mathcal{D}_{l}\bigg{\\}}$
(186)

where
$\mathcal{D}_{x}\equiv\mathcal{D}\psi_{x}\equiv\varepsilon^{-4}(\partial_{\varphi}\psi_{x})$.
By explicitly applying the differentiation and using the wavefunction of the
spin-2 modes
($\partial_{\varphi}\mathcal{D}_{x}=-\mu_{x}^{2}\varepsilon^{-2}\psi_{x}$), we
attain

$\displaystyle(\text{I1})_{(kl)j}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}\Big{[}-3A^{\prime\prime}-6(A^{\prime})^{2}\Big{]}\varepsilon^{+2}\,\mathcal{D}_{k}\mathcal{D}_{l}\,\psi_{j}-\tfrac{1}{2}\varepsilon^{+4}\Big{[}\mu_{k}^{2}\psi_{k}\mathcal{D}_{l}+\mu_{l}^{2}\mathcal{D}_{k}\psi_{l}\Big{]}\mathcal{D}_{j}$
$\displaystyle\hskip
35.0pt+3A^{\prime}\Big{[}\mu_{k}^{2}\psi_{k}\mathcal{D}_{l}+\mu_{l}^{2}\mathcal{D}_{k}\psi_{l}\Big{]}\psi_{j}-\tfrac{1}{2}\varepsilon^{+4}\,\mathcal{D}_{k}\mathcal{D}_{l}\,\mu_{j}^{2}\psi_{j}\bigg{\\}}$
(187)

Next, to attain the desired index structure, we multiply by
$b_{m^{\prime}n^{\prime}j}$ and sum over all $j$:

$\displaystyle\sum_{j=0}^{+\infty}(\text{I1})_{(kl)j}b_{m^{\prime}n^{\prime}j}$
$\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}\Big{[}-3A^{\prime\prime}-6(A^{\prime})^{2}\Big{]}\varepsilon^{+8}\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}\bigg{\\}}$
$\displaystyle\hskip
35.0pt-\dfrac{1}{2}\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}n^{\prime}j}-\dfrac{1}{2}\sum_{j=0}^{+\infty}\Big{[}\mu_{k}^{2}b_{kl^{\prime}j^{\prime}}b_{m^{\prime}n^{\prime}j}+\mu_{l}^{2}b_{k^{\prime}lj^{\prime}}b_{m^{\prime}n^{\prime}j}\Big{]}$
$\displaystyle\hskip 35.0pt+\dfrac{3}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}A^{\prime}\,\varepsilon^{+6}\Big{[}\mu_{k}^{2}\psi_{k}\mathcal{D}_{l}+\mu_{l}^{2}\mathcal{D}_{k}\psi_{l}\Big{]}\mathcal{D}_{m}\mathcal{D}_{n}\bigg{\\}}$
(188)

Because $(\text{I1})_{(kl)j}$ vanishes, this sum vanishes too, as does the
following combination:

$\displaystyle(\text{I1})$
$\displaystyle\equiv\sum_{j=0}^{+\infty}(\text{I1})_{(kl)j}b_{m^{\prime}n^{\prime}j}+b_{k^{\prime}l^{\prime}j}(\text{I1})_{(mn)j}$
(189) $\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}\Big{[}-6A^{\prime\prime}-12(A^{\prime})^{2}\Big{]}\varepsilon^{+8}\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}\bigg{\\}}-\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}n^{\prime}j}$
$\displaystyle\hskip
35.0pt-\dfrac{1}{2}\sum_{j=0}^{+\infty}\Big{[}\mu_{k}^{2}b_{kl^{\prime}j^{\prime}}b_{m^{\prime}n^{\prime}j}+\mu_{l}^{2}b_{k^{\prime}lj^{\prime}}b_{m^{\prime}n^{\prime}j}+\mu_{m}^{2}b_{k^{\prime}l^{\prime}j}b_{mn^{\prime}j^{\prime}}+\mu_{n}^{2}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}nj^{\prime}}\Big{]}$
$\displaystyle\hskip 35.0pt+\dfrac{3}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}A^{\prime}\,\varepsilon^{+6}\Big{[}(\mu_{k}^{2}\psi_{k})\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}+\mathcal{D}_{k}(\mu_{l}^{2}\psi_{l})\mathcal{D}_{m}\mathcal{D}_{n}$
$\displaystyle\hskip
70.0pt+\mathcal{D}_{k}\mathcal{D}_{l}(\mu_{m}^{2}\psi_{m})\mathcal{D}_{n}+\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}(\mu_{n}^{2}\psi_{n})\Big{]}\bigg{\\}}$
(190)

This completes our manipulations of the first integral quantity.

Integral 2: The second integral directly yields a vanishing combination of
quartic quantities, and is defined as

$\displaystyle(\text{I2})_{k(lmn)}$
$\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\partial_{\varphi}\bigg{\\{}\Big{[}\tfrac{3}{8}(\partial_{\varphi}A)\varepsilon^{-2}(\partial_{\varphi}\psi_{k})-\mu_{k}^{2}\psi_{k}\Big{]}\varepsilon^{-6}(\partial_{\varphi}\psi_{l})(\partial_{\varphi}\psi_{m})(\partial_{\varphi}\psi_{n})\bigg{\\}}$
(191) $\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\partial_{\varphi}\bigg{\\{}\tfrac{3}{8}A^{\prime}\,\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}-\mu_{k}^{2}\,\varepsilon^{+6}\,\psi_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}\bigg{\\}}$
(192)

Like the previous integral, we next carry out the differentiation while making
use of the spin-2 mode wavefunction
($\partial_{\varphi}\mathcal{D}_{x}=-\mu_{x}^{2}\varepsilon^{-2}\psi_{x}$) as
we go, and thereby attain

$\displaystyle(\text{I2})_{k(lmn)}$
$\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}\tfrac{1}{4}\Big{[}\tfrac{3}{2}A^{\prime\prime}+12(A^{\prime})^{2}-\mu_{k}^{2}\varepsilon^{+2}\Big{]}\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}$
$\displaystyle\hskip
35.0pt+\tfrac{1}{4}\varepsilon^{+4}(\mu_{k}^{2}\psi_{k})\Big{[}(\mu_{l}^{2}\psi_{l})\mathcal{D}_{m}\mathcal{D}_{n}+\mathcal{D}_{l}(\mu_{m}^{2}\psi_{m})\mathcal{D}_{n}+\mathcal{D}_{l}\mathcal{D}_{m}(\mu_{n}^{2}\psi_{n})\Big{]}$
$\displaystyle\hskip
35.0pt-\tfrac{3}{8}A^{\prime}\,\varepsilon^{+6}\Big{[}5(\mu_{k}^{2}\psi_{k})\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}+\mathcal{D}_{k}(\mu_{l}^{2}\psi_{l})\mathcal{D}_{m}\mathcal{D}_{n}$
$\displaystyle\hskip
70.0pt+\mathcal{D}_{k}\mathcal{D}_{l}(\mu_{m}^{2}\psi_{m})\mathcal{D}_{n}+\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}(\mu_{n}^{2}\psi_{n})\Big{]}\bigg{\\}}$
(193)

Next, we symmetrize over the indices, as to attain all unique combinations of
indices:

$\displaystyle(\text{I2})$
$\displaystyle\equiv(\text{I2})_{k(lmn)}+(\text{I2})_{l(kmn)}+(\text{I2})_{m(kln)}+(\text{I2})_{n(klm)}$
(194) $\displaystyle=\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}\Big{[}\tfrac{3}{2}A^{\prime\prime}+12(A^{\prime})^{2}\Big{]}\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}\bigg{\\}}-\dfrac{1}{4}\vec{\mu}^{\,2}c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
$\displaystyle\hskip
35.0pt+\dfrac{1}{2}\Big{[}\mu_{k}^{2}\mu_{l}^{2}b_{klm^{\prime}n^{\prime}}+\mu_{k}^{2}\mu_{m}^{2}b_{kl^{\prime}mn^{\prime}}+\mu_{k}^{2}\mu_{n}^{2}b_{kl^{\prime}m^{\prime}n}$
$\displaystyle\hskip
70.0pt+\mu_{l}^{2}\mu_{m}^{2}b_{k^{\prime}lmn^{\prime}}+\mu_{l}^{2}\mu_{n}^{2}b_{k^{\prime}lm^{\prime}n}+\mu_{m}^{2}\mu_{n}^{2}b_{k^{\prime}l^{\prime}mn}\Big{]}$
$\displaystyle\hskip 35.0pt-\dfrac{3}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}A^{\prime}\,\varepsilon^{+6}\Big{[}(\mu_{k}^{2}\psi_{k})\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}+\mathcal{D}_{k}(\mu_{l}^{2}\psi_{l})\mathcal{D}_{m}\mathcal{D}_{n}$
$\displaystyle\hskip
70.0pt+\mathcal{D}_{k}\mathcal{D}_{l}(\mu_{m}^{2}\psi_{m})\mathcal{D}_{n}+\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}(\mu_{n}^{2}\psi_{n})\Big{]}\bigg{\\}}$
(195)

Because $(\text{I2})_{k(lmn)}$ vanishes, $(\text{I2})$ vanishes as well. This
completes our manipulations of the second integral.

Combining: We finally add $(\text{I1})$ from Eq. (190) and $(\text{I2})$ from
Eq. (195) to attain a new quantity that, of course, also equals zero. Doing
so, we attain

$\displaystyle 0$ $\displaystyle=(\text{I1})+(\text{I2})$
$\displaystyle=-\dfrac{9}{2\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{\\{}A^{\prime\prime}\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}\bigg{\\}}-\dfrac{1}{4}\vec{\mu}^{\,2}c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}-\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}n^{\prime}j}$
$\displaystyle\hskip
35.0pt-\dfrac{1}{2}\sum_{j=0}^{+\infty}\Big{[}\mu_{k}^{2}b_{kl^{\prime}j^{\prime}}b_{m^{\prime}n^{\prime}j}+\mu_{l}^{2}b_{k^{\prime}lj^{\prime}}b_{m^{\prime}n^{\prime}j}+\mu_{m}^{2}b_{k^{\prime}l^{\prime}j}b_{mn^{\prime}j^{\prime}}+\mu_{n}^{2}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}nj^{\prime}}\Big{]}$
$\displaystyle\hskip
35.0pt+\dfrac{1}{2}\Big{[}\mu_{k}^{2}\mu_{l}^{2}b_{klm^{\prime}n^{\prime}}+\mu_{k}^{2}\mu_{m}^{2}b_{kl^{\prime}mn^{\prime}}+\mu_{k}^{2}\mu_{n}^{2}b_{kl^{\prime}m^{\prime}n}$
$\displaystyle\hskip
70.0pt+\mu_{l}^{2}\mu_{m}^{2}b_{k^{\prime}lmn^{\prime}}+\mu_{l}^{2}\mu_{n}^{2}b_{k^{\prime}lm^{\prime}n}+\mu_{m}^{2}\mu_{n}^{2}b_{k^{\prime}l^{\prime}mn}\Big{]}$
(196)

The two integrals were intentionally weighted as to ensure all terms
containing $A^{\prime}$ exactly cancel between the two expressions. The
resulting expression possesses a couple of important features.

First, Eq. (196) contains the desired sum
$\sum_{j=0}^{+\infty}\mu_{j}^{2}b_{k^{\prime}l^{\prime}j}b_{m^{\prime}n^{\prime}j}$,
which we have already demonstrated generates our ultimate target
$\sum_{j=0}^{+\infty}\mu_{j}^{6}a_{klj}a_{mnj}$ via the B-to-A formulas, as
made explicit in Eq. (184).

Second, nearly all other terms in Eq. (196) can be expressed in terms of
$a_{klmn}$, the $B_{(kl)(mn)}$ and
$c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$ using our existing relations.
The only exceptional term is

$\displaystyle-\dfrac{9}{2\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0ptA^{\prime\prime}\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}$
(197)

which (as we describe now) contains important information about the radion and
the tower of Goldberger-Wise scalars. In the unstabilized model,
$A^{\prime\prime}$ is a sum of brane-localized Dirac deltas (specifically,
$A^{\prime\prime}=(kr_{c}|\varphi|)^{\prime\prime}=2kr_{c}[\delta(\varphi)-\delta(\varphi-\pi)]$)
and thus – because
$\mathcal{D}_{x}=\varepsilon^{-4}\partial_{\varphi}\psi_{x}$ vanishes at the
branes for each spin-2 state – Eq. (197) vanishes in the absence of a
stabilization mechanism. This contrasts the Goldberger-Wise-stabilized
Randall-Sundrum I model, where the equations of motion demand

$\displaystyle(\partial_{\varphi}^{2}A)$
$\displaystyle=\dfrac{1}{12}(\partial_{\varphi}\phi_{0})^{2}+\dfrac{1}{3}\,V_{1}\,r_{c}\,\delta(\varphi)+\dfrac{1}{3}\,V_{2}\,r_{c}\,\delta(\varphi-\pi)$
(198)

such that Eq. (197) yields a nonzero contribution directly originating from
the nonconstant profile of $\phi_{0}$ through the bulk. That is, a nonzero
contribution from Eq. (197) to our calculations directly reflects the
stabilization of the radion.

We may use the equation of motion Eq. (198) to simplify Eq. (197). In
particular,

$\displaystyle-\dfrac{9}{2\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0ptA^{\prime\prime}\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}$
$\displaystyle=-\dfrac{3}{8\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\phi_{0}^{\prime})^{2}\varepsilon^{+8}\,\mathcal{D}_{k}\mathcal{D}_{l}\mathcal{D}_{m}\mathcal{D}_{n}\equiv-\dfrac{3}{8}x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
(199)

In order to finish rewriting Eq. (196), all that remains now is the
application of the B-to-A formulas, the sum relations, Eq. (184), and a lot of
algebra. The next section presents the results of this process and summarizes
the other inelastic sum relations we have derived. The section thereafter
reduces those results to the equivalent elastic rules. Finally, the last
section of these supplementary materials reviews additional unproven rules
necessary to cancel any “bad” high-energy behavior from the tree-level
helicity-zero $(k,l)\rightarrow(m,n)$ matrix element in the stabilized
Randall-Sundrum model.

###  B.6 Summary of Proven Sum Rules (Inelastic)

All B-type couplings
$\\{b_{l^{\prime}m^{\prime}n},b_{k^{\prime}l^{\prime}mn}\\}$ can be eliminated
in favor of A-type couplings $\\{a_{lmn},a_{klmn}\\}$ and new $B_{(kl)(mn)}$
objects via the B-to-A formulas

$\displaystyle b_{l^{\prime}m^{\prime}n}$
$\displaystyle=\dfrac{1}{2}\left[\mu_{l}^{2}+\mu_{m}^{2}-\mu_{n}^{2}\right]a_{lmn}$
(200) $\displaystyle b_{k^{\prime}l^{\prime}mn}$
$\displaystyle=B_{(kl)(mn)}+\dfrac{1}{6}\bigg{[}2(\mu_{k}^{2}+\mu_{l}^{2})-(\mu_{m}^{2}+\mu_{n}^{2})\bigg{]}a_{klmn}$

where the $B_{(kl)(mn)}$ are constrained such that
$B_{(km)(ln)}+B_{(kn)(lm)}+B_{(kl)(mn)}=0$, and are symmetric in each
individual pair $(k,l)$ and $(m,n)$ as well as with respect to the pair swap
replacement $(k,l)\leftrightarrow(m,n)$. These sums are

$\displaystyle\sum_{j=0}a_{klj}\,a_{mnj}$ $\displaystyle=a_{klmn}$ (201)
$\displaystyle\sum_{j=0}\mu_{j}^{2}\,a_{klj}\,a_{mnj}$
$\displaystyle=-2\,B_{(kl)(mn)}+\dfrac{1}{3}\,\vec{\mu}^{\,2}\,a_{klmn}$ (202)
$\displaystyle\sum_{j=0}\mu_{j}^{4}\,a_{klj}\,a_{mnj}$
$\displaystyle=4\,c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}-2\,\vec{\mu}^{\,2}\,B_{(kl)(mn)}+\dfrac{1}{6}\,M^{(2)}_{(kl)(mn)}(1,1,-1)\,a_{klmn}$
(203) $\displaystyle\sum_{j=0}\mu_{j}^{6}\,a_{klj}\,a_{mnj}$
$\displaystyle=5\,\vec{\mu}^{\,2}c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}+M^{(2)}_{(km)(ln)}(1,1,1)\,B_{(km)(ln)}+M^{(2)}_{(kn)(lm)}(1,1,1)\,B_{(kn)(lm)}$
$\displaystyle-M^{(2)}_{(kl)(mn)}(0,1,0)\,B_{(kl)(mn)}+\dfrac{1}{6}\,M^{(3)}_{(kl)(mn)}(1,4,-4,0)\,a_{klmn}-\dfrac{3}{2}\,x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
(204)

where $\vec{\mu}^{\,2}\equiv\mu_{k}^{2}+\mu_{l}^{2}+\mu_{m}^{2}+\mu_{n}^{2}$,
and

$\displaystyle
c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}\equiv\dfrac{1}{\pi}\int
d\varphi\hskip
5.0pt\varepsilon^{-6}(\partial_{\varphi}\psi_{k})(\partial_{\varphi}\psi_{l})(\partial_{\varphi}\psi_{m})(\partial_{\varphi}\psi_{n})$
(205) $\displaystyle
x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}k^{\prime}l^{\prime}m^{\prime}n^{\prime}}\equiv\dfrac{1}{\pi}\int
d\varphi\hskip
5.0pt\varepsilon^{-8}\,(\partial_{\varphi}\phi_{0})^{2}\,(\partial_{\varphi}\psi_{k})(\partial_{\varphi}\psi_{l})(\partial_{\varphi}\psi_{m})(\partial_{\varphi}\psi_{n})$
(206)

The last two sum rules can be combined as to cancel all factors of
$c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$, and thereby yields

$\displaystyle\sum_{j=0}\mu_{j}^{4}\,\bigg{(}\mu_{j}^{2}-\dfrac{5}{4}\,\vec{\mu}^{\,2}\bigg{)}\,a_{klj}\,a_{mnj}$
$\displaystyle=\dfrac{1}{4}\,M^{(2)}_{(kl)(mn)}(5,1,10)\,B_{(kl)(mn)}$
$\displaystyle+M^{(2)}_{(km)(ln)}(1,1,1)\,B_{(km)(ln)}+M^{(2)}_{(kn)(lm)}(1,1,1)\,B_{(kn)(lm)}$
$\displaystyle-\frac{1}{24}\,M^{(3)}_{(kl)(mn)}(1,-1,16,0)\,a_{klmn}-\dfrac{3}{2}\,x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
(207)

These equations extend and generalize the sum rules derived in
Chivukula:2020hvi .

###  B.7 Summary of Proven Sum Rules (Elastic)

Oftentimes, we are particularly interested in the elastic massive spin-2 KK
mode scattering process, wherein $k=l=m=n\,(\neq 0)$ and the relations of the
previous subsections simplify. Consider, for example, the $B$’s constraint in
this context:

$\displaystyle B_{(km)(ln)}+B_{(kn)(lm)}+B_{(kl)(mn)}=0\hskip
15.0pt\stackrel{{\scriptstyle\text{elastic}}}{{\longrightarrow}}\hskip
15.0ptB_{(nn)(nn)}=0$ (208)

such that all of the $B$’s become identical and vanish. The relevant B-to-A
formulas become

$\displaystyle b_{n^{\prime}n^{\prime}j}$
$\displaystyle=\dfrac{1}{2}\left[2\mu_{n}^{2}-\mu_{j}^{2}\right]a_{nnj}\hskip
35.0ptb_{j^{\prime}n^{\prime}n}=\dfrac{1}{2}\,\mu_{j}^{2}\,a_{nnj}\hskip
35.0ptb_{n^{\prime}n^{\prime}nn}=\dfrac{1}{3}\,\mu_{n}^{2}\,a_{nnnn}$ (209)

whereas the sum rules reduce to

$\displaystyle\sum_{j}a_{nnj}^{2}$ $\displaystyle=a_{nnnn}$ (210)
$\displaystyle\sum_{j}\mu_{j}^{2}\,a_{nnj}^{2}$
$\displaystyle=\dfrac{4}{3}\,\mu_{n}^{2}\,a_{nnnn}$ (211)
$\displaystyle\sum_{j}\mu_{j}^{4}\,a_{nnj}^{2}$
$\displaystyle=4\,c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{4}{3}\,\mu_{n}^{4}\,a_{nnnn}$
(212) $\displaystyle\sum_{j}\mu_{j}^{6}a_{nnj}^{2}$
$\displaystyle=20\,\mu_{n}^{2}\,c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{4}{3}\,\mu_{n}^{6}\,a_{nnnn}-\dfrac{3}{2}\,x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}n^{\prime}n^{\prime}n^{\prime}n^{\prime}}$
(213)

with the last two expressions combining to yield

$\displaystyle\sum_{j}\bigg{[}\mu_{j}^{2}-5\mu_{n}^{2}\bigg{]}\mu_{j}^{4}\,a_{nnj}^{2}$
$\displaystyle=-\dfrac{16}{3}\,\mu_{n}^{6}\,a_{nnnn}-\dfrac{3}{2}\,x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}n^{\prime}n^{\prime}n^{\prime}n^{\prime}}$
(214)

###  B.8 Unproven Rules

The aforementioned rules are insufficient on their own for ensuring
cancellations of the $(k,l)\rightarrow(m,n)$ matrix element (which naively
contains $\mathcal{O}(s^{5})$ terms) down to $\mathcal{O}(s)$ growth. This is
true in both the fully elastic ($k=l=m=n$) and more general cases.

In the fully elastic case, only one additional rule is required:

$\displaystyle
3\bigg{[}9\sum_{i=0}^{+\infty}a_{n^{\prime}n^{\prime}(i)}^{2}-\mu_{n}^{2}\mu_{n}^{2}a_{nn0}^{2}\bigg{]}=15\,c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+2\mu_{n}^{4}a_{nnnn}$
(215)

The inelastic case provides a generalization of this rule, as well as two
additional rules we have yet to prove analytically. These analytic rules have
been attained by calculating the full $(k,l)\rightarrow(m,n)$ matrix element
(a nontrivial task), asymptotically series expanding that matrix element in
$s$ down to $\mathcal{O}(s^{3/2})$ (also nontrivial; note odd powers of $s$
automatically vanish for this particular process), applying the sum rules we
previously derived, and demanding coefficients of any $s^{\sigma}$ for
$\sigma>1$ vanish.

Having done so, we find cancellations of “bad” high-energy behavior
additionally require

$\displaystyle 6B_{(kl)(mn)}$
$\displaystyle=(\mu_{k}^{2}-\mu_{m}^{2})(\mu_{l}^{2}-\mu_{n}^{2})\sum_{j>0}\dfrac{a_{kmj}a_{lnj}}{\mu_{j}^{2}}$
$\displaystyle\hskip
35.0pt+(\mu_{k}^{2}-\mu_{n}^{2})(\mu_{l}^{2}-\mu_{m}^{2})\sum_{j>0}\dfrac{a_{knj}a_{lmj}}{\mu_{j}^{2}}$
(216)

to cancel $\mathcal{O}(s^{4})$ growth and, noting the KK indices $(k,l,m)$ are
cycled through from term to term,

$\displaystyle 0$
$\displaystyle=(\mu_{k}^{2}-\mu_{l}^{2})(\mu_{m}^{2}-\mu_{n}^{2})\sum_{j>0}\dfrac{a_{klj}a_{mnj}}{\mu_{j}^{2}}$
$\displaystyle\hskip
35.0pt+(\mu_{l}^{2}-\mu_{m}^{2})(\mu_{k}^{2}-\mu_{n}^{2})\sum_{j>0}\dfrac{a_{lmj}a_{knj}}{\mu_{j}^{2}}$
$\displaystyle\hskip
35.0pt+(\mu_{m}^{2}-\mu_{k}^{2})(\mu_{l}^{2}-\mu_{n}^{2})\sum_{j>0}\dfrac{a_{mkj}a_{lnj}}{\mu_{j}^{2}}$
(217)

to cancel $\mathcal{O}(s^{3})$ growth.

To simplify writing expressions like those above, define

$\displaystyle L_{kl;mn}$
$\displaystyle=(\mu_{k}^{2}-\mu_{l}^{2})(\mu_{m}^{2}-\mu_{n}^{2})\sum_{j>0}\dfrac{a_{klj}a_{mnj}}{\mu_{j}^{2}}$
(218)

$L_{kl;mn}$ is antisymmetric under $k\leftrightarrow l$ and $m\leftrightarrow
n$, and is symmetric under $kl\leftrightarrow mn$. The previously-listed new
sum rules can thus be written succinctly as

$\displaystyle 6B_{(kl)(mn)}$ $\displaystyle=L_{km;ln}+L_{kn;lm}$ (219)
$\displaystyle 0$ $\displaystyle=L_{kl;mn}+L_{lm;kn}+L_{mk;ln}$ (220)

This latter sum rule is mathematically distinct from the defining constraint
of $B_{(kl)(mn)}$ (i.e., that the sum of all unique $B$ vanishes). Note that
$L_{nn;nn}=0$, thus explaining the absence of these relations when deriving
our elastic sum rules.

Lastly, the $\mathcal{O}(s^{3})$ cancellations also necessitates the following
generalization of the elastic radion rule (Eq. (215)):

$\displaystyle
12\bigg{[}9\sum_{i=0}^{+\infty}a_{k^{\prime}l^{\prime}(i)}a_{m^{\prime}n^{\prime}(i)}-\mu_{k}^{2}\mu_{m}^{2}a_{kl0}a_{mn0}\bigg{]}=60c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}+\dfrac{1}{2}M^{(2)}_{(kl)(mn)}(4,-8,5)a_{klmn}$
$\displaystyle\hskip
35.0pt-3(\mu_{k}^{2}-\mu_{l}^{2})^{2}(\mu_{m}^{2}-\mu_{n}^{2})^{2}\sum_{j>0}\dfrac{a_{klj}a_{mnj}}{\mu_{j}^{4}}+2\vec{\mu}^{\,2}\bigg{[}L_{km;ln}+L_{kn;lm}\bigg{]}$
$\displaystyle\hskip
35.0pt-3\bigg{[}(\mu_{k}^{2}+\mu_{l}^{2})\,(\mu_{m}^{2}-\mu_{n}^{2})^{2}+(\mu_{k}^{2}-\mu_{l}^{2})^{2}\,(\mu_{m}^{2}+\mu_{n}^{2})\bigg{]}\sum_{j>0}\dfrac{a_{klj}a_{mnj}}{\mu_{j}^{2}}$
(221)

where $M^{(2)}_{(nn)(nn)}(4,-8,5)=8\mu_{n}^{4}$. In the unstabilized inelastic
calculation, an identical rule is attained, but with the sum
$\sum_{i}a_{(i)k^{\prime}l^{\prime}}a_{(i)m^{\prime}n^{\prime}}$ replaced
simply by $a_{(0)k^{\prime}l^{\prime}}a_{(0)m^{\prime}n^{\prime}}$. This
completes the rules necessary to ensure cancellations down to $\mathcal{O}(s)$
for 2-to-2 spin-2 mode scattering in the stabilized Randall-Sundrum model.

##  Appendix C Eigenvalues and Eigenfunctions of Spin-2 and Spin-0 Modes

The solutions to the Sturm-Liouville (SL) problem for the spin-0 and the
spin-2 parts of the stabilized RS model determine the eigenvalues and
eigenfunctions. Here we outline two related methods of computing the eigen-
values and eigen-functions for both the spin-0 and the spin-2 SL problem in
perturbation theory. In the following we introduced the standard Rayleigh-
Schrödinger perturbation theory in the context of a SL problem. Here, the
perturbed wavefunctions are expressed as an infinite series in unperturbed
wavefunctions. On the other hand, being able to have closed form expressions
for the perturbed wavefunctions are extremely useful, especially for the
numerical part of our analysis. This leads us to solving the perturbed SL
problem directly, by solving an inhomogeneous differential equation. In the
end the normalized wavefunction derived in either of these methods are
identical and calculating the wavefunction using these two methods serves as a
cross-check of our results.

###  C.1 Perturbation theory and a general Sturm-Liouville Problem

Here we discuss the application of Rayleigh-Schrödinger perturbation theory to
a general Sturm-Liouville problem, including one in which the weight function
is also perturbed. We compute the first-order shifts to the eigenvalues and
eigenfunctions, and demonstrate that completeness holds to the appropriate
order. In section C.1.1 we show how the perturbed eigenfunctions can be
calculated as a linear combination of unperturbed eigenfunctions. In practice
performing an infinite sum of wavefunctions to determine the perturbed
wavefunction is not computationally efficient, so in section C.1.2 we outline
an equivalent method of determining the perturbed wavefunctions as closed form
expressions and show how it is related to Rayleigh-Schrödinger perturbation
theory.

####  C.1.1 Mass Corrections

Consider a generic Sturm-Liouville (SL) problem for the Kaluza-Klein modes,
which is of the form

$\tilde{\mathcal{L}}\psi_{n}=-\lambda_{n}\tilde{\rho}\psi_{n}$ (222)

where $\tilde{\mathcal{L}}$ is the SL operator (given appropriate boundary-
conditions) acting on eigenfunctions $\psi_{n}$ with eigenvalues
$\tilde{\lambda}_{n}$ and and a weight factor $\tilde{\rho}$. The solutions to
the SL problem are orthogonal with respect to the weight factor $\tilde{\rho}$

$\dfrac{1}{\pi}\int_{-\pi}^{\pi}d\varphi\hskip
5.0pt\tilde{\rho}(\varphi)\,\psi_{k}(\varphi)\,\psi_{l}(\varphi)=\delta_{k,l}~{}.$
(223)

These solutions then satisfy the completeness relation252525The symmetry of
the $\delta$-function implies that the argument of $\rho$ in the sum could be
either $\varphi$ or $\varphi^{\prime}$.

$\sum_{\ell}\tilde{\rho}(\varphi)\,\psi_{\ell}(\varphi)\,\psi_{\ell}(\varphi^{\prime})=\pi\delta(\varphi-\varphi^{\prime})~{}.$
(224)

Depending on the nature of the SL problem, the boundary conditions can be
Dirichlet or Neumann as pointed out in the main body of the paper. For the
rest of this appendix, we will drop the argument $\varphi$ in wave functions
and weight factors for simplicity.

In perturbation theory, the SL operator and weight function can be expanded as
$\tilde{\mathcal{L}}=\mathcal{L}+\delta\mathcal{L}$ and
$\tilde{\rho}=\rho+\delta\rho$, while we expand the eigenvalue
$\tilde{\lambda}_{n}=\lambda_{n}^{(0)}+\lambda_{n}^{(1)}+\lambda_{n}^{(2)}+\cdots$
. Here both $\mathcal{L}$ and $\delta\mathcal{L}$ are of Sturm-Liouville form

$\displaystyle\mathcal{L}=\dfrac{d}{d\varphi}\left[p\dfrac{d}{d\varphi}\right]+q~{},$
(225) $\displaystyle\delta\mathcal{L}=\dfrac{d}{d\varphi}\left[\delta
p\dfrac{d}{d\varphi}\right]+\delta q~{}.$ (226)

In our problems, the perturbations $\delta p$, $\delta q$, and $\delta\rho$
come from expanding Eqs. (47) and (54) respectively in powers of
$\epsilon$.262626Note that the spin-2 system in Eq. (47) yields a perturbation
expansion in $\epsilon^{2}$, whereas that for the spin-0 system in Eq. (54)
gives an expansion in powers of $\epsilon$. We then expand the eigenfunction
in perturbation theory as,

$\psi_{n}=\psi_{n}^{(0)}+\psi_{n}^{(1)}+\psi_{n}^{(2)}+\cdots$ (227)

As usual in perturbation theory, we expand the first order perturbed wave
function as a sum of unperturbed wave functions,

$\psi_{n}^{(1)}=\sum_{m=0}^{\infty}C_{nm}\psi_{m}^{(0)}\ .$ (228)

The coefficients $C_{nm}$ can be determined using perturbation theory, as we
will be described later. Here, as usual in Rayleigh-Schrödinger perturbation
theory, we assume that $\psi_{n}^{(1)}$ is chosen to be orthogonal to
$\psi_{n}^{(0)}$.

To the lowest order, the SL equation reads

$\mathcal{L}\psi_{n}^{(0)}=-\lambda_{n}^{(0)}\rho\psi_{n}^{(0)}$ (229)

where $\lambda_{n}^{(0)}$ is the lowest order (unperturbed) eigenvalue.
Additionally, $\psi_{n}^{(0)}$ is the lowest order eigenfunction.

Expanding the SL equation to first order, and using the fact the lowest order
SL problem satisfies Eq. (229) , we obtain, to first order,

$\mathcal{L}\psi_{n}^{(1)}+\delta\mathcal{L}\psi_{n}^{(0)}=-\left(\lambda_{n}^{(0)}\delta\rho\psi_{n}^{(0)}+\lambda_{n}^{(1)}\rho\psi_{n}^{(0)}+\lambda_{n}^{(0)}\rho\psi_{n}^{(1)}\right)$
(230)

Multiplying the first order perturbed equation by $\psi_{n}^{0}$ and
integrating,

$\int d\varphi\ \psi_{n}^{(1)}\mathcal{L}\psi_{n}^{(0)}+\int d\varphi\
\delta\mathcal{L}(\psi_{n}^{0})^{2}=-\int
d\varphi\left(\lambda_{n}^{(0)}\delta\rho(\psi_{n}^{(0)})^{2}+\lambda_{n}^{(1)}\rho(\psi_{n}^{(0)})^{2}+\lambda_{n}^{(0)}\rho\psi_{n}^{(1)}\psi_{n}^{(0)}\right)\
.$ (231)

In the first term in the equation above, we have used the fact that the
operator $\mathcal{L}$ is self adjoint. Using Eq. (229) and rearranging, we
obtain,

$\lambda_{n}^{(1)}=-\dfrac{1}{\pi}\left(\int_{-\pi}^{\pi}d\varphi~{}\delta\rho(\psi_{n}^{(0)})^{2}+\int_{-\pi}^{\pi}d\varphi~{}\psi_{n}^{(0)}\delta\mathcal{L}\psi_{n}^{(0)}\right)$
(232)

Since $\delta\mathcal{L}=\frac{d}{dx}\left[\delta p\right]+\delta q$, we can
integrate the above equation by parts to obtain,

$\lambda_{n}^{(1)}=-\dfrac{1}{\pi}\left[-\int_{-\pi}^{\pi}d\varphi~{}\delta
p\left(\frac{d\psi_{n}^{(0)}}{d\varphi}\right)^{2}+\int_{-\pi}^{\pi}d\varphi~{}\delta
q\left(\psi_{n}^{(0)}\right)^{2}+\int_{-\pi}^{\pi}d\varphi~{}\delta\rho\left(\psi_{n}^{(0)}\right)^{2}\right]~{}.$
(233)

Now that we have the perturbed eigenvalue, we can proceed to calculate the
perturbed eigenfunctions. We describe two methods to do this. The first one
involves directly solving the non- homogenous differential equation in Eq.
(230). The second one makes use of standard Rayleigh-Schrödinger perturbation
theory. In the end, both methods lead to the same eigenfunctions and the use
of these two methods serves as a cross-check of our results.

####  C.1.2 Solving the In-homogeneous Differential Equation using Variation
of Parameters

In this first method, one simply solves the non-homgeneous differential
equation that is derived by substituting into Eq. (230), the unperturbed
eigenvalue and eigenfunction

$\left[\frac{d}{d\varphi}\left(p\frac{d}{d\varphi}\right)+q-\lambda_{n}^{(0)}\rho\right]\psi_{n}^{(1)}=\left[-\frac{d}{d\varphi}\left(\delta
p\frac{d}{d\varphi}\right)-\delta
q+\lambda_{n}^{(0)}\delta\rho+\lambda_{n}^{(1)}\rho\right]\psi_{n}^{(0)}~{}.$
(234)

In order to solve this equation for the spin-2 KK modes, we have used the
method of variation of parameters. Using the solution $\psi_{n}^{(1)}$ found
using this method, the wave function $\psi_{n}^{(0)}+\psi_{n}^{(1)}$ must then
be normalized with respect to $\rho+\delta\rho$ as follows

$\displaystyle\tilde{\psi}_{n}^{(\text{normalized})}=\psi_{n}^{(0)}+\left[\psi_{n}^{(1)}-\dfrac{\psi_{n}^{(0)}}{\pi}\int
d\varphi^{\prime}\left(\rho\ \psi_{n}^{(0)}\
\psi_{n}^{(1)}+\frac{1}{2}\delta\rho\ (\psi_{n}^{(0)})^{2}\right)\right]\ .$
(235)

In the following we describe how to determine the same wave function using
Rayleigh-Schrödinger perturbation theory. We reiterate, that the wave
functions determined in either method are identical and that the advantage of
determining the wave function in this way is that we get closed form solutions
for the perturbed wave functions and masses.

####  C.1.3 Wave functions in Rayleigh-Schrödinger perturbation theory

The perturbed wave function is determined as a sum over unperturbed wave
function as shown in Eq. (228). In order to determine the coefficients
$C_{nm}$, we multiply the perturbed SL equation (230) by $\psi_{m}^{(0)}$
($m\neq n$) and integrate,

$-\lambda_{m}^{(0)}\pi
C_{nm}+\int_{-\pi}^{\pi}d\varphi~{}\psi_{m}^{(0)}(\delta\mathcal{L}\psi_{m}^{(0)})=-\left(\lambda_{n}^{(0)}\int_{-\pi}^{\pi}d\varphi~{}\psi_{m}^{(0)}\delta\rho\psi_{n}^{(0)}+\lambda_{n}^{(0)}\pi
C_{nm}\right)$ (236)

leading to

$C_{nm}=-\dfrac{1}{\pi}\frac{\left(\lambda_{n}^{(0)}\int_{-\pi}^{\pi}d\varphi~{}\psi_{m}^{(0)}\delta\rho\psi_{n}^{(0)}+\int_{-\pi}^{\pi}d\varphi~{}\psi_{m}^{(0)}(\delta\mathcal{L}\psi_{m}^{(0)})\right)}{\lambda_{n}^{(0)}-\lambda_{m}^{(0)}}\
.$ (237)

Using the definition of $\delta\mathcal{L}$ and integrating by parts, we get,

$C_{nm}=-\dfrac{1}{\pi}\left[\frac{-\int_{-\pi}^{\pi}d\varphi~{}\delta
p\frac{d\psi_{n}^{(0)}}{d\varphi}\frac{d\psi_{m}^{(0)}}{d\varphi}+\int_{-\pi}^{\pi}d\varphi~{}\delta
q\psi_{n}^{(0)}\psi_{m}^{(0)}+\lambda_{n}^{(0)}\int_{-\pi}^{\pi}d\varphi~{}\delta\rho\psi_{n}^{(0)}\psi_{m}^{(0)}}{\lambda_{n}^{(0)}-\lambda_{m}^{(0)}}\right]\
.$ (238)

In Rayleigh-Schrödinger perturbation theory, we usually assume that the first
order perturbed solutions are orthogonal to the lowest order solution so that
$C_{nn}=0$. However, in the presence of a perturbation to the weight function,
that is no longer the case. Here, to obtain the coefficient $C_{nn}$, we use
the normalization condition,

$\displaystyle\int_{-\pi}^{\pi}d\varphi\,\tilde{\rho}\tilde{\psi}_{n}^{2}=\int_{-\pi}^{\pi}d\varphi(\rho+\delta\rho)\left(\psi^{(0)}_{n}+\sum_{m}C_{nm}\psi^{(0)}_{m}\right)^{2}=\pi~{},$
(239)

which, to first order, implies

$C_{nn}=-\,\frac{1}{2\pi}\int_{-\pi}^{\pi}d\varphi\,\delta\rho(\psi^{(0)}_{n})^{2}~{}.$
(240)

We have checked, numerically, that the wave functions derived using Eq. (228)
and Eq. (238) are identical to the ones derived using Eq. (234) and (235).

###  C.2 Wavefunction and Masses of KK modes in the DFGK model

Here we present the wavefunctions and mass corrections for the spin-2 as well
as the scalar sector for the DFGK model in the stiff-wall limit. These
expressions are derived by solving the differential equations described in
section C and specifically using Eq. (233) and Eq. (234). The expressions
presented here are relevant for the large $kr_{c}$ limit. The general
expressions, valid for all $kr_{c}$, are quite cumbersome and are provided in
supplementary Mathematica files on GitHub.

####  C.2.1 Spin-2 mass and wavefunction corrections

In order to verify sum rules to order $\epsilon^{2}$ we need the spin-2
wavefunction and masses to order $\epsilon^{2}$. We start by expanding the
spin-2 Sturm-Liouville equation in (9) up to order $\epsilon^{2}$. We also
expand the wavefunction and masses, as described earlier, to order
$\epsilon^{2}$

$\displaystyle\psi_{n}$
$\displaystyle=\psi_{n}^{(0)}+\psi_{n}^{(2)}+\cdots~{},$ (241)
$\displaystyle\mu_{n}^{2}$
$\displaystyle=(\mu_{n}^{(0)})^{2}+\delta\mu_{n}^{2}+\cdots~{}.$ (242)

We have dropped the $\psi_{n}^{(1)}$ term since the corrections to the spin-2
Sturm-Liouville problem start at order $\epsilon^{2}$ as can be seen from its
expanded form below

$\displaystyle 0$
$\displaystyle=\bigg{[}\partial_{\varphi}^{2}-4\tilde{k}r_{c}\,\partial_{\varphi}+\left(\mu_{n}^{(0)}\right)^{2}\,e^{2\tilde{k}r_{c}\varphi}\bigg{]}\psi_{n}^{(0)}$
$\displaystyle\hskip
15.0pt+\epsilon^{2}\Bigg{\\{}\bigg{[}-8\varphi\,\partial_{\varphi}+e^{2\tilde{k}r_{c}\varphi}\bigg{(}\delta\mu_{n}^{2}+2\,\varphi^{2}\left(\mu_{n}^{(0)}\right)^{2}\bigg{)}\bigg{]}\psi^{(0)}_{n}$
$\displaystyle\hskip
35.0pt+\bigg{[}\partial_{\varphi}^{2}-4\tilde{k}r_{c}\,\partial_{\varphi}+e^{2\tilde{k}r_{c}\varphi}\left(\mu_{n}^{(0)}\right)^{2}\bigg{]}\psi^{(2)}_{n}\Bigg{\\}}+\mathcal{O}(\epsilon^{3})$
(243)

Here we see that the leading term for $\psi_{n}^{(0)}$is the usual one that we
encounter in the unstabilized limit. Solutions to the leading order
differential equation are well known and can be found in Ref.
Chivukula:2020hvi . We reproduce some of these results here later. After
solving for $\psi_{n}^{(0)}$, we then proceed to solve the above differential
equation at order $\epsilon^{2}$.

####  C.2.2 The massless graviton to order $\epsilon^{2}$

The massless graviton is the easiest, since it does not acquire a mass, its
wavefunction is derived by setting $\mu_{0}^{(0)}=\delta\mu_{0}=0$. The
wavefunction is a constant and any $\epsilon$ dependence comes from the
normalization condition (10). The normalized wavefunction for the massless
graviton to order $\epsilon^{2}$ is of the form

$\displaystyle\psi_{0}$ $\displaystyle=\bigg{[}\dfrac{1}{\pi}\int d\varphi\
e^{-2A}\bigg{]}^{-1/2}$
$\displaystyle=e^{\pi\tilde{k}r_{c}}\sqrt{\frac{\pi\tilde{k}r_{c}}{e^{2\pi\tilde{k}r_{c}}-1}}+\epsilon^{2}\frac{\sqrt{\pi}e^{\pi\tilde{k}r_{c}}\Big{[}-2\pi^{2}\tilde{k}^{2}r_{c}^{2}-2\pi\tilde{k}r_{c}+e^{2\pi\tilde{k}r_{c}}-1\Big{]}}{8r_{c}^{3/2}\Big{[}\tilde{k}(e^{2\pi\tilde{k}r_{c}}-1)\Big{]}^{3/2}}\
+\mathcal{O}(\epsilon^{3}).$ (244)

####  C.2.3 Massive spin-2 modes to order $\epsilon^{2}$

Before we present results on the perturbed wavefunctions and masses of the
massive spin-2 modes, we remind the reader about the unperturbed wave
functions and masses that are identical to the unstabilized case. Since the
full expressions can be quite lengthy, we present only results that are valid
in the large $\tilde{k}r_{c}$ limit and provide the full expressions in
supplementary Mathematica files on GitHub. The wavefunction for the massive
spin-2 modes to order $\epsilon^{0}$ is the same as those derived in the
unstabilized RS model and in the large $\tilde{k}r_{c}$ limit is of the form

$\displaystyle\psi_{n}^{(0)}=e^{2\varphi\tilde{k}r_{c}}\sqrt{\frac{\pi
r_{c}\tilde{k}}{e^{2\tilde{k}r_{c}\pi}\,J_{2}\Big{(}j_{1,n}\Big{)}^{2}-J_{2}\Big{(}e^{-\pi\tilde{k}r_{c}}j_{1,n}\Big{)}^{2}}}\,\,J_{2}\Big{(}e^{\tilde{k}(\varphi
r_{c}-\pi r_{c})}j_{1,n}\Big{)}\ .$ (245)

Here $J_{i}$ are Bessel-J functions. The mass of the spin-2 modes are
determined by solving a transcendental equation that is derived from the
Neummann boundary condition satisfied by the spin-2 modes:

$\displaystyle
J_{1}\bigg{(}\frac{e^{\pi\tilde{k}r_{c}}\mu_{n}}{\tilde{k}r_{c}}\bigg{)}\,Y_{1}\bigg{(}\frac{\mu_{n}}{\tilde{k}r_{c}}\bigg{)}-J_{1}\bigg{(}\frac{\mu_{n}}{\tilde{k}r_{c}}\bigg{)}\,Y_{1}\bigg{(}\frac{e^{\pi\tilde{k}r_{c}}\mu_{n}}{\tilde{k}r_{c}}\bigg{)}=0\
.$ (246)

In the large $kr_{c}$ limit, the solution to the transcendental equation
reduces to a simple form

$\displaystyle
m_{n}=\dfrac{\mu_{n}^{(0)}}{r_{c}}=\tilde{k}e^{-\pi\tilde{k}r_{c}}\,j_{1,n}\
.$ (247)

Here $j_{1,n}$ are the roots of $J_{1}$.

Substituting the above form of the leading order spin-2 wavefunction and
masses into Eq. (243), we end up with a non-homogeneous differential equation
which can be solved using the method of variation of parameters. Alternately,
the same differential equation can also be derived from Eq. (234) and in the
large $\tilde{k}r_{c}$ limit is of the form given below.

$\displaystyle\Big{[}\partial_{\varphi}^{2}-4\tilde{k}r_{c}\partial_{\varphi}+\left(\mu_{n}^{(0)}\right)^{2}e^{2\tilde{k}r_{c}\varphi}\Big{]}\psi^{(2)}_{n}$
(248) $\displaystyle\hskip
35.0pt=2c_{1}e^{3\varphi\tilde{k}r_{c}}\Bigg{[}8\mu_{n}^{(0)}\,\varphi\,J_{1}\bigg{(}\frac{e^{\tilde{k}r_{c}\varphi}\mu_{n}^{(0)}}{\tilde{k}r_{c}}\bigg{)}-e^{\tilde{k}r_{c}\varphi}\Big{(}\delta\mu_{n}^{2}+2(\mu_{n}^{(0)})^{2}\varphi^{2}\Big{)}J_{2}\bigg{(}\frac{e^{\tilde{k}r_{c}\varphi}\mu_{n}^{(0)}}{\tilde{k}r_{c}}\bigg{)}\Bigg{]}$

Here $c_{1}$ corresponds to the normalization of the leading order
wavefunction $\psi_{n}^{(0)}$. Below we write down the perturbation of the
spin-2 wavefunction $\psi_{n}^{(2)}$. The resulting expressions at order
$\epsilon^{2}$ are quite lengthy and are provided also provided in
supplementary Mathematica files on GitHub.

$\displaystyle\psi_{n}^{(2)}=$
$\displaystyle\frac{c_{1}Y_{2}(z)}{384\mu_{n}^{4}}\epsilon^{2}\Bigg{\\{}\mu_{n}^{2}\pi
z^{6}\hskip
5.0pt\,_{3}F_{4}\left(\tfrac{3}{2},2,2;1,3,3,4;-z^{2}\right)\Big{[}1-4\log(\beta
z)\Big{]}$ $\displaystyle+2\pi z^{6}\hskip
5.0pt\,_{2}F_{3}\left(\tfrac{3}{2},2;1,3,4;-z^{2}\right)\Big{[}2\log(\beta
z)-1\Big{]}$ $\displaystyle+384\pi
z^{2}\bigg{[}\bigg{(}\,_{1}F_{2}\left(\tfrac{1}{2};1,1;-z^{2}\right)-2\hskip
5.0pt\,_{1}F_{2}\left(\tfrac{1}{2};1,2;-z^{2}\right)\bigg{)}\log(\beta
z)+\log(\beta)\bigg{]}$ $\displaystyle+\pi z^{6}\hskip
5.0pt\,_{4}F_{5}(\tfrac{3}{2},2,2,2;1,3,3,3,4;-z^{2})$
$\displaystyle+192\sqrt{\pi}z^{2}\left[G_{2,4}^{2,1}\Bigg{(}z^{2}\Bigg{|}\begin{array}[]{c}\frac{1}{2},1\\\
0,0,0,0\\\
\end{array}\Bigg{)}-2\,G_{2,4}^{2,1}\Bigg{(}z^{2}\Bigg{|}\begin{array}[]{c}\frac{1}{2},1\\\
0,0,-1,0\\\ \end{array}\Bigg{)}\right]$ (253)
$\displaystyle-48\pi\mu_{n}^{2}z^{4}J_{2}(z){}^{2}\bigg{[}4\beta^{2}\delta\mu_{n}^{2}+\bigg{(}2\Big{[}\log(\beta
z)-1\Big{]}\log(\beta z)+1\bigg{)}\bigg{]}$
$\displaystyle+48\pi\beta^{2}z^{4}J_{1}(z)J_{3}(z)\bigg{[}4\beta^{2}\delta\mu_{n}^{2}+\bigg{(}2\Big{[}\log(\beta
z)-1\Big{]}\log(\beta z)+1\bigg{)}\bigg{]}\Bigg{\\}}$
$\displaystyle+\frac{z^{2}J_{2}(z)}{8\mu_{n}^{2}}\Bigg{\\{}2\sqrt{\pi}\,G_{3,5}^{2,2}\Bigg{(}z,\tfrac{1}{2}\Bigg{|}\begin{array}[]{c}1,\frac{3}{2},\frac{1}{2}\\\
1,3,-1,0,\frac{1}{2}\\\
\end{array}\Bigg{)}\Big{[}2\beta^{2}\delta\mu_{n}^{2}+\log^{2}(\beta
z)\Big{]}$ (256)
$\displaystyle+2\mu_{n}^{2}\sqrt{\pi}\left[4\,G_{3,5}^{3,1}\Bigg{(}z,\tfrac{1}{2}\Bigg{|}\begin{array}[]{c}\frac{1}{2},-\frac{1}{2},1\\\
0,0,2,-1,-\frac{1}{2}\\\
\end{array}\Bigg{)}-G_{4,6}^{2,3}\Bigg{(}z,\tfrac{1}{2}\Bigg{|}\begin{array}[]{c}1,1,\frac{3}{2},\frac{1}{2}\\\
1,3,-1,0,0,\frac{1}{2}\\\ \end{array}\Bigg{)}\right]\log(\beta z)$ (261)
$\displaystyle+\mu_{n}^{2}\sqrt{\pi}\left[4\,G_{4,6}^{4,1}\Bigg{(}z,\tfrac{1}{2}\Bigg{|}\begin{array}[]{c}\frac{1}{2},-\frac{1}{2},1,1\\\
0,0,0,2,-1,-\frac{1}{2}\\\
\end{array}\Bigg{)}+G_{5,7}^{2,4}\Bigg{(}z,\tfrac{1}{2}\Bigg{|}\begin{array}[]{c}1,1,1,\frac{3}{2},\tfrac{1}{2}\\\
1,3,-1,0,0,0,\frac{1}{2}\\\ \end{array}\Bigg{)}\right]$ (266)
$\displaystyle+8\mu_{n}^{2}\log(z)\Big{[}2\log(\beta)+\log(z)\Big{]}\Bigg{\\}}+\beta^{2}z^{2}\Big{[}c_{3}J_{2}(z)+c_{4}Y_{2}(z)\Big{]}~{}.$
(267)

Here $\beta\equiv\tilde{k}r_{c}/\mu_{n}$,
$z\equiv(\mu_{n}/\tilde{k}r_{c})e^{\tilde{k}r_{c}\varphi}$, $G_{i,j}^{k,l}$
are Meijer-G functions gradshteyn2014table , ${}_{i}F_{j}$ are Hypergeometric
functions. $c_{1}$ is a normalization constant given in Eq. 240 and $c_{3}$
and $c_{4}$ are constants that are determined from the Neummann boundary
condition that the wavefunction must satisfy. Additionally, we provide these
expressions as well as those valid also for arbitrary values of
$\tilde{k}r_{c}$ in supplementary Mathematica files on GitHub.

Correction to the mass to order $\epsilon^{2}$ are calculated using Eq. (233).
In the large $kr_{c}$ limit is given in terms of Bessel functions and
hypergeometric functions as follows:

$\displaystyle\delta\mu_{n}^{2}=$
$\displaystyle-\frac{c_{1}^{2}\epsilon^{2}}{48\tilde{k}}\Bigg{\\{}\left(j_{1,n}\right)^{2}\Bigg{[}16\,J_{0}\left(j_{1,n}\right)^{2}\Big{(}6\pi\tilde{k}r_{c}-1\Big{)}-48\,J_{2}\left(j_{1,n}\right)^{2}\Big{(}2\pi\tilde{k}r_{c}(\pi\tilde{k}r_{c}-1)+1\Big{)}$
$\displaystyle+96\hskip
5.0pt\,_{3}F_{4}\Big{(}1,1,\tfrac{3}{2};2,2,2,2;-\left(j_{1,n}\right)^{2}\Big{)}-96\hskip
5.0pt\,_{3}F_{4}\Big{(}1,1,\tfrac{3}{2};2,2,2,3;-\left(j_{1,n}\right)^{2}\Big{)}$
$\displaystyle+\left(j_{1,n}\right)^{2}\hskip
5.0pt\,_{4}F_{5}\Big{(}\tfrac{3}{2},2,2,2;1,3,3,3,4;-\left(j_{1,n}\right)^{2}\Big{)}\Bigg{]}-32\Big{(}3\,J_{0}\left(j_{1,n}\right)^{2}-3\Big{)}\Bigg{\\}}\
.$ (268)

###  C.3 Spin-0 mass and wave functions in the DFGK model

In order to verify the sum rules to order $\epsilon^{2}$ we need the radion
mass squared and wavefunction corrections to order $\epsilon^{2}$. On the
other hand, due to the normalization condition in Eq. (17) in the stiff-wall
limit, the GW scalar wavefunctions do not have a $\epsilon^{0}$ piece, but
instead start at order $\epsilon$. Therefore it is only necessary to calculate
their masses and wavefunctions to leading order in $\epsilon$. In order to
determine the wavefunctions to the required order in $\epsilon$, we start with
the Sturm-Liouville problem for the scalar modes defined in Eq. (14), and
perform and expansion of the same up to order $\epsilon^{2}$ as follows

$\displaystyle\bigg{[}\partial_{\varphi}^{2}+2\tilde{k}r_{c}\partial_{\varphi}+\left(\mu_{(n)}^{(0)}\right)^{2}\,e^{2\tilde{k}r_{c}\varphi}\bigg{]}\gamma_{n}^{(0)}+\epsilon\bigg{[}\dfrac{4\sqrt{6}}{\phi_{1}}\partial_{\varphi}\bigg{]}\gamma_{n}^{(0)}$
$\displaystyle+\epsilon^{2}\Bigg{\\{}\bigg{[}4\varphi\,\partial_{\varphi}-4+e^{2\tilde{k}r_{c}\varphi}\bigg{(}\delta\mu^{2}_{(n)}+2\varphi^{2}\left(\mu^{(0)}_{(n)}\right)^{2}\bigg{)}\bigg{]}\gamma^{(0)}_{n}+\bigg{[}\partial_{\varphi}^{2}+2\tilde{k}r_{c}\,\partial_{\varphi}+e^{2\tilde{k}r_{c}\varphi}\left(\mu^{(0)}_{(n)}\right)^{2}\bigg{]}\gamma^{(2)}_{n}\Bigg{\\}}=0$
(269)

####  C.3.1 Radion wavefunction and mass to order $\epsilon^{2}$

We expand the wavefunction and mass in perturbation theory as described
earlier. For the massless radion we start the with the ansatz

$\displaystyle\gamma_{n}$
$\displaystyle=\gamma_{n}^{(0)}+\gamma_{n}^{(2)}+\cdots$
$\displaystyle\mu_{(n)}^{2}$
$\displaystyle=\left(\mu_{(n)}^{(0)}\right)^{2}+\delta\mu_{(n)}^{2}+\cdots$
(270)

Note the absence of order $\epsilon$ term in the expansion above although
there is an explicit order $\epsilon$ term in the differential equation. It is
easy to see that up to order $\epsilon$, the radion wavefunction is constant
and only acquires non-trivial dependence at order $\epsilon^{2}$. We can
substitute the above expansion into Eq. (269) and solving for
$\gamma_{n}^{(0)}$, $\gamma_{n}^{(2)}$, $(\mu_{(n)}^{(0)})^{2}$ and
$\delta\mu_{(n)}^{2}$ order by order. This amounts to determining the
unperturbed wavefunction and using Eq. (234) to determine the perturbed
wavefunction by solving the resulting non-homogeneous differential equation

$\displaystyle\bigg{[}\partial_{\varphi}^{2}+2\tilde{k}r_{c}\partial_{\varphi}+\Big{(}\delta\mu_{(0)}^{2}\,e^{2\tilde{k}r_{c}\varphi}-4\Big{)}\bigg{]}\gamma_{0}^{(0)}=0$
(271)

We find the normalized radion mass and wavefunction to order $\epsilon^{2}$ to
be

$\displaystyle\mu_{(0)}$ $\displaystyle=2\,\epsilon\,\sqrt{\frac{2}{1+e^{2\pi
kr_{c}}}}~{},$ (272) $\displaystyle\gamma_{0}=$
$\displaystyle\sqrt{\frac{\pi\tilde{k}r_{c}}{e^{2\pi\tilde{k}r_{c}}-1}}$
$\displaystyle+\epsilon^{2}\frac{\sqrt{\pi\left(e^{2\pi\tilde{k}r_{c}}-1\right)}}{6(\tilde{k}r_{c})^{3/2}\left(e^{4\pi\tilde{k}r_{c}}-1\right){}^{2}}\Bigg{[}-5+6e^{2\varphi\tilde{k}r_{c}}+3e^{2\pi\tilde{k}r_{c}}$
$\displaystyle-6e^{(2\varphi+4\pi)\tilde{k}r_{c}}-6\tilde{k}r_{c}\Big{(}e^{2\pi\tilde{k}r_{c}}+1\Big{)}^{2}\Big{(}e^{2\pi\tilde{k}r_{c}}(\pi^{2}\tilde{k}r_{c}-2\varphi+\pi)+2\varphi\Big{)}$
$\displaystyle+5e^{6\pi\tilde{k}r_{c}}-6e^{2(\pi-\varphi)\tilde{k}r_{c}}+6e^{2(3\pi-\varphi)\tilde{k}r_{c}}-3e^{4\pi\tilde{k}r_{c}}\Bigg{]}+\mathcal{O}(\epsilon^{3})\
.$ (273)

Note that that the expression is valid for arbitrary values of
$\tilde{k}r_{c}$. It is possible to simplify this expression further in the
large $\tilde{k}r_{c}$ limit to

$\displaystyle\gamma_{0}\bigg{|}_{\tilde{k}r_{c}\gg
1}=\gamma_{0}^{(0)}\left\\{1+\frac{\epsilon^{2}}{4\tilde{k}^{2}r_{c}^{2}}\left(-\tilde{k}r_{c}\left(\pi^{2}\tilde{k}r_{c}-2\varphi+\pi\right)+e^{-2\varphi\tilde{k}r_{c}}-e^{2(\varphi-\pi)\tilde{k}r_{c}}+\frac{5}{6}\right)\right\\}+\mathcal{O}(\epsilon^{3})$
(274)

In the large $\tilde{k}r_{c}$ limit, the wavefunction for the unstabilized
radion is $\gamma_{0}^{(0)}=e^{-\pi\tilde{k}r_{c}}\sqrt{\pi\tilde{k}r_{c}}$.

####  C.3.2 GW-scalar mass and wavefunction to leading order

As remarked upon earlier, in order to verify the sumrules to order
$\epsilon^{2}$, we need the GW-scalar wavefunction to order $\epsilon$, which,
due to the normalization condition in Eq. (17), is in fact the leading order
for the massive GW-scalars. Hence we only need to solve the leading term in
Eq. (269). The normalized GW-scalar wavefunction to order $\epsilon$ is

$\displaystyle\gamma_{i}=\dfrac{2\epsilon}{\mu_{(i)}}\,e^{-\tilde{k}r_{c}\varphi}\sqrt{\frac{\tilde{k}r_{c}\pi}{e^{2\pi\tilde{k}r_{c}}\,J_{1}\Big{(}\frac{e^{\pi\tilde{k}r_{c}}\mu_{(i)}}{\tilde{k}r_{c}}\Big{)}^{2}-J_{1}\Big{(}\frac{\mu_{(i)}}{\tilde{k}r_{c}}\Big{)}^{2}}}\hskip
5.0ptJ_{1}\bigg{(}\frac{e^{\tilde{k}r_{c}\varphi}\mu_{(i)}}{\tilde{k}r_{c}}\bigg{)}$
(275)

Looking at the normalization condition of Eq. (17), we see that there is no
$\epsilon^{0}$ piece. Here we have used the large $kr_{c}$ limit and omitted
Bessel-Y functions. The mass of GW-scalars are determined by solutions of the
following transcendental equation:

$\displaystyle
J_{2}\bigg{(}\frac{e^{\tilde{k}r_{c}\pi}\mu_{(i)}}{\tilde{k}r_{c}}\bigg{)}\,Y_{2}\bigg{(}\frac{\mu_{(i)}}{\tilde{k}r_{c}}\bigg{)}-J_{2}\bigg{(}\frac{\mu_{(i)}}{\tilde{k}r_{c}}\bigg{)}\,Y_{2}\bigg{(}\frac{e^{\pi\tilde{k}r_{c}}\mu_{(i)}}{\tilde{k}r_{c}}\bigg{)}=0~{}.$
(276)

In the large $kr_{c}$ limit, this reduces to the simple form

$\displaystyle
m_{(i)}=\dfrac{\mu_{(i)}}{r_{c}}=\tilde{k}e^{-\pi\tilde{k}r_{c}}\,j_{2,i}\ ,$
(277)

where $j_{2,i}$ are roots of $J_{2}$.

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