Proceeding now to Method 2 we obtain

$\lambda b_{4}|_{\mathrm{DS}}=\left(\frac{1}{8}{\bf L}^{2}-\frac{1}{4}{\bf
TE}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,1,1)})+\left(-\frac{1}{2}{\bf
L}^{2}+{\bf TE}-\frac{1}{8}{\bf
S}\right)\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega)\,.$
(150)

In addition the square envelope topology is given by

$\mathrm{SE}=\int\mathrm{d}\sigma_{2}\mathrm{d}\sigma_{3}\mathrm{d}\sigma_{4}\langle\zeta_{\sigma_{1}}\zeta_{\sigma_{2}}\rangle\langle\partial_{1}\xi_{\sigma_{1}}\partial_{1}\xi_{\sigma_{2}}\rangle\otimes\langle\partial_{1}\xi_{\sigma_{2}}\xi_{\sigma_{3}}\rangle\otimes\langle\partial_{1}\xi_{\sigma_{3}}\xi_{\sigma_{4}}\rangle\otimes\langle\partial_{1}\xi_{\sigma_{4}}\partial_{1}\xi_{\sigma_{1}}\rangle\,.$
(151)

Once the contractions are carried out, and basic identities applied, two
tensors only appear. The final result for the square envelopes evaluates to

$\displaystyle\lambda b_{4}|_{\mathrm{SE}}=$
$\displaystyle\frac{1}{2}\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega)[[k_{1}p_{1}(3k_{1}p_{1}p_{0}^{2}-3k_{0}p_{1}^{2}p_{0}+k_{1}p_{1}^{3}-k_{0}p_{0}^{3})]]_{3,1,1}$
$\displaystyle+\frac{1}{4}\operatorname{Tr}(\mathcal{H}^{(1,1,1,1)})[[k_{1}p_{1}^{2}(k_{1}p_{0}^{2}+k_{1}p_{1}^{2}-2k_{0}p_{0}p_{1})]]_{3,1,1}\,.$
(152)

Proceeding with Method 2 we have

$\lambda b_{4}|_{\mathrm{SE}}=\left(\frac{1}{4}{\bf TE}-\frac{1}{4}{\bf I}{\bf
T}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,1,1)})-\left(-{\bf
TE}+\frac{1}{4}{\bf I}{\bf L}+\frac{1}{8}{\bf S}+{\bf I}{\bf
T}\right)\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega)\,.$ (153)

In summation we obtain

$\displaystyle\lambda b_{4}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{8}{\bf L}^{2}-\frac{1}{4}{\bf I}{\bf
T}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,1,1)})-\left(-{\bf
TE}+\frac{1}{4}{\bf I}{\bf L}+\frac{1}{8}{\bf S}+{\bf I}{\bf
T}\right)\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega)$ (154)
$\displaystyle+\left(-\frac{1}{2}{\bf L}^{2}+{\bf TE}-\frac{1}{8}{\bf
S}\right)\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega)\,.$

### $b_{5}$

As it stands, $b_{5}$ contains tensors which, in total, are second-order in
derivatives. We are thus prompted to extract terms quadratic in the external
momentum $q$ from the loop integral. We have

$\displaystyle b_{5}$ $\displaystyle=\frac{i}{2}\langle{\cal B}^{[0]}_{1}{\cal
B}^{[0]}_{1}\rangle=\frac{i}{8}\int\mathrm{d}^{2}\sigma_{2}\mathcal{H}^{(1)}_{AB}(\sigma_{1})\mathcal{H}^{(1)}_{CD}(\sigma_{2})\langle\zeta_{\sigma_{1}}\zeta_{\sigma_{2}}\rangle\langle\partial_{1}\xi^{A}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{1}}\partial_{1}\xi^{C}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{2}}\rangle$
$\displaystyle=\frac{1}{4}\mathcal{H}_{AB}^{(1)}(\sigma_{1})\int\frac{\mathrm{d}^{2}q}{(2\pi)^{2}}\mathcal{H}_{CD}^{(1)}(q)e^{-iq\sigma_{1}}\int\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{p_{1}^{2}k_{1}^{2}\mathcal{H}^{AC}\mathcal{H}^{BD}+p_{1}p_{0}k_{1}k_{0}\eta^{AC}\eta^{BD}}{p^{2}k^{2}[q-(k+p)]^{2}}\,.$
(155)

Now, to second order, the denominator with external momentum insertion is
expanded as

$\frac{1}{(q-(k+p))^{2}}=\dots+4\frac{[(k+p)\cdot
q]^{2}}{[(k+p)^{2}]^{3}}-\frac{q^{2}}{[(k+p)^{2}]^{2}}+\dots\,.$ (156)

In the previous expression, we restrict ourselves to terms involving
$q_{0}^{2}$, as we want to concentrate on external legs $(\partial_{0}y)^{2}$.
In passing from momentum to position space, we turn $q_{0}^{2}$ into
$-\partial_{0}^{2}$ acting upon $\mathcal{H}_{CD}^{(1)}$. However, as a second
derivative would necessarily produce a derivative of $\Omega$, integration by
parts is in order. Specifically we let
$\mathcal{H}_{AB}^{(1)}\partial_{0}^{2}(\mathcal{H}_{CD}^{(1)})=-\partial_{0}(\mathcal{H}^{(1)}_{AB})\partial_{0}(\mathcal{H}^{(1)}_{CD})$.
We eventually arrive at

$\displaystyle b_{5}=$
$\displaystyle-\frac{1}{2}\left(\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega)-\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega)+2\operatorname{Tr}(\mathcal{H}^{(2,1)}\Omega)-\frac{1}{2}\operatorname{Tr}(\mathcal{H}^{(2,2)})\right)$
$\displaystyle\times\left(-[[p_{1}k_{1}(p_{0}k_{0}-p_{1}k_{1})]]_{1,1,2}+4[[(k_{0}+p_{0})^{2}p_{1}k_{1}(p_{0}k_{0}-p_{1}k_{1})]]_{1,1,3}\right)(\partial_{0}y)^{2}\,.$
(157)

With Method 2, the above expression boils down to

$b_{5}=\left(\frac{3}{4}{\bf S}+2{\bf
TE}\right)\left(\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega)-\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega)+2\operatorname{Tr}(\mathcal{H}^{(2,1)}\Omega)-\frac{1}{2}\operatorname{Tr}(\mathcal{H}^{(2,2)})\right)(\partial_{0}y)^{2}\,.$
(158)

### $b_{6}$

Although ${\cal
B}_{2}^{[0]}=-Y\partial_{\mu}\zeta\partial^{\mu}\zeta-\frac{1}{4}\mathcal{H}^{(2)}_{AB}\zeta^{2}\partial_{1}\xi^{A}\partial_{1}\xi^{B}-X_{AB}\partial_{1}\xi^{A}\partial_{1}\xi^{B}$
we can simplify the evaluation of $b_{6}=i\langle{\cal B}^{[1]}_{1}{\cal
B}_{2}^{[0]}\rangle$ by noting the term with $Y$ will not contribute (it is a
disconnected diagram) and the $\zeta$ loop on the term with
$\mathcal{H}^{(2)}$ is evaluated to $i{\bf I}$; hence effectively we use

${\cal B}_{2}^{[0]}=A_{AB}\partial_{1}\xi^{A}\partial_{1}\xi^{B}\,,\quad
A_{AB}=-\frac{i}{4}{\bf I}\mathcal{H}^{(2)}_{AB}-X_{AB}\,.$ (159)

The contraction gives

$\displaystyle b_{6}=$
$\displaystyle-\frac{i}{2}\Omega_{0AB}(\sigma_{1})A_{CD}(\sigma_{2})\langle\partial_{1}\xi^{A}_{\sigma_{1}}\xi^{B}_{\sigma_{1}}\partial_{1}\xi^{C}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{2}}\rangle$
$\displaystyle=$
$\displaystyle-\Omega_{0AB}(\sigma_{1})\int\frac{\mathrm{d}^{2}q}{(2\pi)^{2}}A_{CD}(q)e^{-iq\sigma_{1}}\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}(p-q)^{2}}$
$\displaystyle\times\biggl{(}p_{1}p_{1}(q-p)_{1}\mathcal{H}^{AC}\mathcal{H}^{BD}+p_{0}p_{1}(q-p)_{0}\eta^{AC}\eta^{BD}$
$\displaystyle+p_{1}p_{1}(q-p)_{0}\mathcal{H}^{AC}\eta^{BD}+p_{0}p_{1}(q-p)_{1}\eta^{AC}\mathcal{H}^{BD}\biggr{)}\,.$
(160)

To proceed one simply Taylor expands to extract the linear dependence on
$q_{0}$ (and $q_{1}$, even though we will omit that part here) from the
integrands. In this case it is not even really necessary to use the specific
rules for Method 2, for no $p_{0}^{2}$ appears once the dust settles. Still,
in the language of Method 2 we can rephrase the result as

$b_{6}=\frac{1}{4}{\bf L}({\bf I}-{\bf
P})\operatorname{Tr}(\mathcal{H}^{(2,1)}\Omega)\partial_{0}y\partial_{0}y-\frac{1}{8}{\bf
L}({\bf I}-{\bf
P})\operatorname{Tr}(\mathcal{H}^{(3,1)})\partial_{0}y\partial_{0}y\,.$ (161)

Note that the final contribution from this diagram to
$\partial_{0}y\partial_{1}y$ cancels out.

### $b_{7}$

This is by far the most complicated case and we shall provide the reader with
additional details. The Wick contractions are easily simplified exploiting the
symmetries of the tensorial structure

$\displaystyle b_{7}$
$\displaystyle=-\frac{1}{2}\langle\mathcal{B}_{1}^{[0]}\mathcal{B}_{1}^{[0]}\mathcal{B}_{1}^{[1]}\rangle$
$\displaystyle=\frac{1}{16}\mathcal{H}^{(1)}_{EF}(\sigma_{1})\int\mathrm{d}\sigma_{2}\mathrm{d}\sigma_{3}\,\mathcal{H}^{(1)}_{CD}(\sigma_{2})\Omega_{0AB}(\sigma_{3})\langle\partial_{1}\xi^{A}_{\sigma_{3}}\xi^{B}_{\sigma_{3}}\partial_{1}\xi^{C}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{1}}\partial_{1}\xi^{F}_{\sigma_{1}}\rangle\langle\zeta_{\sigma_{1}}\zeta_{\sigma_{2}}\rangle$
$\displaystyle=\frac{1}{2}\mathcal{H}^{(1)}_{EF}(\sigma_{1})\int\mathrm{d}\sigma_{2}\mathrm{d}\sigma_{3}\,\mathcal{H}^{(1)}_{CD}(\sigma_{2})\Omega_{0AB}(\sigma_{3})\Delta_{12}\partial_{12}\Delta_{12}^{CE}\partial_{23}\Delta_{23}^{AD}\partial_{1}\Delta_{13}^{BF}\,,$
(162)

where we have shortened the expression using
$\Delta_{ij}\equiv\Delta(\sigma_{i}-\sigma_{j})$ and
$\Delta_{ij}^{AB}\equiv\mathcal{H}^{AB}\Delta(\sigma_{i}-\sigma_{j})+\eta^{AB}\theta(\sigma_{i}-\sigma_{j})$.
The momentum routing is slightly subtle. First: two external momenta, $q$ and
$l$ (Fourier partners of $\sigma_{2}$ and $\sigma_{3}$), have to be
introduced. Second: we are entitled to choose the routing that will best suite
our purpose.

Killing every instance of $q_{1}$ and $l_{1}$ (they would eventually produce
some $\partial_{1}y$ leg) we arrive at

$\displaystyle b_{7}=$
$\displaystyle-\frac{i}{2}\mathcal{H}^{(1)}_{EF}(\sigma_{1})\int\frac{\mathrm{d}^{2}q}{(2\pi)^{2}}\Omega_{0AB}(q)e^{-iq\sigma_{1}}\int\frac{\mathrm{d}^{2}l}{(2\pi)^{2}}\mathcal{H}^{(1)}_{CD}(l)e^{-il\sigma_{1}}$
$\displaystyle\times\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{p_{1}(p_{1}+k_{1})}{k^{2}(p+k)^{2}(p+l)^{2}(p+l+q)^{2}}$
$\displaystyle\times\left((k_{1}+p_{1})\mathcal{H}^{CE}+(p_{0}+k_{0})\eta^{CE}\right)\left(p_{1}\mathcal{H}^{AD}+(p_{0}+l_{0})\eta^{AD}\right)\left(p_{1}\mathcal{H}^{BF}+(p+l+q)_{0}\eta^{BF}\right)\,.$
(163)

Judging from the tensorial structures already at our disposal, we need to
extract from the loop integral linear terms in either $q_{0}$ or $l_{0}$

$\displaystyle b_{7}=$
$\displaystyle-\frac{i}{2}\mathcal{H}^{(1)}_{EF}(\sigma_{1})\int\frac{\mathrm{d}^{2}q}{(2\pi)^{2}}\Omega_{0AB}(q)e^{-iq\sigma_{1}}\int\frac{\mathrm{d}^{2}l}{(2\pi)^{2}}\mathcal{H}^{(1)}_{CD}(l)e^{-il\sigma_{1}}$
$\displaystyle\times\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{1}{(p^{2})^{3}}\frac{1}{k^{2}}\frac{p_{1}(p_{1}+k_{1})}{(p+k)^{2}}\left((p_{1}+k_{1})\mathcal{H}^{CE}+(p_{0}+k_{0})\eta^{CE}\right)$
$\displaystyle\times\biggl{(}l_{0}\left[4p_{0}p_{1}^{2}\mathcal{H}^{AD}\mathcal{H}^{BF}+2p_{0}(p_{0}^{2}+p_{1}^{2})\eta^{AD}\eta^{BF}+\left(2p_{0}^{2}p_{1}+p_{1}(p_{0}^{2}+p_{1}^{2})\right)\left(\mathcal{H}^{AD}\eta^{BF}+\eta^{AD}\mathcal{H}^{BF}\right)\right]$
$\displaystyle+q_{0}\left[2p_{0}p_{1}^{2}\mathcal{H}^{AD}\mathcal{H}^{BF}+p_{0}(p_{0}^{2}+p_{1}^{2})\eta^{AD}\eta^{BF}+2p_{0}^{2}p_{1}\eta^{AD}\mathcal{H}^{BF}+p_{1}(p_{0}^{2}+p_{1}^{2})\mathcal{H}^{AD}\eta^{BF}\right]\biggr{)}\,.$
(164)

We shall now pass to position space and turn $l_{0},q_{0}$ into derivatives.
While the former would hit a generalised metric, the latter would act upon
$\Omega$. As we would like to avoid that situation we integrate by parts.
Massaging the expression we obtain a little further, dropping terms linear in
$p_{0}$ or $k_{0}$

$\displaystyle b_{7}=$
$\displaystyle\frac{1}{2}\Omega_{0AB}\left(\mathcal{H}_{EF}^{(1)}\partial_{0}\mathcal{H}^{(1)}_{CD}-\partial_{0}\mathcal{H}^{(1)}_{EF}\mathcal{H}^{(1)}_{CD}\right)\int\frac{\mathrm{d}^{2}p}{(2\pi)^{2}}\frac{\mathrm{d}^{2}k}{(2\pi)^{2}}\frac{1}{(p^{2})^{3}k^{2}(p+k)^{2}}$
$\displaystyle\times\biggl{[}p_{1}^{2}(p_{1}+k_{1})^{2}\left(2p_{0}^{2}\eta^{AD}\mathcal{H}^{BF}+(p_{0}^{2}+p_{1}^{2})\mathcal{H}^{AD}\eta^{BF}\right)\mathcal{H}^{CE}$
$\displaystyle+p_{1}p_{0}(p_{1}+k_{1})(p_{0}+k_{0})\left(2p_{1}^{2}\mathcal{H}^{AD}\mathcal{H}^{BF}+(p_{0}^{2}+p_{1}^{2})\eta^{AD}\eta^{BF}\right)\eta^{CE}\biggr{]}\,.$
(165)

.

We can now perform the tensor contractions. The crucial observation here is
that, when contracted, $\mathcal{H}_{AB}$ anti-commutes with both
$\mathcal{H}^{(1)}_{AB}$ and $\partial_{0}\mathcal{H}^{(1)}_{AB}$. To see
this, it is important to keep in mind that
$\mathcal{H}^{(1)}_{AB}\equiv\mathcal{V}_{A}{}^{I}\mathcal{H}^{(1)}_{IJ}\mathcal{V}_{B}{}^{J}$.
We obtain

$b_{7}=I\left(\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega)-\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega)+\operatorname{Tr}(\mathcal{H}^{(2,1)}\Omega)\right)\partial_{0}y\partial_{0}y\,,$
(166)

where $I$ stands for the integral we are left with, namely

$I=[[p_{1}(k_{1}+p_{1})\left(k_{0}\left(p_{0}^{3}+3p_{0}p_{1}^{2}\right)-3k_{1}p_{0}^{2}p_{1}-p_{1}^{3}(k_{1}+p_{1})+p_{0}^{4}\right)]]_{3,1,1}\,.$
(167)

Using Method 2 the latter becomes

$I=\frac{1}{4}{\bf S}-2{\bf TE}+\frac{1}{2}{\bf LI}+2{\bf TI}\,.$ (168)

##  Appendix C Base $\partial_{0}y\partial_{1}y$ Wick Contractions

In order to evaluate Wick contractions associated to the
$\partial_{0}y\partial_{1}y$ legs we single out the following terms in the
action

$\displaystyle\mathcal{C}_{1}^{[0]}$
$\displaystyle=-\frac{1}{2}\mathcal{H}^{(1)}_{AB}\zeta\partial_{1}\xi^{A}\partial_{1}\xi^{B}\,,$
(169) $\displaystyle\mathcal{C}_{2}^{[0]}$
$\displaystyle=-\frac{1}{4}\mathcal{H}^{(2)}_{AB}\zeta^{2}\partial_{1}\xi^{A}\partial_{1}\xi^{B}-Y\partial_{\mu}\zeta\partial^{\mu}\zeta-
X_{AB}\partial_{1}\xi^{A}\partial_{1}\xi^{B}\,,$ (170)
$\displaystyle\mathcal{C}_{3}^{[1;\tau]}$
$\displaystyle=-2Y^{(1)}\zeta\partial_{0}\zeta\partial_{0}y\,$ (171)
$\displaystyle\mathcal{C}_{3}^{[1;\sigma]}$
$\displaystyle=-\frac{1}{2}\mathcal{H}^{(2)}_{BC}\Omega_{1A}{}^{C}\zeta^{2}\xi^{A}\partial_{1}\xi^{B}+2Y^{(1)}\zeta\partial_{1}\zeta\partial_{1}y-2X_{BC}\Omega_{1A}{}^{C}\xi^{A}\partial_{1}\xi^{B}\,,$
(172) $\displaystyle\mathcal{C}_{2}^{[1;\sigma]}$
$\displaystyle=-\mathcal{H}^{(1)}_{BC}\Omega_{1A}{}^{C}\zeta\xi^{A}\partial_{1}\xi^{B}\,,$
(173) $\displaystyle\mathcal{C}_{1}^{[1;\sigma]}$
$\displaystyle=-\mathcal{H}_{BC}\Omega_{1A}{}^{C}\xi^{A}\partial_{1}\xi^{B}+\frac{1}{2}\Omega_{1AB}\xi^{A}\partial_{0}\xi^{B}\,,$
(174) $\displaystyle\mathcal{C}_{1}^{[1;\tau]}$
$\displaystyle=\frac{1}{2}\Omega_{0AB}\xi^{A}\partial_{1}\xi^{B}\,,$ (175)
$\displaystyle\mathcal{C}_{2}^{[2;\tau,\sigma]}$
$\displaystyle=\frac{1}{2}\Omega_{1AC}\Omega_{0B}{}^{C}\xi^{A}\xi^{B}\,,$
(176)

where $\mathcal{C}_{p}^{[n;\sigma^{\mu}]}$ indicates a term with $p$
derivatives, $n$ external legs of type $\sigma^{\mu}$. They can be used to be
create the following combinations

$\displaystyle c_{1}$
$\displaystyle=-\frac{i}{2}\langle\mathcal{C}_{2}^{[2;\tau,\sigma]}\mathcal{C}_{1}^{[0]}\mathcal{C}_{1}^{[0]}\rangle\,,\quad
c_{2}=-\langle\mathcal{C}_{2}^{[2;\tau,\sigma]}\mathcal{C}_{2}^{[0]}\rangle\,,\quad
c_{3}=\frac{1}{2}\langle\mathcal{C}_{1}^{[1;\tau]}\mathcal{C}_{1}^{[1;\sigma]}\mathcal{C}_{1}^{[0]}\mathcal{C}_{1}^{[0]}\rangle\,,$
(177) $\displaystyle c_{4}$
$\displaystyle=-i\langle\mathcal{C}_{1}^{[1;\tau]}\mathcal{C}_{1}^{[1;\sigma]}\mathcal{C}_{2}^{[0]}\rangle\,,\quad
c_{5}=-i\langle\mathcal{C}_{2}^{[1;\sigma]}\mathcal{C}_{1}^{[1;\tau]}\mathcal{C}_{1}^{[0]}\rangle\,,\quad
c_{6}=-\langle\mathcal{C}_{3}^{[1;\tau]}\mathcal{C}_{1}^{[1;\sigma]}\rangle\,.$
(178)

These should be supplemented with additional terms coming from integral with
external momenta insertion. The relevant basis of tensors is simply

$\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\,,\quad\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\,.$
(179)

### $c_{1}$

In this case

$\displaystyle c_{1}$
$\displaystyle=-\frac{1}{16}\mathcal{H}^{(1)}_{CD}\mathcal{H}^{(1)}_{EF}\Omega_{1AI}\Omega_{0B}{}^{I}\langle\zeta_{\sigma_{2}}\zeta_{\sigma_{3}}\rangle\langle\xi^{A}_{\sigma_{1}}\xi^{B}_{\sigma_{1}}\partial_{1}\xi^{C}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{3}}\partial_{1}\xi^{F}_{\sigma_{3}}\rangle$
$\displaystyle=-\frac{1}{2\lambda}[[k_{1}^{2}(p_{0}^{2}+p_{1}^{2})-2k_{1}k_{0}p_{1}p_{0}]]_{2,1,1}\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle=\frac{1}{2\lambda}\left({\bf LI}-\frac{1}{2}{\bf
S}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(180)

### $c_{2}$

In $c_{2}$ we discard a disconnected contraction and obtain

$\displaystyle c_{2}$
$\displaystyle=-\frac{i}{2}X_{CD}\Omega_{1AI}\Omega_{0B}{}^{I}\langle\xi^{A}_{\sigma_{1}}\xi_{\sigma_{2}}^{B}\partial_{1}\xi^{C}_{\sigma_{2}}\partial_{2}\xi^{D}_{\sigma_{2}}\rangle-\frac{i}{8}\mathcal{H}^{(2)}_{CD}\Omega_{1AI}\Omega_{0B}{}^{I}\langle\zeta_{\sigma_{2}}\zeta_{\sigma_{2}}\rangle\langle\xi^{A}_{\sigma_{1}}\xi^{B}_{\sigma_{1}}\partial_{1}\xi^{C}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{2}}\rangle$
$\displaystyle=-\frac{i}{2}\Omega_{1AI}\Omega_{0B}{}^{I}\left(X+\frac{i}{4}{\bf
I}\mathcal{H}^{(2)}\right)_{CD}\langle\xi^{A}_{1}\xi^{B}_{1}\partial_{1}\xi^{C}_{2}\partial_{1}\xi^{D}_{2}\rangle$
$\displaystyle=\frac{1}{4\lambda}({\bf I}-{\bf
P})[[p_{0}^{2}-p_{1}^{2}]]_{2,0,0}\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle\phantom{=}-\frac{1}{2\lambda}\left({\bf
I}[[p_{1}^{2}]]_{2,0,0}+\frac{1}{2}{\bf
P}[[p_{0}^{2}-p_{1}^{2}]]_{2,0,0}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}\partial_{1}y$
$\displaystyle=-\frac{1}{2\lambda}{\bf I}\left(\bf{L}+\frac{1}{2}{\bf
P}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y+\frac{1}{4\lambda}{\bf
I}\left(\bf{I}-{\bf
P}\right)\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(181)

### $c_{3}$

This combination comprises of square envelope and diamond sunset topologies,
making it the hardest to compute. However, one can show that the diamond
sunset topology does not ultimately contribute to the final result, as the
corresponding integral always contains odd powers of 0-components.

$\displaystyle c_{3}$
$\displaystyle=-\frac{i}{16}\Omega_{0AB}\mathcal{H}^{(1)}_{CD}\mathcal{H}^{(1)}_{EF}\mathcal{H}_{HI}\Omega_{1}{}^{I}{}_{G}\langle\xi^{G}_{\sigma_{1}}\partial_{1}\xi^{H}_{\sigma_{1}}\xi^{A}_{\sigma_{2}}\partial_{1}\xi^{B}_{\sigma_{2}}\partial_{1}\xi^{C}_{\sigma_{3}}\partial_{1}\xi^{D}_{\sigma_{3}}\partial_{1}\xi^{E}_{\sigma_{4}}\partial_{1}\xi^{F}_{\sigma_{4}}\rangle\langle\zeta_{\sigma_{3}}\zeta_{\sigma_{4}}\rangle$
$\displaystyle\phantom{=}-\frac{i}{32}\mathcal{H}^{(1)}_{CD}\mathcal{H}^{(1)}_{EF}\Omega_{1GH}\Omega_{0AB}\langle\xi^{G}_{\sigma_{1}}\partial_{0}\xi^{H}_{\sigma_{1}}\xi^{A}_{\sigma_{2}}\partial_{1}\xi^{B}_{\sigma_{2}}\partial_{1}\xi^{C}_{\sigma_{3}}\partial_{1}\xi^{D}_{\sigma_{3}}\partial_{1}\xi^{E}_{\sigma_{4}}\partial_{1}\xi^{F}_{\sigma_{4}}\rangle\langle\zeta_{\sigma_{3}}\zeta_{\sigma_{4}}\rangle$
$\displaystyle=\frac{1}{\lambda}[[(k_{1}^{2}(p_{0}^{2}+2p_{1}^{2})-k_{0}k_{1}p_{0}p_{1})(p_{0}^{2}-p_{1}^{2})+4k_{1}p_{0}p_{1}^{2}(k_{0}p_{1}-k_{1}p_{0})]]_{3,1,1}\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle=\frac{1}{\lambda}\left(\frac{3}{4}{\bf S}-2{\bf
TE}-\frac{1}{2}{\bf LI}+2{\bf
TI}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(182)

### $c_{4}$

In this case the topologies involved are triangle (with counter-term
insertion) and decorated triangle.

$\displaystyle c_{4}$
$\displaystyle=\frac{1}{2}\mathcal{H}_{DI}\Omega_{1}{}^{I}{}_{C}X_{EF}\Omega_{0AB}\langle\xi^{C}_{\sigma_{1}}\xi^{A}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{3}}\partial_{1}\xi^{F}_{\sigma_{3}}\rangle$
$\displaystyle\phantom{=}+\frac{1}{8}\mathcal{H}^{(2)}_{EF}\mathcal{H}_{DI}\Omega_{1}{}^{I}{}_{C}\Omega_{0AB}\langle\zeta_{\sigma_{3}}^{2}\rangle\langle\xi_{\sigma_{1}}^{C}\xi_{\sigma_{2}}^{A}\partial_{1}\xi^{D}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{3}}\partial_{1}\xi^{F}_{\sigma_{3}}\rangle$
$\displaystyle\phantom{=}+\frac{1}{4}X_{EF}\Omega_{1CD}\Omega_{0AB}\langle\xi_{\sigma_{1}}^{C}\xi_{\sigma_{2}}^{A}\partial_{0}\xi^{D}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{3}}\partial_{1}\xi^{F}_{\sigma_{3}}\rangle$
$\displaystyle\phantom{=}+\frac{1}{16}\mathcal{H}^{(2)}_{EF}\Omega_{1CD}\Omega_{0AB}\langle\zeta_{\sigma_{3}}^{2}\rangle\langle\xi_{\sigma_{1}}^{C}\xi_{\sigma_{2}}^{A}\partial_{0}\xi^{D}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{3}}\partial_{1}\xi^{F}_{\sigma_{3}}\rangle$
$\displaystyle=\frac{1}{\lambda}\left(\frac{1}{2}{\bf
P}[[(p_{0}^{2}-p_{1}^{2})^{2}]]_{3,0,0}-\frac{1}{2}{\bf
I}[[p_{1}^{2}(3p_{1}^{2}+p_{0}^{2})]]_{3,0,0}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle\phantom{=}+\frac{1}{2\lambda}({\bf P}-{\bf
I})[[(p_{0}^{2}-p_{1}^{2})^{2}]]_{3,0,0}\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle=\frac{{\bf I}}{2\lambda}\left({\bf P}-{\bf L}-4{\bf
T}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y+\frac{{\bf
I}}{2\lambda}({\bf P}-{\bf
I})\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(183)

### $c_{5}$

The only topology involved here are triangle envelopes.

$\displaystyle c_{5}$
$\displaystyle=\frac{1}{4}\mathcal{H}^{(1)}_{BL}\mathcal{H}^{(1)}_{EF}\Omega_{1}{}^{L}{}_{A}\Omega_{0CD}\langle\zeta_{\sigma_{1}}\zeta_{\sigma_{3}}\rangle\langle\xi^{A}_{\sigma_{1}}\xi^{C}_{\sigma_{2}}\partial_{1}\xi^{B}_{\sigma_{1}}\partial_{1}\xi^{F}_{\sigma_{2}}\partial_{1}\xi^{E}_{\sigma_{3}}\partial_{1}\xi^{F}_{\sigma_{3}}\rangle$
$\displaystyle=-\frac{1}{\lambda}[[k_{1}^{2}(p_{0}^{2}+p_{1}^{2})-2k_{0}k_{1}p_{0}p_{1}]]_{2,1,1}\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle=\frac{1}{\lambda}\left({\bf LI}-\frac{1}{2}{\bf
S}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(184)

### $c_{6}$

This one contains only loop and decorated loop topologies.

$\displaystyle c_{6}$
$\displaystyle=iX_{DE}\Omega_{0AB}\Omega_{1}^{E}{}_{C}\langle\xi^{C}_{\sigma_{1}}\xi^{A}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{2}}\rangle+\frac{i}{4}\mathcal{H}^{(2)}_{DI}\Omega_{1}{}^{I}{}_{C}\Omega_{0AB}\langle\zeta_{\sigma_{1}}\zeta_{\sigma_{1}}\rangle\langle\xi^{C}_{\sigma_{1}}\xi^{A}_{\sigma_{2}}\partial_{1}\xi^{D}_{\sigma_{1}}\partial_{1}\xi^{B}_{\sigma_{2}}\rangle$
$\displaystyle=-\frac{1}{2\lambda}\left({\bf P}[[p_{0}^{2}]]_{2,0,0}+{\bf
I}[[p_{1}^{2}]]_{2,0,0}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle\phantom{=}+\frac{1}{2\lambda}[[p_{0}^{2}]]_{2,0,0}({\bf I}-{\bf
P})\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle=-\frac{1}{2\lambda}\left({\bf IP}+\bf{LI}+{\bf
LP}\right)\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega)\partial_{0}y\partial_{1}y+\frac{1}{2\lambda}({\bf
I}+{\bf L})({\bf I}-{\bf
P})\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(185)

### Contributions From External Momentum Insertion

There are a few contributors to $\partial_{0}y\partial_{1}y$ legs that
$c_{1},\dots,c_{6}$ have missed. These arise from loop integral with non-
vanishing external momentum. A careful analysis reveals that in fact only the
sunset topology with a linear insertion of external momentum is responsible
for such contributions. Also, we shall only get a correction, call it $c_{7}$,
to $\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)$. We find

$\displaystyle c_{7}$
$\displaystyle=\frac{2}{\lambda}[[k_{1}p_{0}(k_{1}p_{0}-k_{0}p_{1})]]_{2,1,1}\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y$
$\displaystyle=\frac{1}{\lambda}\left(\frac{3}{2}\bf
S-\bf{LI}\right)\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega)\partial_{0}y\partial_{1}y\,.$
(186)

##  Appendix D Results of Wick Contractions

Before we evaluate the loop integrals, we collect and present in this section
the results of the standard Wick contractions. To slightly reduce the length
of expressions, each time a numerator of a loop integrand contains an even
number of $\tau$-components of momenta, these are swapped for the invariant
combination e.g. $p_{0}^{2}\rightarrow p^{2}+p_{1}^{2}$, though at this stage
no assumption about how this holds in $d=2+\epsilon$ is made and, in
particular, no cancellations of factors of $p^{2}$ between numerator and
denominator are employed up to this point.

For the counter-term attached to
$\lambda^{-2}\partial_{1}\mathbb{X}^{I}\partial_{1}\mathbb{X}^{J}$ we obtain

$\displaystyle\mathcal{H}^{(4)}_{\bullet\bullet}:\quad$
$\displaystyle\frac{1}{16}{\bf I}^{2}-\frac{1}{8}{\bf I}{\bf P}$
$\displaystyle\mathcal{H}^{(3,1,0)}_{\bullet\bullet}:\quad$
$\displaystyle\frac{1}{2}({\bf P}-{\bf
I})[[p_{1}^{2}]]_{2,0,0}-\frac{1}{4}{\bf I}{\bf P}$
$\displaystyle\mathcal{H}^{(2,0,2)}_{\bullet\bullet}:\quad$
$\displaystyle\frac{1}{4}{\bf I}{\bf P}+\frac{1}{4}[[p_{1}^{2}]]_{1,1,1}$
$\displaystyle\mathcal{H}^{(2,1,1)}_{\bullet\bullet}:\quad$
$\displaystyle\frac{1}{2}{\bf I}{\bf P}-\frac{1}{2}{\bf
P}[[p_{1}^{2}]]_{2,0,0}+[[p_{1}k_{1}p\cdot k]]_{2,1,1}$
$\displaystyle\mathcal{H}^{(1,2,1)}_{\bullet\bullet}:\quad$
$\displaystyle\frac{1}{8}{\bf I}{\bf P}-\frac{1}{2}{\bf
P}[[p_{1}^{2}]]_{2,0,0}-\frac{1}{2}[[p_{1}^{2}k_{1}^{2}]]_{2,2,0}-\frac{1}{4}({\bf
P}-{\bf I})[[p_{1}^{2}p^{2}]]_{3,0,0}$
$\displaystyle\mathcal{H}^{(1,1,1,1,0)}_{\bullet\bullet}:\quad$
$\displaystyle\frac{1}{2}{\bf
P}[[p_{1}^{2}]]_{2,0,0}+[[p_{1}^{2}k_{1}^{2}]]_{2,2,0}+\frac{1}{2}{\bf
I}[[p_{1}^{4}]]_{3,0,0}+\frac{1}{4}{\bf P}[[p_{1}^{2}p^{2}]]_{3,0,0}$
$\displaystyle-[[p_{1}^{3}k_{1}k\cdot
p]]_{3,1,1}+\frac{1}{2}[[p_{1}^{2}k_{1}^{2}p^{2}]]_{3,1,1}-[[(p_{1}^{2}k_{1}^{2}+p_{1}^{3}k_{1})k^{2}]]_{2,2,1}$
$\displaystyle\mathcal{H}^{(2)}_{\bullet\bullet}\operatorname{Tr}(\mathcal{H}^{(1,1)}):\quad$
$\displaystyle-\frac{1}{32}{\bf
P}[[p^{2}]]_{2,0,0}+\frac{1}{8}[[(p+k)_{1}k_{1}(p\cdot k+k^{2})]]_{2,1,1}$
$\displaystyle\mathcal{H}^{(1,1,0)}_{\bullet\bullet}\operatorname{Tr}(\mathcal{H}^{(1,1)}):\quad$
$\displaystyle\frac{1}{16}{\bf
P}[[p_{1}^{2}p^{2}]]_{3,0,0}-\frac{1}{4}[[(p_{1}+k_{1})p_{1}^{2}k_{1}(p\cdot
k+k^{2})]]_{3,1,1}$ (187)

For counter-term attached to $\partial_{0}y\partial_{0}y$ external legs we
find:

$\displaystyle\operatorname{Tr}(\mathcal{H}^{(2,2)}):\quad$
$\displaystyle[[k_{1}(k_{1}+p_{1})(p^{2}+p_{1}^{2})(k^{2}+k\cdot
p)]]_{3,1,1}-\frac{1}{4}[[k_{1}(k_{1}+p_{1})(k^{2}+k\cdot p)]]_{2,1,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(3,1)}):\quad$
$\displaystyle\frac{1}{8}\left({\bf P}-{\bf I}\right)[[p_{1}^{2}]]_{2,0,0}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1,1,1,1)}):\quad$
$\displaystyle\frac{1}{8}{\bf P}[[p_{1}^{2}p^{2}]]_{3,0,0}+\frac{1}{4}{\bf
I}[[p_{1}^{4}]]_{3,0,0}+\frac{1}{4}[[k_{1}p_{1}^{2}(k_{1}p^{2}-2p_{1}k\cdot
p)]]_{3,1,1}$ $\displaystyle-\frac{1}{16}[[k_{1}^{2}p^{2}-4p_{1}k_{1}k\cdot
p+3p_{1}^{2}k^{2}]]_{2,2,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega):\quad$
$\displaystyle\frac{1}{4}{\bf
I}[[p_{1}^{2}(3p^{2}+4p_{1}^{2})]]_{3,0,0}+\frac{1}{2}[[k_{1}(k\cdot
p(k_{1}-p_{1})+k^{2}(k_{1}+p_{1}))]]_{2,1,1}$
$\displaystyle-\frac{1}{2}[[k_{1}(2p^{2}k_{1}p_{1}^{2}+4k^{2}(k_{1}+p_{1})(p^{2}+p_{1}^{2})+k\cdot
p(4k_{1}p_{1}^{2}+p^{2}(4k_{1}+p_{1})))]]_{3,1,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega):\quad$
$\displaystyle-\frac{1}{2}[[k_{1}(k\cdot
p(k_{1}-p_{1})+k^{2}(k_{1}+p+1))]]_{2,1,1}$ $\displaystyle
2[[k_{1}(k_{1}(k\cdot p+k^{2})p^{2}+p_{1}k^{2}p^{2}+k_{1}p_{1}^{2}(k\cdot
p+k^{2}+p^{2})+p_{1}^{3}(k^{2}-k\cdot p))]]_{3,1,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(2,1)}\Omega):\quad$
$\displaystyle\frac{1}{4}({\bf P}-{\bf
I})[[p_{1}^{2}]]_{2,0,0}-\frac{1}{4}({\bf P}-{\bf
I})[[p_{1}^{2}p^{2}]]_{3,0,0}+[[k_{1}(k^{2}(k_{1}+p_{1})+k_{1}k\cdot
p)]]_{2,1,1}$
$\displaystyle-2[[k_{1}(k_{1}p_{1}^{2}p^{2}+2(k_{1}+p_{1})k^{2}(p^{2}+p_{1}^{2})+(2k_{1}p_{1}^{2}+2k_{1}p^{2}+p_{1}p^{2})k\cdot
p)]]_{3,1,1}$ (188)

For the counter-term attached to mixed derivatives of the background field
$\partial_{0}y\partial_{1}y$ we obtain:

$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1,1,0)}\Omega\Omega):\quad$
$\displaystyle-\frac{1}{4}{\bf P}[[3p^{2}-2p_{1}^{2}]]_{2,0,0}-{\bf
I}[[p_{1}^{2}]]_{2,0,0}+\frac{1}{2}{\bf I}[[(p^{2})^{2}]]_{3,0,0}$
$\displaystyle-\frac{1}{2}{\bf
I}[[p^{2}p_{1}^{2}-4p_{1}^{4}]]_{3,0,0}-\frac{3}{2}[[k_{1}^{2}p^{2}-2k_{1}p_{1}k\cdot
p]]_{2,1,1}$ $\displaystyle+[[k_{1}(k_{1}(p^{2})^{2}+4p_{1}^{3}k\cdot
p-p_{1}p^{2}(k\cdot p+2k_{1}p_{1}))]]_{3,1,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(2)}\Omega\Omega):\quad$
$\displaystyle-\frac{1}{4}({\bf P}-{\bf
I})[[3p^{2}+2p_{1}^{2}]]_{2,0,0}+\frac{1}{2}({\bf P}-{\bf
I})[[(p^{2})^{2}]]_{3,0,0}$
$\displaystyle+2[[k_{1}(k_{1}(p^{2})^{2}-2k_{1}p_{1}^{2}p^{2}+4p_{1}^{3}k\cdot
p)]]_{3,1,1}$ (189)

For the counter-term attached to $\partial_{1}y\partial_{1}y$ we obtain

$\displaystyle\operatorname{Tr}(\mathcal{H}^{(2,2)}):\quad$
$\displaystyle[[k_{1}(k_{1}+p_{1})p_{1}^{2}(k^{2}+k\cdot
p)]]_{3,1,1}+\frac{1}{4}[[k_{1}(k_{1}+p_{1})(k^{2}+k\cdot p)]]_{2,1,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(3,1)}):\quad$
$\displaystyle\frac{1}{8}{\bf I}^{2}-\frac{1}{8}{\bf
P}[[p^{2}-p_{1}^{2}]]_{2,0,0}+\frac{3}{8}{\bf
I}[[p_{1}^{2}]]_{2,0,0}-\frac{1}{2}{\bf P}[[p_{1}^{2}p^{2}]]_{3,0,0}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1,1,1,1)}):\quad$
$\displaystyle\frac{1}{8}{\bf P}[[p^{2}+2p_{1}^{2}]]_{2,0,0}+\frac{1}{4}{\bf
I}[[p_{1}^{2}]]_{2,0,0}-\frac{1}{8}{\bf
P}[[p_{1}^{2}p^{2}+(p^{2})^{2}]]_{3,0,0}+\frac{1}{4}{\bf
I}[[p_{1}^{4}]]_{3,0,0}$
$\displaystyle+\frac{1}{4}[[k_{1}^{2}+k_{1}p_{1}]]_{1,1,1}+\frac{1}{4}[[k_{1}^{2}p^{2}-2p_{1}(p_{1}+2k_{1})k\cdot
p]]_{2,1,1}$ $\displaystyle-\frac{1}{4}[[k_{1}(k_{1}p^{2}-2p_{1}k\cdot
p)(p^{2}-p_{1}^{2})]]_{3,1,1}+\frac{1}{16}[[k_{1}^{2}p^{2}(k\cdot
p+k_{1}p_{1})]]_{2,2,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1,1)}\Omega\Omega):\quad$
$\displaystyle\frac{1}{4}{\bf I}[[p^{2}+6p_{1}^{2}]]_{2,0,0}-\frac{1}{4}{\bf
I}[[(p^{2})^{2}+p_{1}^{2}p^{2}-4p_{1}^{4}]]_{3,0,0}$
$\displaystyle-\frac{1}{2}[[\frac{(k_{1}+p_{1})(p^{2}+k\cdot
p)}{p_{1}}]]_{1,1,1}$
$\displaystyle-\frac{1}{2}[[\frac{k_{1}}{p_{1}}(-2p^{2}k\cdot
p+k_{1}p_{1}(k\cdot p+k^{2}-3p^{2})+5p_{1}^{2}(k^{2}+k\cdot p))]]_{2,1,1}$
$\displaystyle-\frac{1}{2}[[\frac{k_{1}}{p_{1}}((p^{2})^{2}k\cdot
p+2p_{1}k_{1}(p^{2})^{2}+p_{1}^{2}p^{2}k\cdot p]]_{3,1,1}$
$\displaystyle-[[\frac{k_{1}}{p_{1}}(k_{1}p_{1}^{3}(p^{2}+2k\cdot
p+2k^{2})+2k^{2}p_{1}^{4})]]_{3,1,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(1)}\Omega\mathcal{H}^{(1)}\Omega):\quad$
$\displaystyle-[[p^{2}+k\cdot p]]_{1,1,1}$
$\displaystyle+\frac{1}{2}[[(4p_{1}^{2}-k_{1}p_{1}+k_{1}^{2}-2p^{2})k\cdot
p+k_{1}(k^{2}k_{1}+5p_{1}k^{2}-2p_{1}p^{2})]]_{2,1,1}$
$\displaystyle+2[[k_{1}p_{1}(p^{2}k\cdot p)+k_{1}p_{1}(k\cdot
p+k^{2}+p^{2})+(k^{2}-k\cdot p)p_{1}^{2}]]_{3,1,1}$
$\displaystyle-\frac{1}{8}[[k_{1}p_{1}(-2k^{2}p^{2}+k_{1}^{2}p^{2}+7p_{1}^{2}k^{2})]]_{2,2,1}$
$\displaystyle-\frac{1}{8}[[k\cdot
p(k_{1}^{2}(3p^{2}-8p_{1}^{2})+k^{2}(-2p^{2}+5p_{1}^{2}))]]_{2,2,1}$
$\displaystyle\operatorname{Tr}(\mathcal{H}^{(2,1)}\Omega):\quad$
$\displaystyle-\frac{1}{4}{\bf P}[[2p^{2}+p_{1}^{2}]]_{2,0,0}+\frac{1}{4}{\bf
I}[[p^{2}+5p_{1}^{2}]]_{2,0,0}$ $\displaystyle-\frac{1}{4}({\bf P}-{\bf
I})[[(p^{2})^{2}]]_{3,0,0}-\frac{3}{4}{\bf
P}[[p^{2}p_{1}^{2}]]_{3,0,0}-\frac{1}{4}{\bf I}[[p_{1}^{2}p^{2}]]_{3,0,0}$
$\displaystyle+[[k_{1}(k^{2}(k_{1}+p_{1})+k_{1}k\cdot p)]]_{2,1,1}$
$\displaystyle-2[[k_{1}(k_{1}p_{1}^{2}p^{2}+2(k_{1}+p_{1})k^{2}(p^{2}+p_{1}^{2})+(2k_{1}p_{1}^{2}+2k_{1}p^{2}+p_{1}p^{2})k\cdot
p)]]_{3,1,1}$ (190)

##  Appendix E Loop Integrals via $O(d)$-invariance (Method 1)

For a $L$-loop calculation, $2L$ copies of the worldsheet are needed. We shall
label each copy by a number and indicate the $n$-th with $\mathbf{n}$.
Propagators stretching from $\mathbf{n}$ to $\mathbf{m}$ will be reported
schematically as $\mathbf{n}\rightarrow\mathbf{m}$, where $\mathbf{n}$ and
$\mathbf{m}$ are allowed to coincide. Accordingly, a sequence with the same
extrema shall indicate a closed loop.

At one-loop order, two topologies only contribute to the $\beta$-function:
having two copies of the worldsheet, propagators can either stretch from
$\mathbf{1}$ to $\mathbf{2}$ or close on the same copy. All possible loops are
exhausted by $\mathbf{1}\rightarrow\mathbf{1}$ and
$(\mathbf{1}\rightarrow\mathbf{2})^{2}$. We shall name them bubble and loop,
respectively.

At two-loop order, divergences can originate from either i) one-loop integrals
multiplied by a $1/\epsilon$ pole due to one-loop counter-term insertions; ii)
products of one-loop integrals; iii) genuinely new two-loop integrals. New
topologies for one-loop diagrams arise in i) and ii).

For a two-loop calculation, triangle-shaped and square-shaped loops,
corresponding schematically to
$\mathbf{1}\rightarrow\mathbf{2}\rightarrow\mathbf{3}\rightarrow\mathbf{1}$
and
$\mathbf{1}\rightarrow\mathbf{2}\rightarrow\mathbf{3}\rightarrow\mathbf{4}\rightarrow\mathbf{1}$,
are in fact allowed. For the sake of simplicity, we will refer to them as
triangle and square diagrams. On top of that,
$\mathbf{1}\rightarrow\mathbf{1}$ and $(\mathbf{1}\rightarrow\mathbf{2})^{2}$
can still appear, even though their finite $O(\epsilon^{0})$ parts are now to
be kept252525Actually, square diagrams are only required in calculating one-
loop diagrams with external quantum fields. .

Graphs of type ii) can all be seen graphically as dressings of the previous
diagrams (up to the squares) with an extra loop or bubble. When a bubble is
added, we call the resulting diagram “decorated”: for example, adding a bubble
to a triangle results in a decorated triangle.

Scenario iii) is the most intricate as it allows for many different
topologies. The simplest instance of genuine two-loop diagram is the sunset
diagram, corresponding to $(\mathbf{1}\rightarrow\mathbf{2})^{3}$. The only
way to non-trivially extend the triangle diagram to two-loop is by adding an
internal line, resulting in
$(\mathbf{1}\rightarrow\mathbf{2})^{2}\rightarrow\mathbf{3}\rightarrow\mathbf{1}$.
We shall call it triangle envelope diagram. Squares allow for two extensions:
we can either add a line joining two adjacent vertices,
$(\mathbf{1}\rightarrow\mathbf{2})^{2}\rightarrow\mathbf{3}\rightarrow\mathbf{4}$
(square envelope), or let is stretch to the opposite vertex,
$(\mathbf{1}\rightarrow\mathbf{2}\rightarrow\mathbf{3}\rightarrow\mathbf{4}\rightarrow\mathbf{1})(\mathbf{1}\rightarrow\mathbf{3})$
(diamond sunset).

$1$$k$ (a) Bubble

$1$$2$$k$$-k$ (b) Loop

$1$$2$$3$$k$$k$$-k$ (c) Triangle

$1$$2$$3$$4$$k$$k$$k$$-k$ (d) Square

Figure 1: One-loop diagrams.

###  E.1 Combinatorics

The basic integrals in Minkowski space are given by

$I_{n}=\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-m^{2})^{n}}=\frac{(-1)^{n}i}{(4\pi)^{\frac{d}{2}}}\frac{\Gamma(n-\frac{d}{2})}{\Gamma(n)}(m^{2})^{\frac{d}{2}-n}\,.$
(191)

Our aim is to recast divergent 1-loop and 2-loop integrals as combinations of
the basic integrals $I_{n}$, postponing the evaluation of their precise
dependence on the dimensional-regulator $\epsilon$. At 2-loop order we require
only $I_{1},I_{2}$ and $I_{3}$ which are given by

$I_{1}\approx\frac{i}{2\pi\epsilon}+\frac{i\bar{\gamma}}{4\pi}+\frac{i\epsilon\left(6\bar{\gamma}^{2}+\pi^{2}\right)}{96\pi}\,,\quad
m^{2}I_{2}\approx+\frac{i}{4\pi}+\frac{i\epsilon\bar{\gamma}}{8\pi}\,,\quad
m^{4}I_{3}\approx-\frac{i}{8\pi}-\frac{i\epsilon\left(\bar{\gamma}-1\right)}{16\pi}\,,$
(192)

where

$\bar{\gamma}=\gamma_{E}+\log\left(\frac{m^{2}}{4\pi}\right)\,.$ (193)

Integrals with non-positive $n$ can also be dropped: $\Gamma(n)$ has poles for
$n\in\mathbb{Z}^{-}$, thus forcing $I_{n\leq 0}\rightarrow 0$.

Prior to venturing into the explicit evaluation of loop integrals, let us
pause and analyse their combinatorial structure. Fix some integer number
$q\in\mathbb{N}$ and consider the one-loop integral

$I_{\lambda_{1}\dots\lambda_{2q}}=\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{p_{\lambda_{1}}\dots
p_{\lambda_{2q}}}{D}$ (194)

for some denominator $D=D(p^{2})$. Integrals of this form are usually
evaluated assuming $O(d)$ symmetry in the final result; that is, we postulate
that the right-hand side can be recast in the form

$\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{p_{\lambda_{1}}\dots
p_{\lambda_{2q}}}{D}=A(\eta_{\lambda_{1}\lambda_{2}}\dots\eta_{\lambda_{2q-1}\lambda_{2q}}+\mathrm{perms.})\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{(p^{2})^{q}}{D}\,,$
(195)

where the “scalar” integral is in general known, and the prefactor $A$ is to
be fixed by taking appropriate contractions of both the left- and right-hand
side with the $d$-dimensional Minkowski metric.

Let us indicate by $P_{\lambda_{1}\dots\lambda_{2q}}$ the rank $2q$ totally
symmetric tensor made up of all possible permutations of tensor products of
the $d$-dimensional Minkowski metric

$P_{\lambda_{1}\dots\lambda_{2q}}=\eta_{\lambda_{1}\lambda_{2}}\dots\eta_{\lambda_{2q-1}\lambda_{2q}}+\mathrm{perms.}$
(196)

As can be easily seen, a total number of $(2q-1)!!$ terms have to be accounted
for in the permutations. Particularly important is the trace of this object,
$P^{(2q)}$,

$P^{(2q)}\equiv\eta^{\lambda_{1}\lambda_{2}}\dots\eta^{\lambda_{2q-1}\lambda_{2q}}P_{\lambda_{1}\dots\lambda_{2q}}=\prod_{j=0}^{q-1}(d+2j)=2^{q-1}d\left(1+\frac{d}{2}\right)_{q-1}\,.$
(197)

To be concrete, the constant $A$ in (195) would be $1/P^{(2q)}$.

In the case of two-loop integrals, the situation is more involved as there are
two distinct momenta to deal with. A prototypical example is for instance

$\begin{gathered}\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{k_{\mu}k_{\nu}p_{\lambda_{1}}\dots
p_{\lambda_{2q}}}{D}=A\eta_{\mu\nu}(\eta_{\lambda_{1}\lambda_{2}}\dots\eta_{\lambda_{2q-1}\lambda_{2q}}+\mathrm{perms.})\\\
+B(\eta_{\mu\lambda_{1}}\eta_{\nu\lambda_{2}}\eta_{\lambda_{3}\lambda_{4}}\dots\eta_{\lambda_{2q-1}\lambda_{2q}}+\mathrm{perms.})\,,\end{gathered}$
(198)

where $A,B$ will now be combinations of scalar loop integrals and some
combinatorial factors. Contracting both sides with the tensor
$\eta^{\mu\nu}\eta^{\lambda_{1}\lambda_{2}}\dots\eta^{\lambda_{2q-1}\lambda_{2q}}$
we obtain a first equation

$\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{k^{2}(p^{2})^{q}}{D}=P^{(2q)}(dA+2qB)\,.$
(199)

A second equation is deduced in a similar manner by contracting with
$\eta^{\mu\lambda_{1}}\eta^{\nu\lambda_{2}}\dots\eta^{\lambda_{2q-1}\lambda_{2q}}$

$\int\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{(k\cdot
p)^{2}(p^{2})^{q-1}}{D}=P^{(2q)}\left[A+(2q+d-1)B\right]\,.$ (200)

Solving (199) and (200) for $A$ and $B$, one finds the correct rewriting of
(198) in terms of scalar loop integrals and combinatiorial factors.

In a theory with non-manifest Lorentz invariance,
$P_{\lambda_{1}\dots\lambda_{2q}}$ is most often encountered with explicit
values assigned to its indices (either 0 or 1). As we shall make large use of
this formula, let us mention that if $P_{\lambda_{1}\dots\lambda_{2q}}$ comes
with $2q_{0}$ 0-indices and $2q_{1}$ 1-indices (so that $2q=2q_{0}+2q_{1}$),
the tensor specifically evaluates to

$P_{0_{1}\dots 0_{2q_{0}}1_{1}\dots
1_{2q_{1}}}=(2q_{0}-1)!!(2q_{1}-1)!!(\eta_{00})^{q_{0}}(\eta_{11})^{q_{1}}=(2q_{0}-1)!!(2q-2q_{0}-1)!!(\eta_{00})^{q_{0}}(\eta_{11})^{q_{1}}\,.$
(201)

###  E.2 One-loop Integrals

#### Bubble Diagrams

Integrals for bubble diagrams are trivial, as they simply coincide with
$I_{1}$.

#### Loop Diagrams

Loop diagrams are only slightly more involved. In general we are interested in
both the divergent and convergent (at least $O(\epsilon^{0})$) parts, as the
latter is important for counter-terms. The divergent diagrams we shall
encounter are of the form

$\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{2}}=\frac{\eta_{\mu\nu}}{d}(I_{1}+m^{2}I_{2})\,.$
(202)

Diagrams with an odd number of momenta in the numerator vanish by symmetry.
Fully convergent integrals are necessary of the form

$\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-m^{2})^{2}}=I_{2}\,.$
(203)

#### Triangle Diagrams

Divergent integrals stemming from triangle diagrams are of the form

$\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{k_{\mu}k_{\nu}k_{\rho}k_{\sigma}}{(k^{2}-m^{2})^{3}}=\frac{1}{d(d+2)}(\eta_{\mu\nu}\eta_{\rho\sigma}+\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})(I_{1}+2m^{2}I_{2}+m^{4}I_{3})\,,$
(204)

where we have made use of $O(d)$ symmetry to perform the integral. By the
remarks above, the right-hand side is mass independent at zero-th order in
$\epsilon$. Other finite results are

$\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}-m^{2})^{3}}=I_{3}\,,\qquad\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{k_{\mu}k_{\nu}}{(k^{2}-m^{2})^{3}}=\frac{\eta_{\mu\nu}}{d}(I_{2}+m^{2}I_{3})\,.$
(205)

###  E.3 Two-loop Integrals

A two-loop integral with vanishing external momenta is of the form

$\left[\left[\dots\right]\right]_{ijk}=\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{(\dots)}{(p^{2}-m^{2})^{i}(k^{2}-m^{2})^{j}[(k+p)^{2}-m^{2}]^{k}}\,,$
(206)

where the dots will be specified on a case-by-case basis. In performing
manipulation we discard finite terms, for instance:

$[[1]]_{i,j,k}=0\,,\quad\text{for $i,j,k\geq 1$}\,.$ (207)

Typical UV divergent integrals encountered are

$\left[\left[1\right]\right]_{1,0,1}=\left[\left[1\right]\right]_{1,1,0}=\left[\left[1\right]\right]_{0,1,1}=I_{1}^{2}\
,\quad\left[\left[1\right]\right]_{2,0,1}=I_{1}I_{2}\ .$ (208)

$4$$3$$2$$1$$-p$$-p$$k$$p$$-(k+p)$ (a) Square Envelope

$1$$2$$3$$4$$k+p$$p$$k$$-p$$-k$ (b) Diamond Sunset

$1$$2$$k$$p$$-(k+p)$ (c) Sunset

$1$$2$$3$$-(k+p)$$-p$$p$$k$ (d) Triangle Envelope

$1$$k$$p$ (e) Decorated bubble

$1$$2$$k$$-k$$p$ (f) Decorated loop

$1$$2$$3$$p$$k$$k$$-k$ (g) Decorated triangle

$1$$2$$3$$k$$-k$$p$$-p$ (h) Double loop

Figure 2: Two-loop diagrams. Momentum flows are aligned with numerical
ordering of vertices.

#### Sunset Diagrams

Sunset diagrams are equivalent to $[[\dots]]_{1,1,1}$ integrals. Given the
exchange symmetry $k\leftrightarrow p$ in the denominator, the only relevant
integrals are

$[[p^{2}]]_{1,1,1}=I_{1}^{2}\,,\qquad[[(p\cdot
k)]]_{1,1,1}=-\frac{1}{2}I_{1}^{2}\,.$ (209)

It immediately follows that

$[[p_{\mu}p_{\nu}]]_{1,1,1}=\frac{1}{d}\eta_{\mu\nu}I_{1}^{2}\,,\qquad[[p_{\mu}k_{\nu}]]_{1,1,1}=-\frac{1}{2d}\eta_{\mu\nu}I_{1}^{2}\,.$
(210)

#### Diamond Sunset Diagrams

Diamond sunset diagrams correspond to $[[\dots]]_{2,2,1}$. Notice that this
class is symmetric under $k\leftrightarrow p$; hence, we will omit integrals
which can be deduced from symmetry arguments. Again, we begin with the scalar
integrals

$\displaystyle[[(p^{2})^{3}]]_{2,2,1}$
$\displaystyle=I_{1}^{2}+4m^{2}I_{1}I_{2}\,,$ (211a)
$\displaystyle[[(p^{2})^{2}(p\cdot k)]]_{2,2,1}$
$\displaystyle=-I_{1}^{2}-m^{2}I_{1}I_{2}\,,$ (211b)
$\displaystyle[[(p^{2})^{2}k^{2}]]_{2,2,1}$
$\displaystyle=I_{1}^{2}+m^{2}I_{1}I_{2}\,,$ (211c)
$\displaystyle[[p^{2}(p\cdot k)k^{2}]]_{2,2,1}$
$\displaystyle=-\frac{1}{2}I_{1}^{2}\,,$ (211d) $\displaystyle[[p^{2}(p\cdot
k)^{2}]]_{2,2,1}$
$\displaystyle=\frac{3}{4}I_{1}^{2}+\frac{1}{2}m^{2}I_{1}I_{2}\,,$ (211e)
$\displaystyle[[(p\cdot k)^{3}]]_{2,2,1}$
$\displaystyle=\frac{2-3d}{4d}I_{1}^{2}+\frac{2-d}{2d}m^{2}I_{1}I_{2}\,.$
(211f)

Tensorial integrals can now be deduced

$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}p_{\kappa}p_{\lambda}]]_{2,2,1}$
$\displaystyle=F_{1}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+14\,\mathrm{perms})(I_{1}^{2}+4m^{2}I_{1}I_{2})\,,$
(212a)
$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}p_{\kappa}k_{\lambda}]]_{2,2,1}$
$\displaystyle=F_{1}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+14\,\mathrm{perms})\left(-I_{1}^{2}-m^{2}I_{1}I_{2}\right)\,,$
(212b)
$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}k_{\kappa}k_{\lambda}]]_{2,2,1}$
$\displaystyle=\frac{F_{1}}{d-1}(\eta_{\mu\nu}\eta_{\rho\sigma}+2\,\mathrm{perms})\eta_{\kappa\lambda}(dI_{1}^{2}+(d+1)m^{2}I_{1}I_{2})$
$\displaystyle+\frac{F_{1}}{4(d-1)}(\eta_{\mu\kappa}\eta_{\nu\lambda}\eta_{\rho\sigma}+11\,\mathrm{perms})((3d-4)I_{1}^{2}+2(d-2)m^{2}I_{1}I_{2})\,,$
(212c)
$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}k_{\sigma}k_{\kappa}k_{\lambda}]]_{2,2,1}$
$\displaystyle=\frac{F_{1}}{4d(d-1)}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+8\,\mathrm{perms})\biggl{[}-2(d^{2}-2d+2)I_{1}^{2}+4(d-2)m^{2}I_{1}I_{2}\biggr{]}$
$\displaystyle+\frac{F_{1}}{8d(d-1)}(\eta_{\mu\sigma}\eta_{\nu\kappa}\eta_{\rho\lambda}+5\,\mathrm{perms})\biggl{[}-2(3d^{2}-2d-4)I_{1}^{2}-4(d^{2}-4)m^{2}I_{1}I_{2}\biggr{]}\,.$
(212d)

#### Triangle Envelope Diagrams

This class is equivalent to $[[\dots]]_{2,1,1}$. Scalar integrals are

$\displaystyle[[(p^{2})^{2}]]_{2,1,1}$ $\displaystyle=I_{1}^{2}\,,$ (213a)
$\displaystyle[[p^{2}k^{2}]]_{2,1,1}$
$\displaystyle=I_{1}^{2}+m^{2}I_{1}I_{2}\,,$ (213b) $\displaystyle[[(p\cdot
k)^{2}]]_{2,1,1}$
$\displaystyle=\frac{3}{4}I_{1}^{2}+\frac{1}{2}m^{2}I_{1}I_{2}\,,$ (213c)
$\displaystyle[[p^{2}(p\cdot k)]]_{2,1,1}$
$\displaystyle=-\frac{1}{2}I_{1}^{2}\,,$ (213d) $\displaystyle[[(p\cdot
k)k^{2}]]_{2,1,1}$ $\displaystyle=-I_{1}^{2}-m^{2}I_{1}I_{2}\,,$ (213e)
$\displaystyle[[k^{4}]]_{2,1,1}$ $\displaystyle=I_{1}^{2}+3m^{2}I_{1}I_{2}\,.$
(213f)

Adopting the symbol $F_{2}=\frac{1}{d(d+2)}$, tensorial integrals evaluate to

$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}]]_{2,1,1}$
$\displaystyle=F_{2}(\eta_{\mu\nu}\eta_{\rho\sigma}+2\,\mathrm{perms})I_{1}^{2}\,,$
(214a) $\displaystyle[[p_{\mu}p_{\nu}p_{\rho}k_{\sigma}]]_{2,1,1}$
$\displaystyle=-\frac{1}{2}F_{2}(\eta_{\mu\nu}\eta_{\rho\sigma}+2\,\mathrm{perms})I_{1}^{2}\,,$
(214b) $\displaystyle[[p_{\mu}p_{\nu}k_{\rho}k_{\sigma}]]_{2,1,1}$
$\displaystyle=\frac{F_{2}}{2(d-1)}\eta_{\mu\nu}\eta_{\rho\sigma}I_{1}[(2d-1)I_{1}+2dm^{2}I_{2}]$
$\displaystyle+\frac{F_{2}}{4(d-1)}(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho})I_{1}[(3d-4)I_{1}+2(d-2)m^{2}I_{2}]\,,$
(214c) $\displaystyle[[p_{\mu}k_{\nu}k_{\rho}k_{\sigma}]]_{2,1,1}$
$\displaystyle=-F_{2}(\eta_{\mu\nu}\eta_{\rho\sigma}+2\,\mathrm{perms})I_{1}(I_{1}+m^{2}I_{2})\,,$
(214d) $\displaystyle[[k_{\mu}k_{\nu}k_{\rho}k_{\sigma}]]_{2,1,1}$
$\displaystyle=F_{2}(\eta_{\mu\nu}\eta_{\rho\sigma}+2\,\mathrm{perms})(I_{1}^{2}+3m^{2}I_{1}I_{2})\,.$
(214e)

#### Square Envelope Diagrams

Square envelope diagrams correspond to $[[\dots]]_{3,1,1}$, and the scalar
integrals we shall need for this topology are

$\displaystyle[[(p^{2})^{3}]]_{3,1,1}$ $\displaystyle=I_{1}^{2}\,,$ (215a)
$\displaystyle[[(p^{2})^{2}k^{2}]]_{3,1,1}$
$\displaystyle=I_{1}(I_{1}+2m^{2}I_{2}+m^{4}I_{3})\,,$ (215b)
$\displaystyle[[p^{2}(k^{2})^{2}]]_{3,1,1}$
$\displaystyle=I_{1}(I_{1}+4m^{2}I_{2}+3m^{4}I_{3})\,,$ (215c)
$\displaystyle[[(p^{2})^{2}(p\cdot k)]]_{3,1,1}$
$\displaystyle=-\frac{1}{2}I_{1}^{2}\,,$ (215d) $\displaystyle[[p^{2}(p\cdot
k)^{2}]]_{3,1,1}$
$\displaystyle=\frac{3}{4}I_{1}^{2}+m^{2}I_{1}I_{2}+\frac{m^{4}}{2}I_{1}I_{3}\,,$
(215e) $\displaystyle[[p^{2}(p\cdot k)k^{2}]]_{3,1,1}$
$\displaystyle=-I_{1}^{2}-2m^{2}I_{1}I_{2}-m^{4}I_{1}I_{3}\,,$ (215f)
$\displaystyle[[(p\cdot k)^{2}k^{2}]]_{3,1,1}$
$\displaystyle=I_{1}^{2}+\frac{7d+2}{2d}m^{2}I_{1}(I_{2}+m^{2}I_{3})\,,$
(215g) $\displaystyle[[(p\cdot k)^{3}]]_{3,1,1}$
$\displaystyle=-\frac{7}{8}I_{1}^{2}-\frac{3}{2}m^{2}I_{1}I_{2}-\frac{3}{4}m^{4}I_{1}I_{3}\,.$
(215h)

Integrals with “open” Lorentz indices can be built out of these. They all
share a common pre-factor $F_{1}=[d(d+2)(d+4)]^{-1}$ coming from the
contraction of indices. The results are:

$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}p_{\kappa}p_{\lambda}]]_{3,1,1}$
$\displaystyle=F_{1}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+14\,\mathrm{perms})I_{1}^{2}\,,$
(216a)
$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}p_{\kappa}k_{\lambda}]]_{3,1,1}$
$\displaystyle=-\frac{1}{2}F_{1}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+14\,\mathrm{perms})I_{1}^{2}\,,$
(216b)
$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}p_{\sigma}k_{\kappa}k_{\lambda}]]_{3,1,1}$
$\displaystyle=\frac{F_{1}}{d-1}(\eta_{\mu\nu}\eta_{\rho\sigma}+2\,\mathrm{perms})\eta_{\kappa\lambda}I_{1}\left[dI_{1}+m^{2}(d+1)(2I_{2}+m^{2}I_{3})\right]$
$\displaystyle+\frac{F_{1}}{4(d-1)}(\eta_{\mu\kappa}\eta_{\nu\lambda}\eta_{\rho\sigma}+11\,\mathrm{perms})I_{1}\left[\left(3d-4\right)I_{1}+2\left(d-2\right)m^{2}(2I_{2}+m^{2}I_{3})\right]\,,$
(216c)
$\displaystyle[[p_{\mu}p_{\nu}p_{\rho}k_{\sigma}k_{\kappa}k_{\lambda}]]_{3,1,1}$
$\displaystyle=\frac{F_{1}}{4(d-1)}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+8\,\mathrm{perms})I_{1}\left[(3-4d)I_{1}+2(1-2d)m^{2}(2I_{2}+m^{2}I_{3})\right]$
$\displaystyle+\frac{F_{1}}{8(d-1)}(\eta_{\mu\sigma}\eta_{\nu\kappa}\eta_{\rho\lambda}+5\,\mathrm{perms})I_{1}\left[(10-7d)I_{1}+6(2-d)m^{2}(2I_{2}+m^{2}I_{3})\right]\,,$
(216d)
$\displaystyle[[p_{\mu}p_{\nu}k_{\rho}k_{\sigma}k_{\kappa}k_{\lambda}]]_{3,1,1}$
$\displaystyle=\frac{F_{1}}{d(d-1)}(\eta_{\mu\nu}\eta_{\rho\sigma}\eta_{\kappa\lambda}+2\,\mathrm{perms})\biggl{[}d(d-1)I_{1}^{2}+(4d^{2}-2d-4)m^{2}I_{1}I_{2}$
$\displaystyle+(3d^{2}-5d-4)m^{4}I_{1}I_{3}\biggr{]}$
$\displaystyle+\frac{F_{1}}{2(d-1)}(\eta_{\mu\rho}\eta_{\nu\sigma}\eta_{\kappa\lambda}+11\,\mathrm{perms})\biggl{[}2(d-1)I_{1}^{2}+(7d-6)m^{2}I_{1}I_{2}$
$\displaystyle+(7d-4)m^{4}I_{1}I_{3}\biggr{]}\,.$ (216e)

###  E.4 Schwinger Parametrisation

When considering the renormalisation of the $(\partial_{1}y)^{2}$ component of
the metric, the integrals above are not sufficient since we also encounter
diagrams which give rise to integrands in Fourier space wit non-scalar
denominators.

The first integral we shall be concerned with arises from a particular
instance of the sunset topology. Consider

$J_{1}=\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{k^{1}(p^{0})^{2}}{p^{1}}\frac{1}{(k^{2}-m^{2})(p^{2}-m^{2})[(k+p)^{2}-m^{2}]}\,.$
(217)

To address the $p^{1}$ momentum in the denominator we employ a Schwinger
parametrisation

$\frac{1}{p^{1}}=\int_{0}^{\infty}\mathrm{d}u\,e^{-up^{1}}\,.$ (218)

The Lorentz-scalar part of (217) and the momentum integrals are easily dealt
with using the above techniques so we momentarily omit them by defining a
“reduced” integral

$\tilde{J}_{1}=\int_{0}^{\infty}\mathrm{d}u\,e^{-up^{1}}k^{1}(p^{0})^{2}\,.$
(219)

We now perform a series expansion of this exponential which we understand will
give a series of loop integrals we can evaluate using some combinatorics and
formulas in Appendix E. Even powers of $u$ drop out since the numerator of the
corresponding momentum integral will contain an odd number of $\sigma$
components of momentum - this implies that any scalar contraction will come
with a factor of $\eta_{01}$ and hence will drop. So we need retain only

$\tilde{J}_{1}=-\int_{0}^{\infty}\mathrm{d}uu\,\sum_{n=0}^{\infty}\frac{u^{2n}}{(2n+1)!}k^{1}(p^{0})^{2}(p^{1})^{2n+1}\,.$
(220)

Now, the precise replacement stems from the usual combinatorics in loop
integrals

$k^{\mu}p^{\nu_{1}}\dots
p^{\nu_{2n+3}}\rightarrow\frac{1}{P^{(2n+4)}}P^{\mu\nu_{1}\dots\nu_{2n+3}}(k\cdot
p)(p^{2})^{n+1}\,.$ (221)

In a scheme where $\eta^{01}=0$, there are $(2n+1)!!$ non vanishing
contributions in $P^{\mu\nu_{1}\dots\nu_{2n+3}}$ with a choice of indices as
in (220). We then end up with

$\tilde{J}_{1}=\varphi(\epsilon)\psi(\epsilon)(k\cdot
p)p^{2}\int_{0}^{\infty}\mathrm{d}uu\sum_{n=0}^{\infty}\frac{(2n+1)!!}{(2n+1)!P^{(2n+4)}}(-up^{2}\psi(\epsilon))^{n}\,,$
(222)

where we have adopted the parametrisations $\eta^{11}=-\psi(\epsilon)$ and
$\eta^{00}=\varphi(\epsilon)$, with $\psi=1+\mathfrak{g}\epsilon$ and
$\varphi=1+\mathfrak{g}\epsilon$ are both positive definite. Introducing a new
integration variable $z=up^{2}\psi/4$, one can prove that the series can be
resummed to yield the regularised hypergeometric function
${}_{0}\tilde{F}_{1}$

$\tilde{J}_{1}=\frac{\varphi(\epsilon)}{2}\Gamma\left(\frac{d}{2}\right)(k\cdot
p)\int_{0}^{\infty}\mathrm{d}z\,{}_{0}\tilde{F}_{1}\left(2+\frac{d}{2},-z\right)\,.$
(223)

The remaining integral is part of a family of parametric integrals whose
precise evaluation is known and equals

$\int_{0}^{\infty}\mathrm{d}z\,z^{\alpha-1}\,{}_{0}\tilde{F}_{1}(b;-z)=\frac{\Gamma(\alpha)}{\Gamma(b-\alpha)}\,.$
(224)

Adapting this formula to the case at hand we find

$\tilde{J}_{1}=\frac{\varphi(\epsilon)}{d}(k\cdot p)\,,\qquad
J_{1}=-\frac{\varphi(\epsilon)}{2d}I_{1}^{2}\,.$ (225)

Other than $J_{1}$, we have a family of relevant non-invariant integrals
depending on a integer parameter $\alpha\in\mathbb{N}$ reading

$J^{(\alpha)}_{2}=\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d}}\frac{\mathrm{d}^{d}p}{(2\pi)^{d}}\frac{1}{p^{1}}\frac{k^{0}k^{1}(p^{0})^{2\alpha+1}}{(k^{2}-m^{2})(p^{2}-m^{2})^{\alpha+1}[(k+p)^{2}-m^{2}]}\,.$
(226)

Using the same technique the relevant cases evaluate to

$\displaystyle J_{2}^{(0)}$
$\displaystyle=\frac{3d-4}{4d(d-1)}\varphi(\epsilon)I_{1}^{2}\,,$ (227)
$\displaystyle J_{2}^{(1)}$
$\displaystyle=\frac{3(3d-4)}{4(d-1)d(d+2)}\varphi(\epsilon)^{2}I_{1}^{2}+\frac{3(d-2)\varphi(\epsilon)^{2}}{2(d-1)d(d+2)}\varphi(\epsilon)^{2}m^{2}I_{1}I_{2}\,,$
(228) $\displaystyle J_{2}^{(2)}$
$\displaystyle=\frac{15(3d-4)}{4(d-1)d(d+2)(d+4)}\varphi(\epsilon)^{3}I_{1}^{2}+\frac{15(d-2)m^{2}}{(d-1)d(d+2)(d+4)}\varphi(\epsilon)^{3}m^{2}I_{1}\left(I_{2}+\frac{m^{2}}{2}I_{3}\right)\,.$
(229)

##  Appendix F Example

We complement the main presentation with an explicit toy example. This allows
for an independent computerised cross-check of the calculations done elsewhere
in the project. In this toy model we are also able to explicitly examine the
implication of removing the base manifold counter-term at one-loop via a field
redefinition/addition of total derivative.

We begin with a model consisting of a single $S^{1}$ direction of radius
$r=r(y)$ for the fibre with Lagrangian

$\mathcal{L}=\frac{1}{2}(\partial_{1}x\partial_{0}\tilde{x}+\partial_{0}x\partial_{1}\tilde{x})-\frac{1}{2}\left(r^{2}\partial_{1}x\partial_{1}x+r^{-2}\partial_{1}\tilde{x}\partial_{1}\tilde{x}\right)+\frac{1}{2}\partial_{\mu}y\partial^{\mu}y\,.$
(230)

The one-loop renormalisation yields

$\mathcal{L}_{\mathrm{CT}}=\frac{1}{4\pi\epsilon}\frac{r\ddot{r}-\dot{r}^{2}}{r^{4}}(\partial_{1}\tilde{x}\partial_{1}\tilde{x}-r^{4}\partial_{1}x\partial_{1}x)+\frac{1}{4\pi\epsilon}\frac{\dot{r}^{2}}{r^{2}}\partial_{\mu}y\partial^{\mu}y\,,$
(231)

where over-dots denote, as usual, derivatives with respect to $y$. Using
Method 2 followed by Method 1 for final evaluation of integrals, we find the
two-loop contributions

$\displaystyle\widetilde{T}_{2}^{(2)}|_{11}$
$\displaystyle=\frac{1}{8\pi^{2}\epsilon^{2}}\frac{5(r^{(1)})^{4}-8r(r^{(1)})^{2}r^{(2)}+r^{2}(r^{(2)})^{2}+2r^{2}r^{(1)}r^{(3)}}{r^{4}}=\widetilde{T}_{2}^{(2)}|_{00}\,,$
(232) $\displaystyle\widetilde{T}_{2}^{(2)}|_{01}$ $\displaystyle=0\,,$ (233)
$\displaystyle T_{2}^{(2)}|_{xx}$
$\displaystyle=-\frac{1}{16\pi^{2}\epsilon^{2}}\frac{2(r^{(1)})^{4}-6r(r^{(1)})^{2}r^{(2)}+4r^{2}r^{(1)}r^{(3)}+r^{2}((r^{(2)})^{2}-r^{(4)})}{r^{2}}\,,$
(234) $\displaystyle T_{2}^{(2)}|_{\tilde{x}\tilde{x}}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon^{2}}\frac{6(r^{(1)})^{4}-14r(r^{(1)})^{2}r^{(2)}+4r^{2}r^{(1)}r^{(3)}+r^{2}(5(r^{(2)})^{2}-rr^{(4)})}{r^{6}}\,,$
(235) $\displaystyle\widetilde{T}_{1}^{(2)}|_{00}$
$\displaystyle=\frac{\mathfrak{g}}{8\pi^{2}\epsilon}\frac{((r^{(1)})^{2}-rr^{(2)})^{2}}{r^{4}}\,,$
(236) $\displaystyle\widetilde{T}_{1}^{(2)}|_{11}$
$\displaystyle=\frac{1}{8\pi^{2}\epsilon}\frac{(\mathfrak{g}-1)(r^{(1)})^{4}+2(3\mathfrak{g}-1)r(r^{(1)})^{2}r^{(2)}-(3\mathfrak{g}-1)r^{2}(r^{(2)})^{2}}{r^{4}}\,,$
(237) $\displaystyle\widetilde{T}_{1}^{(2)}|_{01}$ $\displaystyle=0\,,$ (238)
$\displaystyle T_{1}^{(2)}|_{xx}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon}\frac{2(1-\mathfrak{g})(r^{(1)})^{4}+(2\mathfrak{g}-1)r(r^{(1)})^{2}r^{(2)}+2(2\mathfrak{g}-1)r^{2}(r^{(2)})^{2}}{r^{2}}\,,$
(239) $\displaystyle T_{1}^{(2)}|_{\tilde{x}\tilde{x}}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon}\frac{2(-4+9\mathfrak{g})(r^{(1)})^{4}-9(2\mathfrak{g}-1)r(r^{(1)})^{2}r^{(2)}+2(2\mathfrak{g}-1)r^{2}(r^{(2)})^{2}}{r^{6}}\,,$
(240)

where $r^{(n)}$ is an alternative notation for the $n$-th derivative of $r$
with respect to $y$. These agree with the results of the main text upon the
substitution of the generalised metric
$\mathcal{H}=\mathrm{diag}(r^{2},r^{-2})$. We remark that the vanishing of
$\widetilde{T}^{(2)}|_{01}$ is due to the triviality of the tensorial
combinations within this example.

Let us note that we can amend the one-loop counter-term through the inclusion
of any piece that vanishes upon integration by parts and application of the
equations of motion

$\mathcal{L}_{\mathrm{on-
shell}}=\dot{f}(y)\partial_{\mu}y\partial^{\mu}y-\frac{1}{2}f(y)\mathcal{H}^{(1)}_{\bullet\bullet}\approx
0\,,$ (241)

for some function $f(y)$. With the choice that
$\dot{f}=-\frac{1}{2}\widetilde{T}^{(1)}$ the base divergence of the one-loop
counter can be removed, considering instead

$\mathcal{L}_{\mathrm{CT}}+\mathcal{L}_{\mathrm{on-
shell}}=\frac{f(y)}{4\pi\epsilon}\left(r\dot{r}\,\partial_{1}x\partial_{1}x-\frac{\dot{r}}{r^{3}}\partial_{1}\tilde{x}\partial_{1}\tilde{x}\right)+\frac{1}{4\pi\epsilon}\frac{r\ddot{r}-\dot{r}^{2}}{r^{4}}(\partial_{1}\tilde{x}\partial_{1}\tilde{x}-r^{4}\partial_{1}x\partial_{1}x)\,.$
(242)

We may now proceed to re-calculate the two-loop divergences using this
modified one-loop counter term. One might anticipate that this resolves some
of the discrepancies between $\widetilde{T}^{(2)}_{1}|_{00}$ and
$\widetilde{T}^{(2)}_{1}|_{11}$ seen in the above. However, an explicit
calculation yields

$\displaystyle\widetilde{T}_{1}^{(2)}|_{00}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon}\frac{(1-\mathfrak{g})(r^{(1)})^{4}-2\mathfrak{g}r(r^{(1)})^{2}r^{(2)}+\mathfrak{g}r^{2}(r^{(2)})^{2}}{r^{4}}$
$\displaystyle+\frac{1}{16\pi^{2}\epsilon}\frac{(-1+2\mathfrak{g})frr^{(1)}((r^{(1)})^{2}-rr^{(2)})}{r^{4}}+\dots\,,$
(243) $\displaystyle\widetilde{T}_{1}^{(2)}|_{11}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon}\frac{(5-14\mathfrak{g})(r^{(1)})^{4}+4(1-3\mathfrak{g})r(r^{(1)})^{2}r^{(2)}+2(1-3\mathfrak{g})r^{2}(r^{(2)})^{2}}{r^{4}}$
$\displaystyle+\frac{1}{8\pi^{2}\epsilon}\frac{3(-1+2\mathfrak{g})frr^{(1)}((r^{(1)})^{2}-rr^{(2)})}{r^{4}}+\dots\,,$
(244)

where ellipsis indicate terms proportional to $\frac{\bar{\gamma}}{\epsilon}$
(which do not now vanish). Far from ameliorating the situation, we still have
Lorentz violation from the base counter terms and, moreover, un-cancelled
$\bar{\gamma}$ terms.

## References

  * (1)
  * (2)
  * (3) D. Friedan, Nonlinear Models in Two Epsilon Dimensions, Phys. Rev. Lett. 45, 1057 (1980)
  * (4) C. G. Callan, Jr., E. J. Martinec, M. J. Perry and D. Friedan, Strings in Background Fields, Nucl. Phys. B 262, 593-609 (1985)
  * (5) A. A. Tseytlin, Phys. Lett. B 268, 175-178 (1991) doi:10.1016/0370-2693(91)90800-6
  * (6) C. M. Hull, Lectures on nonlinear simga models and strings, PRINT-87-0480 (CAMBRIDGE).
  * (7) C. G. Callan, Jr. and L. Thorlacius, Sigma Models and String Theory, Print-89-0232 (PRINCETON).
  * (8) C. M. Hull and P. K. Townsend, The Two Loop Beta Function for $\sigma$ Models With Torsion, Phys. Lett. B 191 (1987), 115-121
  * (9) R. R. Metsaev and A. A. Tseytlin, Order alpha-prime (Two Loop) Equivalence of the String Equations of Motion and the Sigma Model Weyl Invariance Conditions: Dependence on the Dilaton and the Antisymmetric Tensor, Nucl. Phys. B 293 (1987), 385-419
  * (10) I. Jack, D. R. T. Jones and N. Mohammedi, A Four Loop Calculation of the Metric Beta Function for the Bosonic $\sigma$ Model and the String Effective Action, Nucl. Phys. B 322, 431-470 (1989)
  * (11) T. H. Buscher, A Symmetry of the String Background Field Equations, Phys. Lett. B 194 (1987) 59.
  * (12) T. H. Buscher, Path Integral Derivation of Quantum Duality in Nonlinear Sigma Models, Phys. Lett. B 201 (1988) 466.
  * (13) A. A. Tseytlin, Duality Symmetric Formulation Of String worldsheet Dynamics, Phys. Lett. B 242 (1990) 163.
  * (14) A. A. Tseytlin, Duality symmetric closed string theory and interacting chiral scalars, Nucl. Phys. B 350 (1991) 395.
  * (15) D. S. Berman and D. C. Thompson, Phys. Lett. B 662, 279-284 (2008) doi:10.1016/j.physletb.2008.03.012 [arXiv:0712.1121 [hep-th]].
  * (16) W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 arXiv:hep-th/9305073
  * (17) W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 arXiv:hep-th/9302036
  * (18) C. Hull and B. Zwiebach, Double Field Theory, JHEP 0909 (2009) 099 arXiv:0904.4664 [hep-th]
  * (19) O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010), 008 arXiv:1006.4823 [hep-th]
  * (20) D. S. Berman and M. J. Perry, Generalized Geometry and M theory, JHEP 1106 (2011) 074 arXiv:1008.1763 [hep-th]
  * (21) O. Hohm and H. Samtleben, Exceptional Form of D=11 Supergravity, Phys. Rev. Lett. 111 (2013) 231601 arXiv:1308.1673 [hep-th]
  * (22) K. A. Meissner, Symmetries of higher order string gravity actions, Phys. Lett. B 392, 298-304 (1997) arXiv:hep-th/9610131
  * (23) O. Hohm, W. Siegel and B. Zwiebach, Doubled $\alpha^{\prime}$-geometry, JHEP 02 (2014), 065 arXiv:1306.2970 [hep-th]
  * (24) A. Coimbra, R. Minasian, H. Triendl and D. Waldram, Generalised geometry for string corrections, JHEP 11 (2014), 160 arXiv:1407.7542 [hep-th]
  * (25) K. Lee, Quadratic $\alpha$’-corrections to heterotic double field theory, Nucl. Phys. B 899 (2015), 594-616 arXiv:1504.00149 [hep-th]
  * (26) D. Marques and C. A. Nunez, T-duality and $\alpha$’-corrections, JHEP 10 (2015), 084 arXiv:1507.00652 [hep-th]
  * (27) O. Hohm and B. Zwiebach, Double metric, generalized metric, and $\alpha$’ -deformed double field theory, Phys. Rev. D 93 (2016) no.6, 064035 arXiv:1509.02930 [hep-th]
  * (28) W. H. Baron, J. J. Fernandez-Melgarejo, D. Marques and C. Nunez, The Odd story of $\alpha$’-corrections, JHEP 04 (2017), 078 arXiv:1702.05489 [hep-th]
  * (29) E. Lescano, $\alpha^{\prime}$-corrections and their double formulation, arXiv:2108.12246 [hep-th]
  * (30) M. J. Duff, Duality Rotations in String Theory, Nucl. Phys. B 335 (1990) 610.
  * (31) J. Maharana and J. H. Schwarz, ‘Noncompact symmetries in string theory,” Nucl. Phys. B 390 (1993), 3-32 arXiv:hep-th/9207016
  * (32) C. M. Hull, A Geometry for non-geometric string backgrounds, JHEP 0510 (2005) 065 arXiv:hep-th/0406102.
  * (33) C. M. Hull, Doubled Geometry and T-Folds, JHEP 0707 (2007) 080 arXiv:hep-th/0605149.
  * (34) C. M. Hull and R. A. Reid-Edwards, Gauge symmetry, T-duality and doubled geometry, JHEP 0808 (2008) 043 arXiv:0711.4818 [hep-th]
  * (35) C. M. Hull and R. A. Reid-Edwards, Non-geometric backgrounds, doubled geometry and generalised T-duality, JHEP 0909 (2009) 014 arXiv:0902.4032 [hep-th]
  * (36) R. Floreanini and R. Jackiw, Selfdual Fields as Charge Density Solitons, Phys. Rev. Lett. 59 (1987), 1873
  * (37) D. S. Berman, N. B. Copland and D. C. Thompson, Background Field Equations for the Duality Symmetric String, Nucl. Phys. B 791 (2008), 175-191 arXiv:0708.2267 [hep-th]
  * (38) D. S. Berman and D. C. Thompson, Duality Symmetric Strings, Dilatons and O(d,d) Effective Actions, Phys. Lett. B 662 (2008), 279-284 arXiv:0712.1121 [hep-th]
  * (39) N. B. Copland, Connecting T-duality invariant theories, Nucl. Phys. B 854 (2012), 575-591 arXiv:1106.1888 [hep-th]
  * (40) N. B. Copland, A Double Sigma Model for Double Field Theory, JHEP 04 (2012), 044 arXiv:1111.1828 [hep-th]
  * (41) C. Klimcik and P. Severa, Dual nonAbelian duality and the Drinfeld double, Phys. Lett. B 351 (1995), 455-462 arXiv:hep-th/9502122
  * (42) S. D. Avramis, J. P. Derendinger and N. Prezas, Conformal chiral boson models on twisted doubled tori and non-geometric string vacua, Nucl. Phys. B 827 (2010), 281-310 arXiv:0910.0431 [hep-th]
  * (43) K. Sfetsos, K. Siampos and D. C. Thompson, Renormalization of Lorentz non-invariant actions and manifest T-duality, Nucl. Phys. B 827 (2010), 545-564 arXiv:0910.1345 [hep-th]
  * (44) S. Groot Nibbelink, F. Kurz and P. Patalong, Renormalization of a Lorentz invariant doubled worldsheet theory, JHEP 10 (2014), 114 arXiv:1308.4418 [hep-th]
  * (45) R. Bonezzi, T. Codina and O. Hohm, Beta functions for the duality-invariant sigma model, arXiv:2103.15931 [hep-th]
  * (46) L. Alvarez-Gaume, D. Z. Freedman and S. Mukhi, The Background Field Method and the Ultraviolet Structure of the Supersymmetric Nonlinear Sigma Model, Annals Phys. 134 (1981), 85
  * (47) S. Mukhi, The Geometric Background Field Method, Renormalization and the Wess-Zumino Term in Nonlinear Sigma Models, Nucl. Phys. B 264, 640-652 (1986)
  * (48) P. S. Howe, G. Papadopoulos and K. S. Stelle, The Background Field Method and the Nonlinear $\sigma$ Model, Nucl. Phys. B 296 (1988), 26-48
  * (49) C. M. Hull and P. K. Townsend, Finiteness and Conformal Invariance in Nonlinear $\sigma$ Models, Nucl. Phys. B 274 (1986), 349-362
  * (50) R. Bonezzi, F. Diaz-Jaramillo and O. Hohm, Old Dualities and New Anomalies, Phys. Rev. D 102, no.12, 126002 (2020) arXiv:2008.06420 [hep-th]