$\epsilon$ and $v_{j}$ but not on $u$. This follows from the fact that all
dependence on the regulators enters through $\hat{\beta}_{j}$, and
$\hat{\beta}_{j}$ depends only on $\epsilon$ and $v_{j}$. Actually, in the
cases we analyze in this paper ($\beta_{j}$ non-integral and positive) the
coefficients are smooth as $\epsilon\to 0$. It follows that

$\mathcal{I}_{(\sigma_{j},n)}=\mathcal{I}^{0}_{(\sigma_{j},n)}+\epsilon\sum_{k=1}^{3}\mathcal{I}^{1}_{(\sigma_{j},n)k}v_{k},$
(224)

where both $\mathcal{I}^{0}_{(\sigma_{j},n)}$ and
$\mathcal{I}^{1}_{(\sigma_{j},n)k}$ are independent of $u$ and $v_{j}$.

As shown in Bzowski:2015yxv , all divergences and finite $u$ and
$v_{j}$-dependent terms for 3-point amplitudes (8) are contained in
$i^{\text{div}}$ defined by

$\displaystyle i^{\text{div}}(u,v_{j};\mu)$
$\displaystyle=\sum_{\\{L_{0}=0\\}}\int_{0}^{\mu^{-1}}\mathrm{d}z\,\mathcal{I}_{(\sigma_{j},n)}z^{-1+\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}}}=\sum_{\\{L_{0}=0\\}}\frac{\mathcal{I}_{(\sigma_{j},n)}\mu^{-\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}}}}{\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}}},$ (225)

where the sum is over all signs $\sigma_{j}$ and $n$ such that $L_{0}=0$ and
$\mu>0$ is an arbitrary number. Using (223), the condition $L_{0}=0$ may be
solved to determine $n$ in terms of the signs $\sigma_{j}$, $n=n(\sigma_{j})$,
and we define

$\mathcal{I}_{\sigma_{1}\sigma_{2}\sigma_{3}}=\left\\{\begin{array}[]{cc}\mathcal{I}_{(\sigma_{j},n(\sigma_{j}))}&\qquad
n(\sigma_{j})=0,1,2,\ldots,\\\ 0&\qquad{\rm otherwise}\,.\end{array}\right.$
(226)

Then

$i^{\text{div}}(u,v_{j};\mu)=\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\frac{\mathcal{I}_{\sigma_{1}\sigma_{2}\sigma_{3}}\mu^{-\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}}}}{\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}}},$ (227)

where the sum is now over all signs $\sigma_{j}$. Note that in all cases
discussed in this paper there is only one combination of signs that yields a
non-zero $\mathcal{I}_{\sigma_{1}\sigma_{2}\sigma_{3}}$, and thus in all cases
the sum in (227) reduces to a single term. Furthermore, the signs that
contribute are either all minus, or one plus and the rest minus. This can be
seen from Table 2: as discussed at length in Bzowski:2015pba , anomalies are
associated with $(---)$ singularities and beta functions with $(--+)$ ones.

In Bzowski:2015yxv  we have shown the following procedure for scheme change:

###### Fact 1

If we already know the amplitude $\hat{i}(\bar{u},\bar{v}_{j})$ regulated in
_some_ scheme with parameters $\bar{u}$ and $\bar{v}_{j}$, we can use (227) to
obtain the amplitude in _any_ regularization scheme,

$\displaystyle\hat{i}(u,v_{j})=\hat{i}(\bar{u},\bar{v}_{j})+\left[i^{\text{div}}(u,v_{j})-i^{\text{div}}(\bar{u},\bar{v}_{j})\right]+O(\epsilon).$
(228)

From now on, we will concentrate on the amplitudes we are interested in for
this paper, i.e., $d=3$ and all conformal dimensions $\Delta_{j}$ equal to $2$
or $3$. For these $d$ and $\Delta_{j}$ the 3-point functions are at most
linearly divergent. The divergence thus can only emerge from the explicit
divergence in (227). In particular, the coefficients
$\mathcal{I}_{\sigma_{1}\sigma_{2}\sigma_{3}}$ are finite and continuous in
the $\epsilon\rightarrow 0$ limit.

####  Scheme change

Equation (228) can be greatly simplified and generalized when we start from
the half-integer scheme with $\bar{v}_{j}=0$. First note that, given an
amplitude regulated in the half-integer scheme with $u=1$ and $v_{j}=0$, we
can obtain for free the amplitude regulated with general $u$ simply by
rescaling $\epsilon\mapsto u\epsilon$. Thus, knowing $\hat{i}(1,0)$ to all
orders in $\epsilon$, we also know $\hat{i}(u,0)$ to all orders in $\epsilon$.

Let us also define $\hat{i}|_{\epsilon^{n}}$ to represent the terms of order
$\epsilon^{n}$ in a regulated amplitude $\hat{i}$, so that up to linear order
a linearly divergent amplitude is given by

$\displaystyle\hat{i}(u,v_{j})=\frac{\hat{i}(u,v_{j})|_{\epsilon^{-1}}}{\epsilon}+\hat{i}(u,v_{j})|_{\epsilon^{0}}+\epsilon\,\hat{i}(u,v_{j})|_{\epsilon^{1}}+O(\epsilon^{2}).$
(229)

With this notation in place, one can show the following:

###### Fact 2

Consider a linearly divergent amplitude
$\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ regulated in the general
$(u,v_{j})$-scheme. The expansion coefficients are given by

$\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{-1}}$
$\displaystyle=i^{\text{div}}\left(u,v_{j};1\right)|_{\epsilon^{-1}},$ (230)
$\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{0}}$
$\displaystyle=\hat{i}(1,0)|_{\epsilon^{0}}+i^{\text{div}}\left(u,v_{j};1\right)|_{\epsilon^{0}},$
(231) $\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{1}}$
$\displaystyle=\hat{i}(u,0)|_{\epsilon^{1}}+\sum_{j=1}^{3}\epsilon
v_{j}\frac{\partial\hat{i}}{\partial\beta_{j}}(u,0)|_{\epsilon^{1}}+i^{\text{div}}\left(u,v_{j};1\right)|_{\epsilon^{1}}.$
(232)

Not only can we evaluate the finite part for general $(u,v_{j})$, we can also
evaluate the subleading terms of order $\epsilon$ provided we know the
derivatives of the regulated amplitude with respect to the Bessel indices
$\beta_{j}$. We will need the expansion of the amplitude to order
$\epsilon^{1}$ when we discuss the scheme change for exchange 4-point
functions.

####  Example

To illustrate our discussion consider the amplitude $\hat{i}_{[222]}$
regulated in a scheme with general $u$ and $v_{[2]}$ for all dimensions,
$\hat{\Delta}_{j}=2+(u+v_{[2]})\epsilon$ for $j=1,2,3$. First, let us recall
the expression we have already computed in the $(1,0)$ scheme,

$\displaystyle\hat{i}_{[222]}(1,0)$
$\displaystyle=\operatorname{Div}_{1}(k_{t})=\Gamma(\epsilon)k_{t}^{-\epsilon}$
$\displaystyle=\frac{1}{\epsilon}-(\log
k_{t}+\gamma_{E})+\frac{\epsilon}{2}\left((\log
k_{t}+\gamma_{E})^{2}+\frac{\pi^{2}}{6}\right)+O(\epsilon^{2}),$ (233)

where $k_{t}=k_{1}+k_{2}+k_{3}$.

To work out the change of scheme we need $i^{\text{div}}_{[222]}$. In this
case, there is only one choice of signs for which $L_{0}=0$ yields a non-
negative integer $n$, namely all minus, which yields $n=0$ and
$L_{---}=u-3v_{[2]}$ so that

$\displaystyle i^{\text{div}}_{[222]}(u,v_{[2]};\mu)$
$\displaystyle=\int_{0}^{\mu^{-1}}\mathrm{d}z\,z^{-1+\epsilon(u-3v_{[2]})}=\frac{\mu^{-\epsilon(u-3v_{[2]})}}{\epsilon\,(u-3v_{[2]})}$
$\displaystyle=\frac{1}{\epsilon\,(u-3v_{[2]})}+\log\mu+\frac{1}{2}(u-3v_{[2]})\log^{2}\mu\,\epsilon+O(\epsilon^{2}).$
(234)

To use Fact 2, we evaluate at $\mu=1$ to obtain

$\displaystyle i^{\text{div}}_{[222]}(u,v_{[2]};1)|_{\epsilon^{-1}}$
$\displaystyle=\frac{1}{\epsilon\,(u-3v_{[2]})},\qquad
i^{\text{div}}_{[222]}(u,v_{[2]};1)|_{\epsilon^{n}}=0,\ n\geq 0.$ (235)

We also need $\hat{i}_{[222]}(u,0)$ which may be obtained from (5.2.3) by
$\epsilon\mapsto u\epsilon$,

$\displaystyle\hat{i}_{[222]}(u,0)$
$\displaystyle=\frac{1}{u\epsilon}-\left(\log
k_{t}+\gamma_{E}\right)+\frac{u\epsilon}{2}\left[(\log
k_{t}+\gamma_{E})^{2}+\frac{\pi^{2}}{6}\right]+O(\epsilon^{2}).$ (236)

The final and the most difficult step is to calculate the terms involving the
derivatives with respect to the Bessel order. Using (491) and (497) we find

$\displaystyle\left.\epsilon
v_{[2]}\frac{\partial\hat{i}_{[222]}}{\partial\beta_{1}}\right|_{\beta_{1}=\frac{1}{2}}$
$\displaystyle\qquad=\epsilon
v_{[2]}\int_{0}^{\infty}\mathrm{d}z\,e^{-zk_{t}}z^{-1+u\epsilon}\left(e^{2zk_{1}}E_{1}(2zk_{1})+\log
k_{1}+\log 2+\gamma_{E}\right)$
$\displaystyle\qquad=\frac{v_{[2]}}{u^{2}\epsilon}+\left[-\operatorname{Li}_{2}\left(\frac{-k_{1}+k_{2}+k_{3}}{k_{t}}\right)+\frac{1}{2}\left(\log
k_{t}+\gamma_{E}\right)^{2}+\frac{\pi^{2}}{12}\right]v_{[2]}\epsilon+O(\epsilon^{2}).$
(237)

We can put everything together to find the amplitude $\hat{i}_{[222]}$ to
linear order in $\epsilon$ and in an arbitrary regularization scheme.
According to Fact 2,

$\displaystyle\hat{i}_{[222]}(u,v_{j})|_{\epsilon^{-1}}$
$\displaystyle=\frac{1}{u-3v_{[2]}},$ (238)
$\displaystyle\hat{i}_{[222]}(u,v_{j})|_{\epsilon^{0}}$ $\displaystyle=-\log
k_{t}-\gamma_{E},$ (239)
$\displaystyle\hat{i}_{[222]}(u,v_{j})|_{\epsilon^{1}}$
$\displaystyle=\frac{u+3v_{[2]}}{2}\left[(\log
k_{t}+\gamma_{E})^{2}+\frac{\pi^{2}}{6}\right]$ $\displaystyle\qquad-
v_{[2]}\left[\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)+\operatorname{Li}_{2}\left(\frac{l_{13-}}{k_{t}}\right)+\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)\right],$
(240)

where $l_{12-}=k_{1}+k_{2}-|\bm{k}_{1}+\bm{k}_{2}|=k_{1}+k_{2}-k_{3}$ and
similarly for $l_{13-}$ and $l_{23-}$.

This example is slightly degenerate as it does not contain any parameter-
dependent finite piece. To illustrate this point consider the case
$\hat{i}_{[332]}$, and to simplify the discussion, we consider the two
operator of dimension three as identical so that
$\hat{\Delta}_{j}=3+(u+v_{[3]})\epsilon$ for $j=1,2$ and
$\hat{\Delta}_{2}=2+(u+v_{[2]})\epsilon$. The $L_{0}=0$ condition is satisfied
with all signs negative and $n=1$ so that $L_{---}=u-2v_{[3]}-v_{[2]}$.
Following the same steps, we find

$\displaystyle i^{\text{div}}_{[332]}(u,v_{[2]},v_{[3]};\mu)$
$\displaystyle=\int_{0}^{\mu^{-1}}\mathrm{d}z\,z^{-1+\epsilon\,L_{---}}\frac{1}{2}\left[\frac{k_{3}^{2}}{1-2v_{[2]}\epsilon}-\frac{k_{1}^{2}+k_{2}^{2}}{1+2v_{[3]}\epsilon}\right]$
$\displaystyle=\frac{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}{2\epsilon\,L_{---}}+\frac{(k_{1}^{2}+k_{2}^{2})v_{[3]}+k_{3}^{2}v_{[2]}}{L_{---}}$
$\displaystyle\qquad\qquad\qquad-\frac{1}{2}(k_{3}^{2}-k_{1}^{2}-k_{2}^{2})\log\mu+O(\epsilon).$
(241)

The first term is an unambiguous divergence, which for $u=1$ and $v_{[j]}=0$
matches $\hat{i}_{[322]}$. The second term is a finite, scheme-dependent piece
and depends only on $v_{j}$ but not $u$. The last term contains the cut-off
scale, but otherwise is scheme-independent. This is the general structure,
which holds in the generic case. Using Fact 2, we find

$\displaystyle\hat{i}_{[332]}(u,v_{j})=\frac{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}{2(u-v_{[2]}-2v_{[3]})\,\epsilon}+\hat{i}_{[332]}(1,0)|_{\epsilon^{0}}+\frac{(k_{1}^{2}+k_{2}^{2})v_{[3]}+k_{3}^{2}v_{[2]}}{u-v_{[2]}-2v_{[3]}}+O(\epsilon).$
(242)

The finite part here requires the expansion of the regulated amplitude (25) to
order $\epsilon^{0}$, yielding

$\displaystyle\hat{i}_{[332]}(1,0)|_{\epsilon^{0}}$
$\displaystyle=\frac{1}{2}(k_{1}^{2}+k_{2}^{2}-k_{3}^{2})\left(\log
k_{t}+\gamma_{E}\right)-\frac{1}{4}(k_{1}+k_{2}-3k_{3})(k_{1}+k_{2}+k_{3}).$
(243)

This provides the expression for $\hat{i}_{[332]}$ regulated in an arbitrary
scheme through order $\epsilon^{0}$. The $\epsilon^{1}$ terms may be computed
as in the previous example. The final result through order $\epsilon^{1}$ is
given in (309)-(314).

###  4-point contact amplitudes

The scheme change procedure for 4-point (and in fact, any $n$-point) contact
amplitudes is completely analogous to that of 3-point amplitudes. We define
the analogue of (223) with four signs,

$\displaystyle L_{0}(\sigma_{j},n)$
$\displaystyle=d+\sum_{j=1}^{4}\sigma_{j}\beta_{j}+2n,\qquad
L_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}(u,v_{j})=u+\sum_{j=1}^{4}\sigma_{j}v_{j}$
(244)

and then (227) becomes

$\displaystyle i^{\text{div}}(u,v_{j};\mu)$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}=\pm
1}\frac{\mathcal{I}_{{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}}(v_{j})\mu^{-\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}}}{\epsilon
L_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}},$ (245)

where $\mathcal{I}_{{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}}(v_{j})$ is
defined by the analogous equation to (226). As in the case of 3-point
functions, in all the cases we discuss in this paper only one combination
signs gives a non-zero
$\mathcal{I}_{{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}}$, and in all cases
these signs are either all minus (corresponding to anomalies), or one plus and
three minus (corresponding to beta functions). The analogue of Fact 2 then
holds:

###### Fact 3

Consider a linearly divergent amplitude
$\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ regulated in an
arbitrary $u,v_{j}$-scheme. The expansion coefficients can be extracted as
follows,

$\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{-1}}$
$\displaystyle=i^{\text{div}}\left(u,v_{j};1\right)|_{\epsilon^{-1}},$ (246)
$\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{0}}$
$\displaystyle=\hat{i}(1,0)|_{\epsilon^{0}}+i^{\text{div}}\left(u,v_{j};1\right)|_{\epsilon^{0}},$
(247) $\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{1}}$
$\displaystyle=\hat{i}(u,0)|_{\epsilon^{1}}+\sum_{j=1}^{4}\epsilon
v_{j}\frac{\partial\hat{i}}{\partial\beta_{j}}(u,0)|_{\epsilon^{1}}+i^{\text{div}}\left(u,v_{j};1\right)|_{\epsilon^{1}}.$
(248)

The generalization to $n$-point contact amplitudes is straightforward.

###  4-point exchange diagrams

In this subsection, we present a procedure that allows one to evaluate the
4-point exchange amplitude in the general regularization scheme (2). The idea
is the same as in our discussion of 3-point functions: we would like to split
the regulated exchange amplitude $\hat{i}$ into two parts, with one of them,
$i^{\text{div}}$, containing all scheme-dependent terms, and then apply the
analogue of Fact 1. We will present such a split, but our construction is not
optimal. There is a natural ambiguity in that one may always add to a given
$i^{\text{div}}$ additional scheme-independent terms, and such terms would not
contribute to $\hat{i}(u,v_{j})$. Ideally, $i^{\text{div}}$ should not contain
any scheme-independent terms, but our procedure will produce $i^{\text{div}}$
containing (a large number of) such scheme-independent terms.

####  Definitions

We begin with a number of definitions. Since the bulk-to-bulk propagator (6)
contains a Bessel-$I$ function, we will need 3-point-like amplitudes with one
Bessel-$K$ replaced by a Bessel-$I$. To do this, we first define

$\displaystyle\operatorname{\mathcal{I}}_{d,\Delta}(z,k)=2^{\Delta-\frac{d}{2}-1}\Gamma\left(\Delta-\frac{d}{2}\right)k^{\Delta-\frac{d}{2}}z^{\frac{d}{2}}I_{\Delta-\frac{d}{2}}(kz)$
(249)

so that we can rewrite the bulk-to-bulk propagator as

$\displaystyle\operatorname{\mathcal{G}}_{d,\Delta}(z,k;\zeta)=k^{-(2\Delta-d)}\times\left\\{\begin{array}[]{ll}\operatorname{\mathcal{I}}_{\Delta-\frac{d}{2}}(kz)\operatorname{\mathcal{K}}_{\Delta-\frac{d}{2}}(k\zeta)&\text{
for }z<\zeta,\\\
\operatorname{\mathcal{K}}_{\Delta-\frac{d}{2}}(kz)\operatorname{\mathcal{I}}_{\Delta-\frac{d}{2}}(k\zeta)&\text{
for }z>\zeta.\end{array}\right.$ (252)

Next, we define the 3-point amplitude (8) with
$\hat{\operatorname{\mathcal{K}}}_{[\Delta_{3}]}$ replaced by
$\hat{\operatorname{\mathcal{I}}}_{[\Delta_{3}]}$ (i.e., the regulated
amplitude $\operatorname{\mathcal{I}}_{d,\Delta}$) as

$\displaystyle\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}=\int_{0}^{\infty}\mathrm{d}z\,z^{-\hat{d}-1}\,\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(z,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(z,k_{2})\hat{\operatorname{\mathcal{I}}}_{[\Delta_{3}]}(z,k_{3}).$
(253)

The properties of these amplitudes are very similar to those of the actual
3-point amplitudes (8). In particular
$j^{\text{div}}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$ is defined analogously to
(227) and Fact 2 holds as well.

Furthermore, we generalize the amplitudes $\hat{i}$ and $\hat{j}$ by leaving
the upper limits of integration open,

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(z;k_{1},k_{2},k_{3})$
$\displaystyle=\int_{0}^{z}\mathrm{d}\zeta\,\zeta^{-\hat{d}-1}\,\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(\zeta,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(\zeta,k_{2})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{3}]}(\zeta,k_{3}),$
(254)
$\displaystyle\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(z;k_{1},k_{2},k_{3})$
$\displaystyle=\int_{0}^{z}\mathrm{d}\zeta\,\zeta^{-\hat{d}-1}\,\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(\zeta,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(\zeta,k_{2})\hat{\operatorname{\mathcal{I}}}_{[\Delta_{3}]}(\zeta,k_{3}),$
(255)

so that
$\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}=\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(\infty)$
and
$\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}=\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(\infty)$.
Finally, we promote the explicit power of the integration variable to another
parameter,

$\displaystyle\hat{i}_{\alpha[\Delta_{1}\Delta_{2}\Delta_{3}]}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}\zeta\,\zeta^{\alpha}\,\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(\zeta,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(\zeta,k_{2})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{3}]}(\zeta,k_{3}),$
(256) $\displaystyle\hat{j}_{\alpha[\Delta_{1}\Delta_{2};\Delta_{3}]}$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}\zeta\,\zeta^{\alpha}\,\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(\zeta,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(\zeta,k_{2})\hat{\operatorname{\mathcal{I}}}_{[\Delta_{3}]}(\zeta,k_{3}),$
(257)

so that
$\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}=\hat{i}_{-\hat{d}-1,[\Delta_{1}\Delta_{2}\Delta_{3}]}$
and
$\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}=\hat{j}_{-\hat{d}-1,[\Delta_{1}\Delta_{2};\Delta_{3}]}$.

####  Change of scheme

Using the bulk-to-bulk propagator (252) and
$\int_{z}^{\infty}=\int_{0}^{\infty}-\int_{0}^{z}$ we can rewrite the 4-point
exchange amplitude (• ‣ 2) as

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=s^{-2\hat{\beta}_{x}}\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)\,\hat{i}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(k_{3},k_{4},s)$
$\displaystyle\qquad-s^{-2\hat{\beta}_{x}}\int_{0}^{\infty}\mathrm{d}z\,z^{-\hat{d}-1}\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(z,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(z,k_{2})\hat{\operatorname{\mathcal{I}}}_{[\Delta_{x}]}(z,s)\hat{i}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(z;k_{3},k_{4},s)$
$\displaystyle\qquad+s^{-2\hat{\beta}_{x}}\int_{0}^{\infty}\mathrm{d}z\,z^{-\hat{d}-1}\hat{\operatorname{\mathcal{K}}}_{[\Delta_{1}]}(z,k_{1})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{2}]}(z,k_{2})\hat{\operatorname{\mathcal{K}}}_{[\Delta_{x}]}(z,s)\hat{j}_{[\Delta_{3}\Delta_{4};\Delta_{x}]}(z;k_{3},k_{4},s).$
(258)

The $z$-dependent amplitudes
$\hat{i}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(z;k_{3},k_{4},s)$ and
$\hat{j}_{[\Delta_{3}\Delta_{4};\Delta_{x}]}(z;k_{3},k_{4},s)$ can now be
computed after power expanding the integrands using (487) and (489).
Analogously to (219), let $\mathcal{I}$ and $\mathcal{J}$ denote the
integrands in (254) and (255) respectively. Furthermore, let
$\mathcal{I}_{(\sigma_{j},n)}$ denote the coefficient of $\zeta^{L-1}$ in the
power expansion of $\mathcal{I}$, and
$\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}$ the coefficient of
$\zeta^{L^{\prime}-1}$ in the power expansion of $\mathcal{J}$. Then

$\displaystyle\hat{i}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(z;k_{3},k_{4},s)$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{n=0}^{\infty}\frac{\mathcal{I}_{(\sigma_{j},n)}(k_{3},k_{4},s)z^{L(\sigma_{j},n)}}{L(\sigma_{j},n)},$
(259)
$\displaystyle\hat{j}_{[\Delta_{3}\Delta_{4};\Delta_{x}]}(z;k_{3},k_{4},s)$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2}=\pm
1}\sum_{n=0}^{\infty}\frac{\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}(k_{3},k_{4},s)z^{L^{\prime}(\sigma_{1},\sigma_{2},n)}}{L^{\prime}(\sigma_{1},\sigma_{2},n)}.$
(260)

The definition of $\mathcal{I}_{(\sigma_{j},n)}(k_{3},k_{4},s)$ and the value
of $L$ are given in (221), and $\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}$ is
defined analogously. For the values of $L^{\prime}$, notice that the series
expansion of the Bessel-$I$ function in (487) is similar to that of the
Bessel-$K$ function (489) but there are only $z^{\beta}$ terms, so relative to
Bessel-$K$ we have $\sigma=1$. Thus, the values of $L^{\prime}$ are identical
to $L$, but with $\sigma_{3}=1$ fixed,

$\displaystyle
L^{\prime}(\sigma_{1},\sigma_{2},n)=L(\sigma_{1},\sigma_{2},+1,n)=\frac{\hat{d}}{2}+\sum_{j=1}^{2}\sigma_{j}\hat{\beta}_{j}+\hat{\beta}_{3}+2n,$
(261)

and we define

$L^{\prime}=L^{\prime}_{0}+\epsilon L^{\prime}_{\sigma_{1}\sigma_{2}},\quad
L^{\prime}_{0}=\frac{d}{2}+\sum_{j=1}^{2}\sigma_{j}\beta_{j}+\beta_{3}+2n,\quad
L^{\prime}_{\sigma_{1}\sigma_{2}}=u+\sum_{j=1}^{2}\sigma_{j}v_{j}+v_{3}\,.$
(262)

With this notation in place, we rewrite (5.4.2) as

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=s^{-2\hat{\beta}_{x}}\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)\,\hat{i}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(k_{3},k_{4},s)$
$\displaystyle\qquad-s^{-2\hat{\beta}_{x}}\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{n=0}^{\infty}\frac{\mathcal{I}_{(\sigma_{j},n)}(k_{3},k_{4},s)}{L(\sigma_{j},n)}\,\hat{j}_{-\hat{d}-1+L,[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)$
$\displaystyle\qquad+s^{-2\hat{\beta}_{x}}\sum_{\sigma_{1},\sigma_{2}=\pm
1}\sum_{n=0}^{\infty}\frac{\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}(k_{3},k_{4},s)}{L^{\prime}(\sigma_{1},\sigma_{2},n)}\,\hat{i}_{-\hat{d}-1+L^{\prime},[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s).$
(263)

Thus, the divergent terms $i^{\text{div}}$ for the exchange 4-point function
can be extracted from our knowledge of the scheme-dependent terms for all
relevant 3-point functions. We arrive at:

###### Fact 4

The piece $i^{\rm
div}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$ containing all
divergences and all finite scheme-dependent terms for the exchange 4-point
amplitudes with $d=3$ and $\Delta_{j}=2$ or $3$ can be computed as

$\displaystyle s^{2\hat{\beta}_{x}}i^{\rm
div}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}=\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)\,\hat{i}_{[\Delta_{3}\Delta_{4}\Delta_{x}]}(k_{3},k_{4},s)$
$\displaystyle\qquad-\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{\begin{subarray}{c}n=0\\\ L_{0}\neq
0\end{subarray}}^{\infty}\frac{\mathcal{I}_{(\sigma_{j},n)}(k_{3},k_{4},s)}{L(\sigma_{j},n)}\,j^{\rm
div}_{-\hat{d}-1+L,[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)$
$\displaystyle\qquad-\sum_{\begin{subarray}{c}\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1\\\
\\{L_{0}=0\\}\end{subarray}}\frac{\mathcal{I}_{{\sigma_{1}\sigma_{2}\sigma_{3}}}(k_{3},k_{4},s)}{\epsilon\,L_{\sigma_{1}\sigma_{2}\sigma_{3}}}\,\hat{j}_{-\hat{d}-1+\epsilon\,L_{\sigma_{1}\sigma_{2}\sigma_{3}},[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)$
$\displaystyle\qquad+\sum_{\sigma_{1},\sigma_{2}=\pm
1}\sum_{\begin{subarray}{c}n=0\\\ L^{\prime}_{0}\neq
0\end{subarray}}^{\infty}\frac{\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}(k_{3},k_{4},s)}{L^{\prime}(\sigma_{1},\sigma_{2},n)}\,i^{\rm
div}_{-\hat{d}-1+L^{\prime},[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s)$
$\displaystyle\qquad+\sum_{\begin{subarray}{c}\sigma_{1},\sigma_{2}=\pm 1\\\
\\{L^{\prime}_{0}=0\\}\end{subarray}}\frac{\mathcal{J}_{\sigma_{1}\sigma_{2}}(k_{3},k_{4},s)}{\epsilon\,L^{\prime}_{\sigma_{1}\sigma_{2}}}\,\hat{i}_{-\hat{d}-1+\epsilon\,L^{\prime}_{\sigma_{1}\sigma_{2}},[\Delta_{1}\Delta_{2};\Delta_{x}]}(k_{1},k_{2},s),$
(264)

where $\mathcal{I}_{{\sigma_{1}\sigma_{2}\sigma_{3}}}(k_{3},k_{4},s)$ is
defined in (226) and $\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}(k_{3},k_{4},s)$
has an analogous definition. The summation in the second and fourth terms is
over all signs and all integers such that $L_{0}$ and $L_{0}^{\prime}$ are not
equal to zero, respectively. These summations are finite, since only
amplitudes with

$\displaystyle
L\leq\hat{\beta}_{1}+\hat{\beta}_{2}-\hat{\beta}_{x}-\frac{\hat{d}}{2},$
$\displaystyle
L^{\prime}\leq\hat{\beta}_{1}+\hat{\beta}_{2}+\hat{\beta}_{x}-\frac{\hat{d}}{2}$
(265)

can be divergent. The summations in the third and fifth terms are over the
locus of $L_{0}=0$ and $L_{0}^{\prime}=0$, respectively.

Thus, if we already know the amplitude $\hat{i}(\bar{u},\bar{v}_{j})$
regulated in _some_ scheme with parameters $\bar{u}$ and $\bar{v}_{j}$, we can
use it to obtain the amplitude in _any_ regularization scheme,

$\displaystyle\hat{i}(u,v_{j})=\hat{i}(\bar{u},\bar{v}_{j})+\left[i^{\rm
div}(u,v_{j})-i^{\rm div}(\bar{u},\bar{v}_{j})\right]+O(\epsilon).$ (266)

In order to use this fact we must be careful to evaluate (4) including all
finite terms of order $\epsilon^{0}$. This means that one generally needs to
evaluate all 3-point amplitudes up to and including terms of order
$\epsilon^{1}$. For example, in the first line both 3-point amplitudes may be
linearly divergent. Thus, in order to extract correctly the finite $u$ and
$v_{j}$-dependent terms we have to multiply the divergence of one amplitude by
terms of order $\epsilon^{1}$ in the other. This is the reason why we computed
the order $\epsilon$ parameter-dependent terms in 3-point amplitudes in Fact
2.

####  Example

Let us discuss an example of the scheme change for the amplitude
$\hat{i}_{[33,22x2]}$. To make the expressions slightly shorter, let us use
only two parameters $v_{[2]}$ and $v_{[3]}$ to regulate the corresponding
dimensions, i.e., for $\Delta=2$ the regulated dimension is
$\hat{\Delta}=2+(u+v_{[2]})\epsilon$, while for $\Delta=3$ we take
$\hat{\Delta}=3+(u+v_{[3]})\epsilon$.

First, we find out which 3-point amplitudes are needed in (4). The first term
is the product of $\hat{i}_{[222]}$ and $\hat{j}_{[33;2]}$. Since the
amplitude $\hat{j}_{[33;2]}$ is finite (see Table 4), it is enough to know
$\hat{i}_{[222]}$ only up to finite order. On the other hand, we need all $u$
and $v_{j}$-dependent terms, including terms of order $\epsilon$ in
$\hat{j}_{[33;2]}$ as they get multiplied by the divergence of
$\hat{i}_{[222]}$.

For the next two terms we must power expand the integrands of
$\hat{i}_{[222]}$ and $\hat{j}_{[22;2]}$. We must include all terms of order
$z^{L-1}$ with $L$ and $L^{\prime}$ bounded by (265). This means $L\leq 1$ and
$L^{\prime}\leq 2$. We find

$\displaystyle\mathcal{I}$ $\displaystyle=\zeta^{-1+\epsilon(u-3v_{[2]})}$
$\displaystyle\qquad+\zeta^{\epsilon(u-v_{[2]})}\frac{\Gamma(-\tfrac{1}{2}-\epsilon
v_{[2]})}{2^{1+2\epsilon v_{[2]}}\Gamma(\tfrac{1}{2}+\epsilon
v_{[2]})}\left(k_{3}^{1+2\epsilon v_{[2]}}+k_{4}^{1+2\epsilon
v_{[2]}}+s^{1+2\epsilon v_{[2]}}\right)$ $\displaystyle\qquad+O(\zeta),$ (267)
$\displaystyle\mathcal{J}$
$\displaystyle=\zeta^{\epsilon(u-v_{[2]})}\frac{\Gamma(\tfrac{1}{2}+\epsilon
v_{[2]})}{2\Gamma(\tfrac{3}{2}+\epsilon v_{[2]})}s^{1+2\epsilon v_{[2]}}$
$\displaystyle\qquad+\zeta^{1+\epsilon(u+v_{[2]})}\frac{\Gamma(-\tfrac{1}{2}-\epsilon
v_{[2]})}{2^{2+2\epsilon v_{[2]}}\Gamma(\tfrac{3}{2}+\epsilon
v_{[2]})}\left[(k_{3}s)^{1+2\epsilon v_{[2]}}+(k_{4}s)^{1+2\epsilon
v_{[2]}}\right]$ $\displaystyle\qquad+O(\zeta^{2}).$ (268)

When integrated term-by-term, only the first term in $\mathcal{I}$ directly
produces a singularity at $\epsilon=0$. This means that the scheme-dependent
terms $i^{\text{div}}_{[33,22x2]}$ are given by

$\displaystyle s^{1+2v_{[2]}\epsilon}i^{\text{div}}_{[33,22x2]}$
$\displaystyle=\hat{j}_{[33;2]}(k_{1},k_{2},s)\hat{i}_{[222]}(k_{3},k_{4},s)$
$\displaystyle\qquad-\frac{1}{\epsilon(u-3v_{[2]})}\times\hat{j}_{-4-\epsilon(u+3v_{[2]}),[33;2]}(k_{1},k_{2},s)$
$\displaystyle\qquad-\frac{\Gamma(-\tfrac{1}{2}-\epsilon
v_{[2]})}{2^{1+2\epsilon
v_{[2]}}(1+\epsilon(u-v_{[2]}))\Gamma(\tfrac{1}{2}+\epsilon
v_{[2]})}\left(k_{3}^{1+2\epsilon v_{[2]}}+k_{4}^{1+2\epsilon
v_{[2]}}+s^{1+2\epsilon v_{[2]}}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\times
j^{\text{div}}_{-3-\epsilon(u+v_{[2]}),[33;2]}(k_{1},k_{2},s)$
$\displaystyle\qquad+\frac{\Gamma(\tfrac{1}{2}+\epsilon
v_{[2]})}{2(1+\epsilon(u-v_{[2]}))\Gamma(\tfrac{3}{2}+\epsilon
v_{[2]})}s^{1+2\epsilon
v_{[2]}}i^{\text{div}}_{-3-\epsilon(u+v_{[2]}),[332]}(k_{1},k_{2},s)$
$\displaystyle\qquad+\frac{\Gamma(-\tfrac{1}{2}-\epsilon
v_{[2]})}{2^{2+2\epsilon
v_{[2]}}(2+\epsilon(u+v_{[2]}))\Gamma(\tfrac{3}{2}+\epsilon
v_{[2]})}\left[(k_{3}s)^{1+2\epsilon v_{[2]}}+(k_{4}s)^{1+2\epsilon
v_{[2]}}\right]\times$ $\displaystyle\qquad\qquad\qquad\qquad\times
i^{\text{div}}_{-2-\epsilon(u-v_{[2]}),[332]}(k_{1},k_{2},s).$ (269)

To complete the calculation, we must be able to evaluate the amplitudes on the
right-hand side. The divergent terms can always be evaluated by means of the
definition (227). For example, to calculate
$j^{\text{div}}_{-3-\epsilon(u+v_{[2]}),[33;2]}(k_{1},k_{2},s)$ we power
expand the integrand in (257) and pick the term of order
$\zeta^{-1+O(\epsilon)}$. To be precise, the power expansion reads

$\displaystyle\zeta^{-3-\epsilon(u+v_{[2]})}\hat{\operatorname{\mathcal{K}}}_{[3]}(\zeta,k_{1})\hat{\operatorname{\mathcal{K}}}_{[3]}(\zeta,k_{2})\hat{\operatorname{\mathcal{I}}}_{[2]}(\zeta,s)=\frac{s^{1+2\epsilon
v_{[2]}}\Gamma(\tfrac{1}{2}+\epsilon v_{[2]})}{2\Gamma(\tfrac{3}{2}+\epsilon
v_{[2]})}\zeta^{-1+2\epsilon(u-v_{[3]})}+O(\zeta^{0}).$ (270)

Thus,

$\displaystyle j^{\text{div}}_{-3-\epsilon(u+v_{[2]}),[33;2]}(k_{1},k_{2},s)$
$\displaystyle=\frac{s^{1+2\epsilon v_{[2]}}\Gamma(\tfrac{1}{2}+\epsilon
v_{[2]})}{2\Gamma(\tfrac{3}{2}+\epsilon
v_{[2]})}\int_{0}^{\mu^{-1}}\mathrm{d}\zeta\,\zeta^{-1+2\epsilon(u-v_{[3]})}$
$\displaystyle=\frac{s^{1+2\epsilon v_{[2]}}\Gamma(\tfrac{1}{2}+\epsilon
v_{[2]})}{2\Gamma(\tfrac{3}{2}+\epsilon
v_{[2]})}\times\frac{\mu^{-2\epsilon(u-v_{[3]})}}{2\epsilon\,(u-v_{[3]})}.$
(271)

In the same fashion we find

$\displaystyle i^{\text{div}}_{-3-\epsilon(u+v_{[2]}),[332]}(k_{1},k_{2},s)$
$\displaystyle=\frac{s^{1+2\epsilon
v_{[2]}}\mu^{-2\epsilon(u-v_{[3]})}\Gamma(-\tfrac{1}{2}-\epsilon
v_{[2]})}{2^{2+2\epsilon
v_{[2]}}\epsilon\,(u-v_{[3]})\Gamma(\tfrac{1}{2}+\epsilon v_{[2]})},$ (272)
$\displaystyle i^{\text{div}}_{-2-\epsilon(u-v_{[2]}),[332]}(k_{1},k_{2},s)$
$\displaystyle=\frac{\mu^{-2\epsilon(u-v_{[3]})}}{2\epsilon\,(u-v_{[3]})}.$
(273)

The remaining term can be evaluated including terms of order $\epsilon^{1}$,
since $\hat{j}_{-4-\epsilon(u+3v_{[2]}),[33;2]}$ equals $\hat{j}_{[33;2]}$
evaluated in the scheme with the $u$-parameter equal to $2u-3v_{[2]}$, i.e.,

$\displaystyle\hat{j}_{-4-\epsilon(u+3v_{[2]}),[33;2]}(u;v_{[2]},v_{[3]})=\hat{j}_{[33;2]}(2u-3v_{[2]};v_{[2]},v_{[3]}).$
(274)

When these results are substituted back to (5.4.3) and power expanded we
arrive at

$\displaystyle i^{\text{div}}_{[33,22x2]}$ $\displaystyle=s^{-1-2\epsilon
v_{[2]}}\left[\frac{\hat{j}^{2u-3v_{[2]}\\{v_{[3]}v_{[3]}v_{[2]}\\}}_{[33;2]}(k_{1},k_{2},s)}{\epsilon\,(-u+3v_{[2]})}+\hat{j}_{[33;2]}(k_{1},k_{2},s)\hat{i}_{[222]}(s,k_{3},k_{4})\right]$
$\displaystyle\qquad+\frac{k_{3}}{2(u-v_{[3]})}\left[\frac{1}{2\epsilon}+v_{[2]}(\log
k_{3}+\gamma_{E}+\log 2-\tfrac{7}{4})+\frac{1}{4}(-3u+4v_{[2]})\right]$
$\displaystyle\qquad+\frac{k_{4}}{2(u-v_{[3]})}\left[\frac{1}{2\epsilon}+v_{[2]}(\log
k_{4}+\gamma_{E}+\log
2-\tfrac{7}{4})+\frac{1}{4}(-3u+4v_{[2]})\right]+O(\epsilon).$ (275)

In the first term we used the convention where, in the superscript, we
indicate the value of the $u$ and $v_{j}$ parameters at which the 3-point
amplitude should be evaluated,

$\displaystyle\hat{i}^{u\\{v_{1}v_{2}v_{3}\\}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3})$
$\displaystyle=\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3};\hat{d}=3+2u\epsilon,\hat{\Delta}_{j}=\Delta_{j}+(u+v_{j})\epsilon),$
(276)
$\displaystyle\hat{j}^{u\\{v_{1}v_{2}v_{3}\\}}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(k_{1},k_{2},k_{3})$
$\displaystyle=\hat{i}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(k_{1},k_{2},k_{3};\hat{d}=3+2u\epsilon,\hat{\Delta}_{j}=\Delta_{j}+(u+v_{j})\epsilon).$
(277)

All results are listed in Section 5.5. There, we list first the 3-point
amplitudes evaluated to order $\epsilon^{1}$, then the regulated 4-point
amplitudes evaluated in an arbitrary scheme.

####  Proofs

Our goal in this subsection is to prove Fact 2 and Fact 4. The starting point
is the analysis of singularities of the regulated 3-point amplitude (8). The
full analysis is carried out in Bzowski:2015yxv , but here we will limit our
attention to the cases at hand, i.e., we assume unregulated $\beta_{j}$
parameters that are non-integral. This implies that all expansion coefficients
in (490) are continuous at $\epsilon=0$, meaning one can split the integral in
(8) into two regions. Choosing $\mu>0$, we write

$\displaystyle\hat{i}$
$\displaystyle=\int_{\mu^{-1}}^{\infty}\mathrm{d}z\,\mathcal{I}+\int_{0}^{\mu^{-1}}\mathrm{d}z\sum_{L}\mathcal{I}_{L}z^{L-1}.$
(278)

Due to the exponential fall-off of the Bessel functions, the first integral
converges for all $\alpha$ and $\beta_{j}$. By the dominated convergence
theorem, one can pass the $\epsilon\rightarrow 0$ limit through the integral.
For the second integral, recall that the regulated amplitude $\hat{i}$ is
defined by the analytic continuation of the parameters $\alpha$ and
$\beta_{j}$ to the region of convergence. In other words, we can assume that
the powers of $z$ are all greater than $-1$ and at least the integral of each
term separately exists. Thus, to commute the sum and the integral it is enough
to show that the series converges to an integrable function when the
$\mathcal{I}_{L}$ are replaced by their absolute values, $|\mathcal{I}_{L}|$.
This is indeed the case, since almost all coefficients $a_{n}(\nu)$ are
positive, i.e., $a_{n}(\nu)>0$ for all $n\geq\nu$. Since the radius of
convergence of (489) is infinite, we can reverse the order of summation and
integration in (278) by the Fubini’s theorem. This shows the following:

###### Fact 5

The amplitude $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ is divergent at
$\epsilon\rightarrow 0$ if and only if there exist independent choices of
signs $\sigma_{1},\sigma_{2},\sigma_{3}=\pm 1$ and a non-negative integer $n$
such that $L_{0}(\sigma_{j},n)=0$, i.e.,

$\displaystyle\frac{d}{2}+\sigma_{1}\beta_{1}+\sigma_{2}\beta_{2}+\sigma_{3}\beta_{3}=-2n.$
(279)

The same argument holds for the amplitudes
$\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$, except that the corresponding
condition is

$\displaystyle
L^{\prime}_{0}(\sigma_{1},\sigma_{2},n)=L_{0}(\sigma_{1},\sigma_{2},+1,n)=0.$
(280)

Since the sign of $\sigma_{3}$ is fixed to $+1$, the amplitudes
$\hat{j}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ cannot be more singular than
$\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$. The following table presents the
rank of the divergence at $\epsilon=0$ of the 3-point amplitudes:

Dimensions | $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ | $\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$
---|---|---
$[22;2]$ | $1$ | $0$
$[22;3]$ | $1$ | $0$
$[32;2]$ | $1$ | $1$
$[32;3]$ | $1$ | $0$
$[33;2]$ | $1$ | $0$
$[33;3]$ | $1$ | $1$
Table 4: Degrees of singularities of 3-point amplitudes.

The reasoning above is sufficient to prove Fact 1. Indeed, as we can see, all
terms in (278) have a finite $\epsilon=0$ limit except the ones connected with
$L_{0}=0$. Thus, all parameter-dependent terms are contained in
$i^{\text{div}}$. This is sufficient to prove the first two equations in Fact
2. However, since we also want to derive the scheme-dependent terms in terms
of order $\epsilon^{1}$, we must first prove the following:

###### Fact 6

Consider a regulated amplitude $\hat{i}(u,v_{j})$ exhibiting a first order
pole as $\epsilon\rightarrow 0$. As far as the $u$ and $v_{j}$-dependence of
the amplitude is concerned, one has the decomposition

$\displaystyle i^{\rm div}(u,v_{j})$
$\displaystyle=\frac{1}{\epsilon}\,\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\frac{A_{\sigma_{1}\sigma_{2}\sigma_{3}}}{L_{\sigma_{1}\sigma_{2}\sigma_{3}}}$
$\displaystyle\qquad+B+\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{j=1}^{3}\frac{B_{\sigma_{1}\sigma_{2}\sigma_{3}}^{(j)}v_{j}}{L_{\sigma_{1}\sigma_{2}\sigma_{3}}}$
$\displaystyle\qquad+\epsilon\left[Cu+\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{j=1}^{3}C_{\sigma_{1}\sigma_{2}\sigma_{3}}^{(j)}v_{j}+\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{i,j=1}^{3}\frac{C^{(ij)}_{\sigma_{1}\sigma_{2}\sigma_{3}}v_{i}v_{j}}{L_{\sigma_{1}\sigma_{2}\sigma_{3}}}\right]+O(\epsilon^{2}),$
(281)

where $L_{\sigma_{1}\sigma_{2}\sigma_{3}}$ is given by (223) and
$A_{\sigma_{1}\sigma_{2}\sigma_{3}},B,B_{\sigma_{1}\sigma_{2}\sigma_{3}}^{(j)},C,C_{\sigma_{1}\sigma_{2}\sigma_{3}}^{(j)},C^{(ij)}_{\sigma_{1}\sigma_{2}\sigma_{3}}$
are all $u$\- and $v_{j}$-independent. The same structure holds for
$\hat{i}(u,v_{j})$.

This fact follows from the structure of terms in (278). All terms in (278),
except (227), are finite as $\epsilon\to 0$. The upper integral in (278) is
finite and scheme-independent. The finite terms of the lower integral are
given by

$i^{\rm finite}=\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{\begin{subarray}{c}n=0\\\ L_{0}\neq
0\end{subarray}}^{\infty}\frac{\mathcal{I}_{(\sigma_{j},n)}\mu^{-L(\sigma_{j},n)}}{L(\sigma_{j},n)}.$
(282)

As shown in (224), $\mathcal{I}_{(\sigma_{j},n)}$ is finite and scheme-
independent as $\epsilon\to 0$, and the order $\epsilon$ term is linear in
$v_{j}$ but independent of $u$ (more generally, the $n$th order term in their
Taylor expansion around $\epsilon=0$ is a polynomial of order $n$ in $v_{j}$
and has no $u$ dependence). Similarly, the $\epsilon,u,v_{j}$ dependence of
terms depending on $L(\sigma_{j},n)$ follow from (222)-(223). When $L_{0}\neq
0$, such terms are finite and scheme-independent as $\epsilon\to 0$, and the
$n$th order term in their Taylor expansion around $\epsilon=0$ is a polynomial
of order $n$ in $u,v_{j}$. It follows that in the absence of divergences the
$\epsilon$-expansion of any amplitude takes the form (6) with
$A_{\sigma_{1}\sigma_{2}\sigma_{3}}=B_{\sigma_{1}\sigma_{2}\sigma_{3}}^{(j)}=C^{(ij)}_{\sigma_{1}\sigma_{2}\sigma_{3}}=0$.

Divergent terms appear only from (227) and have the form presented. Expanding
these in $\epsilon$, we see that we can obtain a finite scheme-dependent term
from the order $\epsilon$ part of $\mathcal{I}_{(\sigma_{j},n)}$, which thus
depends only on $v_{j}$, and their expansion to order $\epsilon$ yields terms
that linear in $u$ and quadratic in $v_{j}$. Thus (6) contains all divergent
and all scheme-dependent terms through order $\epsilon$.

Using the above fact we can derive Fact 2. Consider the finite terms first.
Notice that $B$ is scheme-independent and the remaining term in the second
line vanishes at $v_{j}=0$. Thus $B$ is the finite part of $\hat{i}(1,0)$
evaluated in the half-integer scheme. To recover
$B^{(j)}_{\sigma_{1}\sigma_{2}\sigma_{3}}$ we use (228) with $\bar{u}=1$ and
$\bar{v}_{j}=0$, which tells us that

$\displaystyle\left[\hat{i}(u,v_{j})-\hat{i}(1,0)\right]_{\epsilon^{0}}=\left[i^{\text{div}}(u,v_{j})-i^{\text{div}}(1,0)\right]_{\epsilon^{0}}.$
(283)

Since only $i^{\text{div}}(u,v_{j})$ is $v_{j}$-dependent on the right hand
side, we can find $B^{(j)}_{\sigma_{1}\sigma_{2}\sigma_{3}}$ by retrieving the
coefficient of $v_{j}/L_{\sigma_{1}\sigma_{2}\sigma_{3}}$ in
$i^{\text{div}}(u,v_{j})$. Indeed, using (224) we see that $i^{\text{div}}$
contains precise a terms of this form. In conclusion,

$\displaystyle\hat{i}(u,v_{j})|_{\epsilon^{0}}=\hat{i}(1,0)|_{\epsilon^{0}}+\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{j=1}^{3}i^{\text{div}}(u,v_{j};1)|_{\epsilon^{0}}.$ (284)

An analogous argument holds for terms of order $\epsilon^{1}$. Since for
$v_{j}=0$ only the first term $C$ survives, its value is equal to the term of
order $\epsilon$ in $\hat{i}(1,0)$. The term linear in $v_{j}$ is similarly
determined by the derivative $\partial\hat{i}/\partial\beta_{j}$, and the last
term, with $L_{\sigma_{1}\sigma_{2}\sigma_{3}}$ in the denominator, is
accounted for by $i^{\text{div}}$. We simply power expand (227) to order
$\epsilon$ and keep terms quadratic in the $v_{j}$. This determines
$C^{(ij)}_{\sigma_{1}\sigma_{2}\sigma_{3}}$. All in all, we have found a
simple prescription to calculate any 3-point amplitude $\hat{i}(u,v_{j})$ to
order $\epsilon^{1}$ in any regularization scheme.

Let us now discuss Fact 4. To prove Fact 4, it suffices to show that
$\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}-i^{\text{div}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$
is finite and scheme-independent. The difference is equal to

$\displaystyle
s^{2\hat{\beta}_{x}}\left(\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}-i^{\rm
div}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}\right)=$ (285)
$\displaystyle\qquad-\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}\sum_{\begin{subarray}{c}n=0\\\ L_{0}\neq
0\end{subarray}}^{\infty}\frac{\mathcal{I}_{(\sigma_{j},n)}(k_{3},k_{4},s)}{L(\sigma_{j},n)}\,\left(\hat{j}_{-\hat{d}-1+L,[\Delta_{1}\Delta_{2};\Delta_{x}]}-j^{\rm
div}_{-\hat{d}-1+L,[\Delta_{1}\Delta_{2};\Delta_{x}]}\right)$
$\displaystyle\qquad+\sum_{\sigma_{1},\sigma_{2}=\pm
1}\sum_{\begin{subarray}{c}n=0\\\ L^{\prime}_{0}\neq
0\end{subarray}}^{\infty}\frac{\mathcal{J}_{(\sigma_{1},\sigma_{2},n)}(k_{3},k_{4},s)}{L^{\prime}(\sigma_{1},\sigma_{2},n)}\,\left(\hat{i}_{-\hat{d}-1+L^{\prime},[\Delta_{1}\Delta_{2};\Delta_{x}]}-i^{\rm
div}_{-\hat{d}-1+L^{\prime},[\Delta_{1}\Delta_{2};\Delta_{x}]}\right)$ (286)

By construction, $(\hat{j}-j^{\rm div})$ and $(\hat{i}-i^{\text{div}})$ are
finite and scheme-independent, and we we also showed earlier that
$\mathcal{I}_{(\sigma_{j},n)},L(\sigma_{j},n),\mathcal{J}_{(\sigma_{1},\sigma_{2},n)},L^{\prime}(\sigma_{1},\sigma_{2},n)$
are finite and scheme-independent as $\epsilon\to 0$. It follows that the
right hand side of (285) is indeed finite and scheme-independent.

###  List of results

In this section we list:

  * •

All 3-point amplitudes $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(u,v_{j})$
and $\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(u,v_{j})$ for $\Delta_{j}=2$
or $3$ in an arbitrary regularization scheme, up to and including terms of
order $\epsilon^{1}$.

  * •

All 4-point amplitudes
$\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}(u,v_{j})$ and
$\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(u,v_{j})$
regulated in an arbitrary regularization scheme.

To do so, we introduce the following definitions.

####  Definitions

  * •

The amplitudes $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ and
$\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$ are defined in (8) and (253). We
consider the general regularization scheme (2).

  * •

3-point amplitudes evaluated up to and including order $\epsilon^{1}$ are
decomposed as follows:

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$
$\displaystyle=i^{\text{div}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}+i^{\text{fin}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}+\epsilon\left[u\,\hat{i}^{u\epsilon}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}+\sum_{j=1}^{3}v_{j}\hat{i}^{v_{j}\epsilon}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}\right]+O(\epsilon^{2}),$
(287) $\displaystyle\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$
$\displaystyle=j^{\text{div}}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}+j^{\text{fin}}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}+\epsilon\left[u\,\hat{j}^{u\epsilon}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}+\sum_{j=1}^{3}v_{j}\hat{j}^{v_{j}\epsilon}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}\right]+O(\epsilon^{2}),$
(288)

where

$\displaystyle i^{\text{div}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2},\sigma_{3}=\pm
1}i^{\sigma_{1}\sigma_{2}\sigma_{3}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]},$
(289) $\displaystyle j^{\text{div}}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$
$\displaystyle=\sum_{\sigma_{1},\sigma_{2}=\pm
1}j^{\sigma_{1}\sigma_{2}+}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}.$ (290)

Only the non-vanishing terms
$i^{\sigma_{1}\sigma_{2}\sigma_{3}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ and
$j^{\sigma_{1}\sigma_{2}+}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ are listed.

  * •

The divergences of the amplitude are contained entirely in the divergent parts
$i^{\text{div}}$ and $j^{\text{div}}$. All remaining terms are parameter-
independent.

  * •

3-point amplitudes are treated as functions of the momentum magnitudes
$k_{1},k_{2},k_{3}$. We also use the notation

$\displaystyle k_{t}$ $\displaystyle=k_{1}+k_{2}+k_{3},$ $\displaystyle v_{t}$
$\displaystyle=v_{1}+v_{2}+v_{3},$ $\displaystyle l_{12-}$
$\displaystyle=k_{1}+k_{2}-k_{3},$ (291)

and analogously for $l_{13-}$ and $l_{23-}$. This follows from (470) as
$|\bm{k}_{1}+\bm{k}_{2}|=k_{3}$.

  * •

For 4-point functions, we use the notations

$\displaystyle k_{T}$ $\displaystyle=k_{1}+k_{2}+k_{3}+k_{4},$ $\displaystyle
v_{T}$ $\displaystyle=v_{1}+v_{2}+v_{3}+v_{4}.$ (292)

  * •

An exchange 4-point function
$\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(u,v_{1},v_{2},v_{3},v_{4},v_{x})$
can be evaluated in the regularization scheme (2) by means of the formula

$\displaystyle\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(u,v_{j})$
$\displaystyle=\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(1,0)$
$\displaystyle\quad+\left[i^{\text{div}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(u,v_{j})-i^{\text{div}}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(1,0)\right].$
(293)

The amplitudes
$\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}(1,0)$ are
evaluated in the half-integer scheme and they are listed in Section 3. The
divergent terms are then listed in Section 5.5.5.

  * •

The divergent terms for the 4-point amplitudes contain 3-point amplitudes
evaluated in some specific schemes. To avoid clutter, we use the following
notation,

$\displaystyle\hat{i}^{u\\{v_{1}v_{2}v_{3}\\}}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3})$
$\displaystyle=\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}(k_{1},k_{2},k_{3};\hat{d}=3+2u\epsilon,\hat{\Delta}_{j}=\Delta_{j}+(u+v_{j})\epsilon),$
(294)
$\displaystyle\hat{j}^{u\\{v_{1}v_{2}v_{3}\\}}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(k_{1},k_{2},k_{3})$
$\displaystyle=\hat{i}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}(k_{1},k_{2},k_{3};\hat{d}=3+2u\epsilon,\hat{\Delta}_{j}=\Delta_{j}+(u+v_{j})\epsilon),$
(295)

where the subscripts indicate the values of the $u$ and $v_{j}$-parameters.

  * •

Certain expressions are long and unwieldy – despite our best efforts to avoid
typos, when in doubt, we recommend using the results stored in the Mathematica
notebooks.

####  3-point amplitudes $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$

The expansions of the amplitudes $\hat{i}_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$
to order $\epsilon^{1}$ are as follows:

  * •

For $\hat{i}_{[222]}$:

$\displaystyle i^{---}_{[222]}$ $\displaystyle=\frac{1}{\epsilon\,(u-v_{t})},$
(296) $\displaystyle i^{\text{fin}}_{[222]}$ $\displaystyle=-\log
k_{t}-\gamma_{E},$ (297) $\displaystyle\hat{i}^{u\epsilon}_{[222]}$
$\displaystyle=\frac{1}{2}(\log k_{t}+\gamma_{E})^{2}+\frac{\pi^{2}}{12},$
(298) $\displaystyle\hat{i}^{v_{1}\epsilon}_{[222]}$
$\displaystyle=-\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)-\frac{1}{2}(\log
k_{t}+\gamma_{E})^{2}+\frac{\pi^{2}}{12},$ (299)
$\displaystyle\hat{i}^{v_{2}\epsilon}_{[222]}$
$\displaystyle=\hat{i}^{v_{1}\epsilon}_{[222]}(k_{1}\leftrightarrow k_{2}),$
(300) $\displaystyle\hat{i}^{v_{3}\epsilon}_{[222]}$
$\displaystyle=\hat{i}^{v_{1}\epsilon}_{[222]}(k_{1}\leftrightarrow k_{3}).$
(301)

  * •

For $\hat{i}_{[322]}$:

$\displaystyle i^{-+-}_{[322]}$
$\displaystyle=\frac{k_{2}}{u-v_{1}+v_{2}-v_{3}}\left[-\frac{1}{\epsilon}-2v_{2}(\log
k_{2}+\gamma_{E}+\log 2-1)\right.$
$\displaystyle\qquad\qquad\qquad\left.-2v_{2}^{2}\epsilon\left((\log
k_{2}+\gamma_{E}+\log 2-1)^{2}+1\right)\right],$ (302) $\displaystyle
i^{--+}_{[322]}$ $\displaystyle=i^{-+-}_{[322]}(k_{2}\leftrightarrow
k_{3},v_{2}\leftrightarrow v_{3}),$ (303) $\displaystyle
i^{\text{fin}}_{[322]}$ $\displaystyle=(k_{2}+k_{3})(\log
k_{t}+\gamma_{E}-1)-k_{1},$ (304) $\displaystyle\hat{i}^{u\epsilon}_{[322]}$
$\displaystyle=-\frac{k_{2}+k_{3}}{2}\left[(\log
k_{t}+\gamma_{E}-1)^{2}+1+\frac{\pi^{2}}{6}\right]+k_{1}\left(\log
k_{t}+\gamma_{E}-1\right),$ (305)
$\displaystyle\hat{i}^{v_{1}\epsilon}_{[322]}$
$\displaystyle=\frac{k_{2}+k_{3}}{2}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-1)^{2}+1-\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad+k_{1}\left(\log k_{t}-2\log k_{1}-\gamma_{E}-2\log
2+3\right),$ (306) $\displaystyle\hat{i}^{v_{2}\epsilon}_{[322]}$
$\displaystyle=\frac{k_{3}-k_{2}}{2}\left[2\operatorname{Li}_{2}\left(\frac{l_{13-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-1)^{2}+1-\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad+2k_{2}\left[(\log k_{t}+\gamma_{E}-1)(\log
k_{2}+\gamma_{E}+\log 2-1)+1\right]$ $\displaystyle\qquad-k_{1}(\log
k_{t}+\gamma_{E}-1),$ (307) $\displaystyle\hat{i}^{v_{3}\epsilon}_{[322]}$
$\displaystyle=\hat{i}^{v_{2}\epsilon}_{[322]}(k_{2}\leftrightarrow k_{3}).$
(308)

  * •

For $\hat{i}_{[332]}$:

$\displaystyle i^{---}_{[332]}$
$\displaystyle=\frac{1}{u-v_{t}}\left[\frac{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}{2\epsilon}+(v_{1}k_{1}^{2}+v_{2}k_{2}^{2}+v_{3}k_{3}^{2})\right.$
$\displaystyle\qquad\qquad\qquad\left.-2\epsilon(v_{1}^{2}k_{1}^{2}+v_{2}^{2}k_{2}^{2}-v_{3}^{2}k_{3}^{2})\right],$
(309) $\displaystyle i^{\text{fin}}_{[332]}$
$\displaystyle=\frac{1}{2}(k_{1}^{2}+k_{2}^{2}-k_{3}^{2})\left(\log
k_{t}+\gamma_{E}\right)-\frac{1}{4}k_{t}(k_{1}+k_{2}-3k_{3}),$ (310)
$\displaystyle\hat{i}^{u\epsilon}_{[332]}$
$\displaystyle=-\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{4}\left[(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}+\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad+\frac{1}{2}(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})(k_{1}k_{2}-k_{1}k_{3}-k_{2}k_{3}-k_{3}^{2})$
$\displaystyle\qquad+\frac{k_{3}(k_{1}+k_{2})}{2}-\frac{k_{1}^{2}+k_{2}^{2}-9k_{3}^{2}}{16},$
(311) $\displaystyle\hat{i}^{v_{1}\epsilon}_{[332]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}+\frac{1}{4}-\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad+k_{1}(k_{3}-k_{2})(\log k_{1}+\gamma_{E}+\log 2)$
$\displaystyle\qquad-\frac{1}{2}(k_{1}+k_{3})(2k_{1}-k_{2}-k_{3})(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})$
$\displaystyle\qquad+\frac{1}{2}(3k_{1}k_{2}-4k_{1}k_{3}-k_{2}k_{3}-k_{3}^{2}),$
(312) $\displaystyle\hat{i}^{v_{2}\epsilon}_{[332]}$
$\displaystyle=\hat{i}^{v_{1}\epsilon}_{[332]}(k_{1}\leftrightarrow k_{2}),$
(313) $\displaystyle\hat{i}^{v_{3}\epsilon}_{[332]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}+\frac{1}{4}-\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad-\frac{1}{2}(k_{1}+k_{3})(k_{2}+k_{3})(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})$ $\displaystyle\qquad+(k_{1}+k_{2})k_{3}(\log
k_{3}+\gamma_{E}+\log 2)+\frac{3k_{3}^{2}}{2}.$ (314)

  * •

For $\hat{i}_{[333]}$:

$\displaystyle i^{+--}_{[333]}$
$\displaystyle=\frac{k_{1}^{3}}{u+v_{1}-v_{2}-v_{3}}\left[\frac{1}{3\epsilon}+\frac{2v_{1}}{3}(\log
k_{1}+\gamma_{E}+\log 2-\tfrac{7}{3})\right.$
$\displaystyle\qquad\qquad\qquad\left.\frac{2v_{1}^{2}\epsilon}{3}\left((\log
k_{1}+\gamma_{E}+\log 2-\tfrac{7}{3})^{2}+\frac{19}{9}\right)\right],$ (315)
$\displaystyle i^{-+-}_{[333]}$
$\displaystyle=i^{+--}_{[333]}(k_{1}\leftrightarrow k_{2},v_{1}\leftrightarrow
v_{2}),$ (316) $\displaystyle i^{--+}_{[333]}$
$\displaystyle=i^{+--}_{[333]}(k_{1}\leftrightarrow k_{3},v_{1}\leftrightarrow
v_{3}),$ (317) $\displaystyle i^{\text{fin}}_{[333]}$
$\displaystyle=-\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}}{3}(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})+\frac{1}{3}(\sigma_{(1)123}\sigma_{(2)123}-4\sigma_{(3)123}),$
(318) $\displaystyle\hat{i}^{u\epsilon}_{[333]}$
$\displaystyle=\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}}{6}\left[(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})^{2}+\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad-\frac{1}{3}(\sigma_{(1)123}\sigma_{(2)123}-4\sigma_{(3)123})(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})$
$\displaystyle\qquad+\frac{5(k_{1}^{3}+k_{2}^{3}+k_{3}^{3})+9k_{1}k_{2}k_{3}}{27},$
(319) $\displaystyle\hat{i}^{v_{1}\epsilon}_{[333]}$
$\displaystyle=\frac{k_{1}^{3}-k_{2}^{3}-k_{3}^{3}}{6}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})^{2}+\frac{10}{9}-\frac{\pi^{2}}{6}\right]$
$\displaystyle\qquad-\frac{2}{3}k_{1}^{3}(\log k_{1}+\gamma_{E}+\log
2-\tfrac{7}{3})(\log k_{t}+\gamma_{E}-\tfrac{4}{3})$
$\displaystyle\qquad+\frac{2}{3}k_{1}(k_{2}^{2}+k_{3}^{2}-k_{2}k_{3})(\log
k_{1}+\gamma_{E}+\log 2-\tfrac{7}{3})$
$\displaystyle\qquad+\frac{1}{3}\left[\sigma_{(1)123}\sigma_{(2)123}-2\sigma_{(3)123}-2k_{1}(k_{2}^{2}+k_{3}^{2})\right](\log
k_{t}+\gamma_{E}-\tfrac{4}{3})$
$\displaystyle\qquad-\frac{k_{1}}{27}\left[20k_{1}^{2}+9k_{2}k_{3}+18k_{1}(k_{2}+k_{3})\right],$
(320) $\displaystyle\hat{i}^{v_{2}\epsilon}_{[333]}$
$\displaystyle=\hat{i}^{v_{1}\epsilon}_{[333]}(k_{1}\leftrightarrow k_{2}),$
(321) $\displaystyle\hat{i}^{v_{3}\epsilon}_{[333]}$
$\displaystyle=\hat{i}^{v_{1}\epsilon}_{[333]}(k_{1}\leftrightarrow k_{3}).$
(322)

####  3-point amplitudes $\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$

The expansions of the amplitudes $\hat{j}_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$
to order $\epsilon^{1}$ are as follows:

  * •

For $\hat{j}_{[22;2]}$:

$\displaystyle j^{\pm\pm\pm}_{[22;2]}$ $\displaystyle=0,$ (323) $\displaystyle
j^{\text{fin}}_{[22;2]}$
$\displaystyle=-\frac{1}{2}\log\left(\frac{l_{12-}}{k_{t}}\right),$ (324)
$\displaystyle\hat{j}^{u\epsilon}_{[22;2]}$ $\displaystyle=\frac{1}{4}(\log
l_{12-}+\gamma_{E})^{2}-\frac{1}{4}(\log k_{t}+\gamma_{E})^{2},$ (325)
$\displaystyle\hat{j}^{v_{1}\epsilon}_{[22;2]}$
$\displaystyle=\frac{1}{4}\Big{[}2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)-2\operatorname{Li}_{2}\left(-\frac{l_{13-}}{l_{12-}}\right)$
$\displaystyle\qquad\qquad+(\log k_{t}+\gamma_{E})^{2}-(\log
l_{12-}+\gamma_{E})^{2}\Big{]},$ (326)
$\displaystyle\hat{j}^{v_{2}\epsilon}_{[22;2]}$
$\displaystyle=\hat{j}^{v_{1}\epsilon}_{[22;2]}(k_{1}\leftrightarrow k_{2}),$
(327) $\displaystyle\hat{j}^{v_{3}\epsilon}_{[22;2]}$
$\displaystyle=\frac{1}{2}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)-\log\left(\frac{l_{12-}}{k_{t}}\right)(\log
k_{t}-\gamma_{E}-2\log 2)\right]-\frac{\pi^{2}}{6}.$ (328)

  * •

For $\hat{j}_{[32;2]}$:

$\displaystyle j^{--+}_{[32;2]}$
$\displaystyle=\frac{k_{3}}{u-v_{1}-v_{2}+v_{3}}\left[\frac{1}{\epsilon}+2v_{3}(\log
k_{3}-1)\right.$
$\displaystyle\qquad\qquad\qquad\left.+2v_{3}^{2}\epsilon\,(\log^{2}k_{3}-2\log
k_{3}+2)\right],$ (329) $\displaystyle j^{\text{fin}}_{[32;2]}$
$\displaystyle=-\frac{k_{2}+k_{3}}{2}(\log
k_{t}+\gamma_{E})+\frac{k_{2}-k_{3}}{2}(\log l_{12-}+\gamma_{E})+k_{3},$ (330)
$\displaystyle\hat{j}^{u\epsilon}_{[32;2]}$
$\displaystyle=\frac{k_{2}+k_{3}}{4}(\log
k_{t}+\gamma_{E}-1)^{2}+\frac{k_{3}-k_{2}}{4}(\log l_{12-}+\gamma_{E}-1)^{2}$
$\displaystyle\qquad+\frac{k_{1}}{2}\log\left(\frac{l_{12-}}{k_{t}}\right)+\frac{k_{3}}{2}\left(1+\frac{\pi^{2}}{6}\right),$
(331) $\displaystyle\hat{j}^{v_{1}\epsilon}_{[32;2]}$
$\displaystyle=-\frac{k_{2}+k_{3}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad+\frac{k_{2}-k_{3}}{4}\left[2\operatorname{Li}_{2}\left(-\frac{l_{13-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad+\frac{k_{1}}{2}\log\left(\frac{l_{12-}}{k_{t}}\right)+\frac{k_{3}}{2}\left(-1+\frac{\pi^{2}}{6}\right),$
(332) $\displaystyle\hat{j}^{v_{2}\epsilon}_{[32;2]}$
$\displaystyle=\frac{k_{2}-k_{3}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{13-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad-\frac{k_{2}+k_{3}}{4}\left[2\operatorname{Li}_{2}\left(-\frac{l_{23-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad+\log\left(\frac{l_{12-}}{k_{t}}\right)\left[-\frac{k_{1}}{2}+k_{2}(\log
k_{2}+\gamma_{E}+\log 2-1)\right]$
$\displaystyle\qquad-\frac{k_{3}}{2}\left(1-\frac{\pi^{2}}{6}\right),$ (333)
$\displaystyle\hat{j}^{v_{3}\epsilon}_{[32;2]}$
$\displaystyle=-\frac{k_{2}}{2}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad+\frac{k_{2}+k_{3}}{2}(\log k_{t}+\gamma_{E}-1)(\log
l_{12-}+\gamma_{E}-1)$ $\displaystyle\qquad-k_{3}(\log k_{3}-1)(\log
k_{t}+\log l_{12-}+2\gamma_{E}-2)$
$\displaystyle\qquad-\left[\frac{k_{1}}{2}+k_{2}(\gamma_{E}+\log
2)\right]\log\left(\frac{l_{12-}}{k_{t}}\right)-\frac{3}{2}\left(k_{3}-\frac{\pi^{2}}{18}(2k_{2}+k_{3})\right).$
(334)

  * •

For $\hat{j}_{[22;3]}$:

$\displaystyle j^{\pm\pm\pm}_{[22;3]}$ $\displaystyle=0,$ (335) $\displaystyle
j^{\text{fin}}_{[22;3]}$
$\displaystyle=-\frac{k_{1}+k_{2}}{2}\log\left(\frac{l_{12-}}{k_{t}}\right)-k_{3},$
(336) $\displaystyle\hat{j}^{u\epsilon}_{[22;3]}$
$\displaystyle=-\frac{k_{1}+k_{2}}{4}(\log
k_{t}+\gamma_{E}-1)^{2}+\frac{k_{1}+k_{2}}{4}(\log l_{12-}+\gamma_{E}-1)^{2}$
$\displaystyle\qquad+\frac{k_{3}}{2}(\log l_{12-}+\log k_{t}+2\gamma_{E}-2),$
(337) $\displaystyle\hat{j}^{v_{1}\epsilon}_{[22;3]}$
$\displaystyle=\frac{k_{2}-k_{1}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad+\frac{k_{1}-k_{2}}{4}\left[2\operatorname{Li}_{2}\left(-\frac{l_{13-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-1)^{2}\right]$
$\displaystyle\qquad-\log\left(\frac{l_{12-}}{k_{t}}\right)k_{1}(\log
k_{1}+\gamma_{E}+\log 2-1)$ $\displaystyle\qquad-\frac{k_{3}}{2}\left(\log
k_{t}+\log l_{12-}+2\gamma_{E}-2\right),$ (338)
$\displaystyle\hat{j}^{v_{2}\epsilon}_{[22;3]}$
$\displaystyle=\hat{j}^{v_{1}\epsilon}_{[22;3]}(k_{1}\leftrightarrow k_{2}),$
(339) $\displaystyle\hat{j}^{v_{3}\epsilon}_{[22;3]}$
$\displaystyle=\frac{k_{1}+k_{2}}{2}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)+(\log
k_{t}-\gamma_{E}-2\log 2+3)^{2}\right]$
$\displaystyle\qquad-\frac{k_{1}+k_{2}}{2}(\log k_{t}-\gamma_{E}-2\log
2+3)(\log l_{12-}-\gamma_{E}-2\log 2+3)$
$\displaystyle\qquad+\frac{k_{3}}{2}(\log k_{t}+\log l_{12-}-4\log
k_{3}+2\gamma_{E}-2)-\frac{\pi^{2}}{6}(k_{1}+k_{2}).$ (340)

  * •

For $\hat{j}_{[33;2]}$:

$\displaystyle j^{\pm\pm\pm}_{[33;2]}$ $\displaystyle=0,$ (341) $\displaystyle
j^{\text{fin}}_{[33;2]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{4}\log\left(\frac{l_{12-}}{k_{t}}\right)-\frac{k_{3}(k_{1}+k_{2})}{2},$
(342) $\displaystyle\hat{j}^{u\epsilon}_{[33;2]}$
$\displaystyle=\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{8}(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}-\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{8}(\log
l_{12-}+\gamma_{E}-\tfrac{1}{2})^{2}$
$\displaystyle\qquad+\frac{1}{4}(k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3}-k_{3}^{2})\log
l_{12-}+\frac{1}{4}(-k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3}+k_{3}^{2})\log k_{t}$
$\displaystyle\qquad+\frac{-3+2\gamma_{E}}{4}k_{3}(k_{1}+k_{2}),$ (343)
$\displaystyle\hat{j}^{v_{1}\epsilon}_{[33;2]}$
$\displaystyle=-\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{8}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{3}{2})^{2}\right]$
$\displaystyle\qquad+\frac{k_{1}^{2}+k_{2}^{2}-k_{3}^{2}}{8}\left[2\operatorname{Li}_{2}\left(-\frac{l_{13-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-\tfrac{3}{2})^{2}\right]$
$\displaystyle\qquad+\frac{-k_{1}^{2}+k_{2}^{2}+k_{1}k_{2}}{4}\log\left(\frac{l_{12-}}{k_{t}}\right)$
$\displaystyle\qquad+\frac{k_{3}(k_{1}-k_{2})}{4}(\log k_{t}+\log
l_{12-}+2\gamma_{E}-3)$ $\displaystyle\qquad-k_{1}k_{3}(\log
k_{1}+\gamma_{E}+\log 2-\tfrac{5}{2}),$ (344)
$\displaystyle\hat{j}^{v_{2}\epsilon}_{[33;2]}$
$\displaystyle=\hat{j}^{v_{1}\epsilon}_{[33;2]}(k_{1}\leftrightarrow k_{2}),$
(345) $\displaystyle\hat{j}^{v_{3}\epsilon}_{[33;2]}$
$\displaystyle=\frac{-k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)-\log\left(\frac{l_{12-}}{k_{t}}\right)(\log
k_{t}-\gamma_{E}-2\log 2-\tfrac{1}{2})\right]$
$\displaystyle\qquad+\frac{-k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3}-k_{3}^{2}}{4}\log
l_{12-}+\frac{k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3}+k_{3}^{2}}{4}\log k_{t}$
$\displaystyle\qquad-\frac{k_{3}(k_{1}+k_{2})}{4}(4\log
k_{3}-2\gamma_{E}+1)+\frac{\pi^{2}}{12}(k_{1}^{2}+k_{2}^{2}-k_{3}^{2}).$ (346)

  * •

For $\hat{j}_{[32;3]}$:

$\displaystyle j^{\pm\pm\pm}_{[32;3]}$ $\displaystyle=0,$ (347) $\displaystyle
j^{\text{fin}}_{[32;3]}$
$\displaystyle=-\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{4}\log\left(\frac{l_{12-}}{k_{t}}\right)+\frac{k_{3}(-k_{1}+k_{2})}{2},$
(348) $\displaystyle\hat{j}^{u\epsilon}_{[32;3]}$
$\displaystyle=-\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{8}(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}+\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{8}(\log
l_{12-}+\gamma_{E}-\tfrac{1}{2})^{2}$
$\displaystyle\qquad+\frac{1}{4}(k_{1}k_{2}+k_{1}k_{3}-k_{2}k_{3}+k_{2}^{2})\log
l_{12-}+\frac{1}{4}(-k_{1}k_{2}+k_{1}k_{3}-k_{2}k_{3}-k_{2}^{2})\log k_{t}$
$\displaystyle\qquad+\frac{2\gamma_{E}-1}{4}k_{1}k_{3}-\frac{2\gamma_{E}-3}{4}k_{2}k_{3},$
(349) $\displaystyle\hat{j}^{v_{1}\epsilon}_{[32;3]}$
$\displaystyle=\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{8}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}\right]$
$\displaystyle\qquad-\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{8}\left[2\operatorname{Li}_{2}\left(-\frac{l_{13-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-\tfrac{1}{2})^{2}\right]$
$\displaystyle\qquad-\frac{1}{4}(k_{1}+k_{2})(2k_{1}-k_{2}-k_{3})(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})$
$\displaystyle\qquad+\frac{1}{4}(k_{1}+k_{2})(2k_{1}-k_{2}+k_{3})(\log
l_{12-}+\gamma_{E}-\tfrac{1}{2})$ $\displaystyle\qquad-k_{1}k_{3}(\log
k_{1}+\gamma_{E}+\log 2-\tfrac{3}{2})-\frac{k_{2}k_{3}}{2},$ (350)
$\displaystyle\hat{j}^{v_{2}\epsilon}_{[32;3]}$
$\displaystyle=\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{8}\left[2\operatorname{Li}_{2}\left(\frac{l_{13-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})^{2}\right]$
$\displaystyle\qquad-\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{8}\left[2\operatorname{Li}_{2}\left(-\frac{l_{23-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-\tfrac{1}{2})^{2}\right]$
$\displaystyle\qquad+\frac{(k_{1}+k_{2})(k_{2}-k_{3})}{4}(\log
l_{12-}+\gamma_{E}-\tfrac{1}{2})$
$\displaystyle\qquad-\frac{(k_{1}+k_{2})(k_{2}+k_{3})}{4}(\log
k_{t}+\gamma_{E}-\tfrac{1}{2})+k_{2}k_{3}(\log k_{2}+\gamma_{E}+\log 2),$
(351) $\displaystyle\hat{j}^{v_{3}\epsilon}_{[32;3]}$
$\displaystyle=\frac{k_{1}^{2}-k_{2}^{2}+k_{3}^{2}}{4}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)-\log\left(\frac{l_{12-}}{k_{t}}\right)(\log
k_{t}-\gamma_{E}-2\log 2)\right]$
$\displaystyle\qquad-\frac{1}{8}(7k_{1}^{2}+2k_{1}k_{2}-5k_{2}^{2}+3k_{3}^{2})\log\left(\frac{l_{12-}}{k_{t}}\right)$
$\displaystyle\qquad+\frac{k_{3}(k_{1}-k_{2})}{4}(\log k_{t}+\log
l_{12-}-4\log k_{3}+2\gamma_{E}-3)$
$\displaystyle\qquad-\frac{k_{2}k_{3}}{2}-\frac{\pi^{2}}{12}(k_{1}^{2}-k_{2}^{2}+k_{3}^{2}).$
(352)

  * •

For $\hat{j}_{[33;3]}$:

$\displaystyle j^{--+}_{[33;3]}$
$\displaystyle=\frac{k_{3}^{3}}{u-v_{1}-v_{2}+v_{3}}\left[\frac{1}{3\epsilon}+\frac{2}{9}v_{3}(3\log
k_{3}-1)+\frac{2v_{3}^{2}\epsilon}{27}(9\log^{2}k_{3}-6\log k_{3}+2)\right],$
(353) $\displaystyle j^{\text{fin}}_{[33;3]}$
$\displaystyle=\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}}{6}\log\left(\frac{l_{12-}}{k_{t}}\right)-\frac{k_{3}^{3}}{3}(\log
l_{12-}+\gamma_{E}-\tfrac{4}{3})+\frac{1}{3}k_{3}(k_{1}^{2}+k_{2}^{2}-k_{1}k_{2}),$
(354) $\displaystyle\hat{j}^{u\epsilon}_{[33;3]}$
$\displaystyle=\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}}{12}(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})^{2}-\frac{k_{1}^{3}+k_{2}^{3}-k_{3}^{3}}{12}(\log
l_{12-}+\gamma_{E}-\tfrac{4}{3})^{2}$
$\displaystyle\qquad+\frac{1}{6}(-\sigma_{(1)123}\sigma_{(2)123}+4\sigma_{(3)123})(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})$
$\displaystyle\qquad+\frac{1}{6}(\sigma_{(1)123}\sigma_{(2)123}-2\sigma_{(3)123}-2k_{3}(k_{1}^{2}+k_{2}^{2}))(\log
l_{12-}+\gamma_{E}-\tfrac{4}{3})$
$\displaystyle\qquad+\frac{k_{1}k_{2}k_{3}}{3}+\frac{5k_{3}^{3}}{27}\left(1+\frac{3\pi^{2}}{20}\right),$
(355) $\displaystyle\hat{j}^{v_{1}\epsilon}_{[33;3]}$
$\displaystyle=\frac{k_{1}^{3}-k_{2}^{3}-k_{3}^{3}}{12}\left[2\operatorname{Li}_{2}\left(\frac{l_{23-}}{k_{t}}\right)+(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})^{2}\right]$
$\displaystyle\qquad+\frac{-k_{1}^{3}+k_{2}^{3}-k_{3}^{3}}{12}\left[2\operatorname{Li}_{2}\left(-\frac{l_{13-}}{l_{12-}}\right)+(\log
l_{12-}+\gamma_{E}-\tfrac{4}{3})^{2}\right]$
$\displaystyle\qquad-\frac{1}{6}(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})\left[2k_{1}^{3}(\log k_{1}+\gamma_{E}+\log
2-\tfrac{7}{3})\right.$
$\displaystyle\qquad\qquad\qquad\left.-(k_{1}^{2}k_{2}+k_{1}^{2}k_{3}+k_{2}^{2}k_{3}+k_{2}k_{3}^{2}-k_{1}k_{2}^{2}-k_{1}k_{3}^{2}+k_{1}k_{2}k_{3})\right]$
$\displaystyle\qquad+\frac{1}{6}(\log
l_{12-}+\gamma_{E}-\tfrac{4}{3})\left[2k_{1}^{3}(\log k_{1}+\gamma_{E}+\log
2-\tfrac{7}{3})\right.$
$\displaystyle\qquad\qquad\qquad\left.+(-k_{1}^{2}k_{2}+k_{1}^{2}k_{3}+k_{2}^{2}k_{3}-k_{2}k_{3}^{2}+k_{1}k_{2}^{2}+k_{1}k_{3}^{2}+k_{1}k_{2}k_{3})\right]$
$\displaystyle\qquad-\frac{2}{3}k_{1}k_{2}k_{3}(\log k_{1}+\gamma_{E}+\log
2-\tfrac{11}{6})-\frac{2}{3}k_{1}^{2}k_{3}-\frac{5k_{3}^{3}}{27}\left(1-\frac{3\pi^{2}}{20}\right),$
(356) $\displaystyle\hat{j}^{v_{2}\epsilon}_{[33;3]}$
$\displaystyle=\hat{j}^{v_{1}\epsilon}_{[33;3]}(k_{1}\leftrightarrow k_{2}),$
(357) $\displaystyle\hat{j}^{v_{3}\epsilon}_{[33;3]}$
$\displaystyle=-\frac{k_{1}^{3}+k_{2}^{3}}{6}\left[2\operatorname{Li}_{2}\left(\frac{l_{12-}}{k_{t}}\right)-\log\left(\frac{l_{12-}}{k_{t}}\right)(\log
k_{t}-\gamma_{E}-2\log 2+\tfrac{8}{3})\right]$
$\displaystyle\qquad+\frac{k_{3}^{3}}{6}(\log
k_{t}+\gamma_{E}-\tfrac{2}{3})(\log l_{12-}+\gamma_{E}-\tfrac{2}{3})$
$\displaystyle\qquad+\frac{1}{6}(\log
k_{t}+\gamma_{E}-\tfrac{4}{3})\left[-2k_{3}^{3}\log
k_{3}+\sigma_{(1)123}\sigma_{(2)123}-2\sigma_{(3)123}-2k_{3}(k_{1}^{2}+k_{2}^{2})\right]$
$\displaystyle\qquad+\frac{1}{6}(\log
l_{12-}+\gamma_{E}-\tfrac{4}{3})\left[-2k_{3}^{3}\log
k_{3}-\sigma_{(1)123}\sigma_{(2)123}+4\sigma_{(3)123}\right]$
$\displaystyle\qquad+\frac{2}{3}k_{3}(k_{1}^{2}+k_{2}^{2}-k_{1}k_{2})(\log
k_{3}-\tfrac{1}{3})$
$\displaystyle\qquad-\frac{k_{1}k_{2}k_{3}}{3}-\frac{17}{27}k_{3}^{3}+\frac{\pi^{2}}{36}(2k_{1}^{3}+2k_{2}^{3}+k_{3}^{3}).$
(358)

####  4-point contact amplitudes

The 4-point contact amplitudes regulated in the general scheme read

$\displaystyle\hat{i}_{[2222]}$ $\displaystyle=\frac{1}{k_{T}}+O(\epsilon),$
(359) $\displaystyle\hat{i}_{[3222]}$
$\displaystyle=\frac{1}{(2u-v_{T})\,\epsilon}+\left[-\log
k_{T}+\frac{k_{1}}{k_{T}}-\gamma_{E}\right]+O(\epsilon),$ (360)
$\displaystyle\hat{i}_{[3322]}$
$\displaystyle=-\frac{k_{3}}{2u-v_{1}-v_{2}+v_{3}-v_{4}}\left[\frac{1}{\epsilon}+2v_{3}(\log
k_{3}+\gamma_{E}+\log 2-1)\right]$
$\displaystyle\qquad-\frac{k_{4}}{2u-v_{1}-v_{2}-v_{3}+v_{4}}\left[\frac{1}{\epsilon}+2v_{4}(\log
k_{4}+\gamma_{E}+\log 2-1)\right]$ $\displaystyle\qquad+(k_{3}+k_{4})(\log
k_{T}+\gamma_{E})+\frac{k_{1}k_{2}}{k_{T}}-k_{T}+O(\epsilon),$ (361)
$\displaystyle\hat{i}_{[3332]}$
$\displaystyle=-\frac{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-k_{4}^{2}}{2\epsilon\,(2u-v_{T})}+\left[\frac{k_{1}^{2}+k_{2}^{2}+k_{3}^{2}-k_{4}^{2}}{2}(\log
k_{T}+\gamma_{E})\right.$
$\displaystyle\qquad\qquad\left.+\frac{k_{1}k_{2}k_{3}}{k_{T}}+k_{T}\left(k_{4}-\frac{k_{T}}{4}\right)+\frac{1}{2u-v_{T}}\sum_{j=1}^{4}v_{j}k_{j}^{2}\right]+O(\epsilon),$
(362) $\displaystyle\hat{i}_{[3333]}$
$\displaystyle=\frac{k_{1}^{3}}{2u+v_{1}-v_{2}-v_{3}-v_{4}}\left[\frac{1}{3\epsilon}+\frac{2v_{1}}{3}(\log
k_{1}+\gamma_{E}+\log 2-\tfrac{7}{3})\right]$
$\displaystyle\qquad+[(k_{1},v_{1})\leftrightarrow(k_{2},v_{2})]+[(k_{1},v_{1})\leftrightarrow(k_{3},v_{3})]+[(k_{1},v_{1})\leftrightarrow(k_{4},v_{4})]$
$\displaystyle\qquad-\frac{k_{1}^{3}+k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{3}(\log
k_{T}+\gamma_{E})+\frac{k_{1}k_{2}k_{3}k_{4}}{k_{T}}+k_{T}\left(-\sigma_{(2)1234}+\frac{4}{9}k_{T}^{2}\right)+O(\epsilon).$
(363)

####  4-point exchange diagrams

To evaluate the exchange 4-point amplitudes in an arbitrary regularization
scheme one uses (• ‣ 5.5.1). Here, we list the necessary divergent terms:

$\displaystyle i^{\text{div}}_{[32,22x2]}=s^{-1-2\epsilon
v_{x}}\left[\frac{\hat{j}^{2u-v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}_{[32;2]}(k_{1},k_{2},s)}{\epsilon\,(-u+v_{3}+v_{4}+v_{x})}+\hat{j}_{[32;2]}(k_{1},k_{2},s)\hat{i}_{[222]}(s,k_{3},k_{4})\right]$
$\displaystyle\qquad\qquad\quad+\frac{1}{2u-v_{T}}\left[\frac{1}{\epsilon}-(u-v_{3}-v_{4}+3v_{x})\right]+O(\epsilon),$
(364) $\displaystyle i^{\text{div}}_{[33,22x2]}$
$\displaystyle=s^{-1-2\epsilon
v_{x}}\left[\frac{\hat{j}^{2u-v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}_{[33;2]}(k_{1},k_{2},s)}{\epsilon\,(-u+v_{3}+v_{4}+v_{x})}+\hat{j}_{[33;2]}(k_{1},k_{2},s)\hat{i}_{[222]}(s,k_{3},k_{4})\right]$
$\displaystyle\qquad+\frac{k_{3}}{(2u-v_{1}-v_{2}+v_{3}-v_{4})}\left[\frac{1}{2\epsilon}+v_{3}(\log
k_{3}+\gamma_{E}+\log 2-\tfrac{7}{4})+\frac{1}{4}(-3u+3v_{4}+v_{x})\right]$
$\displaystyle\qquad+\frac{k_{4}}{(2u-v_{1}-v_{2}-v_{3}+v_{4})}\left[\frac{1}{2\epsilon}+v_{4}(\log
k_{4}+\gamma_{E}+\log 2-\tfrac{7}{4})+\frac{1}{4}(-3u+3v_{3}+v_{x})\right]$
$\displaystyle\qquad+O(\epsilon),$ (365) $\displaystyle
i^{\text{div}}_{[32,32x2]}$ $\displaystyle=s^{-1-2\epsilon
v_{x}}\hat{j}_{[32;2]}(k_{1},k_{2},s)\hat{i}_{[232]}(s,k_{3},k_{4})$
$\displaystyle\qquad-\frac{2^{-1-2\epsilon v_{4}}\Gamma(-\tfrac{1}{2}-\epsilon
v_{4})}{\epsilon\,(u-v_{3}+v_{4}-v_{x})\Gamma(\tfrac{1}{2}+\epsilon
v_{4})}s^{-1-2\epsilon v_{x}}k_{4}^{1+2\epsilon
v_{4}}\hat{j}_{[32;2]}^{2u-v_{3}+v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)$
$\displaystyle\qquad-\frac{2^{-1-2\epsilon v_{x}}\Gamma(-\tfrac{1}{2}-\epsilon
v_{x})}{\epsilon\,(u-v_{3}-v_{4}+v_{x})\Gamma(\tfrac{1}{2}+\epsilon
v_{x})}\hat{j}_{[32;2]}^{2u-v_{3}-v_{4}+v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)$
$\displaystyle\qquad+\frac{\Gamma(\tfrac{1}{2}+\epsilon
v_{x})}{2\epsilon\,(u-v_{3}-v_{4}+v_{x})\Gamma(\tfrac{3}{2}+\epsilon
v_{x})}\hat{i}_{[322]}^{2u-v_{3}-v_{4}+v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)$
$\displaystyle\qquad-\frac{k_{2}}{2u-v_{1}+v_{2}-v_{3}-v_{4}}\left[\frac{1}{\epsilon}+2v_{2}(\log
k_{2}+\gamma_{E}+\log 2-1)-(-u+v_{3}+v_{4}+3v_{x})\right]$
$\displaystyle\qquad-\frac{k_{4}}{2u-v_{1}-v_{2}-v_{3}+v_{4}}\left[\frac{1}{\epsilon}+2v_{4}(\log
k_{4}+\gamma_{E}+\log 2-1)+(-u+v_{3}-v_{4}-3v_{x})\right],$ (366)
$\displaystyle i^{\text{div}}_{[32,33x2]}=s^{-1-2\epsilon
v_{x}}\left[\hat{j}_{[32;2]}(k_{1},k_{2},s)\hat{i}_{[233]}(s,k_{3},k_{4})\right.$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2\epsilon\,(u-v_{3}-v_{4}-v_{x})}\left(\frac{k_{3}^{2}}{1+2\epsilon
v_{3}}+\frac{k_{4}^{2}}{1+2\epsilon v_{4}}-\frac{s^{2}}{1-2\epsilon
v_{x}}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\times\hat{j}^{2u-v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}_{[32;2]}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad+\frac{1}{4(2u-v_{T})}\left[\frac{k_{1}^{2}-k_{2}^{2}-2k_{3}^{2}-2k_{4}^{2}-s^{2}}{\epsilon}+\tfrac{1}{2}s^{2}(5(-u+v_{3}+v_{4})-11v_{x})\right.$
$\displaystyle\qquad\qquad\quad+k_{1}^{2}(\tfrac{3}{2}u-2v_{1}-\tfrac{3}{2}v_{3}-\tfrac{3}{2}v_{4}+\tfrac{1}{2}v_{x})+k_{2}^{2}(-\tfrac{3}{2}u-2v_{2}+\tfrac{3}{2}v_{3}+\tfrac{3}{2}v_{4}-\tfrac{1}{2}v_{x})$
$\displaystyle\qquad\qquad\quad\left.+2k_{3}^{2}(u+v_{3}-v_{4}+3v_{x})+2k_{4}^{2}(u-v_{3}+v_{4}+3v_{x})\right],$
(367) $\displaystyle i^{\text{div}}_{[33,33x2]}$
$\displaystyle=s^{-1-2\epsilon
v_{x}}\left[\hat{j}_{[33;2]}(k_{1},k_{2},s)\hat{i}_{[233]}(s,k_{3},k_{4})\right.$
$\displaystyle\qquad\qquad\qquad+\frac{1}{2\epsilon\,(u-v_{3}-v_{4}-v_{x})}\left(\frac{k_{3}^{2}}{1+2\epsilon
v_{3}}+\frac{k_{4}^{2}}{1+2\epsilon v_{4}}-\frac{s^{2}}{1-2\epsilon
v_{x}}\right)\times$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\times\hat{j}^{2u-v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}_{[33;2]}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad-\frac{k_{1}^{3}}{3(2u+v_{1}-v_{2}-v_{3}-v_{4})}\left[\frac{1}{2\epsilon}+v_{1}(\log
k_{1}+\gamma_{E}+\log
2-\tfrac{7}{3})+\tfrac{1}{4}(3u-3v_{3}-3v_{4}+v_{x})\right]$
$\displaystyle\qquad-\frac{k_{2}^{3}}{3(2u-v_{1}+v_{2}-v_{3}-v_{4})}\left[\frac{1}{2\epsilon}+v_{2}(\log
k_{2}+\gamma_{E}+\log
2-\tfrac{7}{3})+\tfrac{1}{4}(3u-3v_{3}-3v_{4}+v_{x})\right]$
$\displaystyle\qquad-\frac{k_{3}^{3}}{3(2u-v_{1}-v_{2}+v_{3}-v_{4})}\left[\frac{1}{2\epsilon}+v_{3}(\log
k_{3}+\gamma_{E}+\log
2-\tfrac{7}{3})-\tfrac{1}{4}(3u+3v_{3}-3v_{4}-v_{x})\right]$
$\displaystyle\qquad-\frac{k_{4}^{3}}{3(2u-v_{1}-v_{2}-v_{3}+v_{4})}\left[\frac{1}{2\epsilon}+v_{4}(\log
k_{4}+\gamma_{E}+\log
2-\tfrac{7}{3})-\tfrac{1}{4}(3u-3v_{3}+3v_{4}-v_{x})\right].$ (368)
$\displaystyle i^{\text{div}}_{[32,22x3]}$ $\displaystyle=s^{-3-2\epsilon
v_{x}}\left[\hat{j}_{[32;3]}(k_{1},k_{2},s)\hat{i}_{[322]}(s,k_{3},k_{4})\right.$
$\displaystyle\qquad\qquad\qquad-\frac{2^{-1-2\epsilon
v_{3}}\Gamma(-\tfrac{1}{2}-\epsilon
v_{3})}{\epsilon\,(u+v_{3}-v_{4}-v_{x})\Gamma(\tfrac{1}{2}+\epsilon
v_{3})}k_{3}^{1+2\epsilon
v_{3}}\hat{j}_{[32;3]}^{2u+v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)$
$\displaystyle\qquad\qquad\qquad\left.-\frac{2^{-1-2\epsilon
v_{4}}\Gamma(-\tfrac{1}{2}-\epsilon
v_{4})}{\epsilon\,(u-v_{3}+v_{4}-v_{x})\Gamma(\tfrac{1}{2}+\epsilon
v_{4})}k_{4}^{1+2\epsilon
v_{4}}\hat{j}_{[32;3]}^{2u-v_{3}+v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad+\frac{1}{2(2u-v_{T})}\left[\frac{1}{\epsilon}+\frac{1}{2}(u-v_{3}-v_{4}-3v_{x})\right]+O(\epsilon).$
(369) $\displaystyle i^{\text{div}}_{[33,22x3]}$
$\displaystyle=s^{-3-2\epsilon
v_{x}}\left[\hat{j}_{[33;3]}(k_{1},k_{2},s)\hat{i}_{[322]}(s,k_{3},k_{4})\right.$
$\displaystyle\qquad\qquad\qquad-\frac{2^{-1-2\epsilon
v_{3}}\Gamma(-\tfrac{1}{2}-\epsilon
v_{3})}{\epsilon\,(u+v_{3}-v_{4}-v_{x})\Gamma(\tfrac{1}{2}+\epsilon
v_{3})}k_{3}^{1+2\epsilon
v_{3}}\hat{j}_{[33;3]}^{2u+v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)$
$\displaystyle\qquad\qquad\qquad\left.-\frac{2^{-1-2\epsilon
v_{4}}\Gamma(-\tfrac{1}{2}-\epsilon
v_{4})}{\epsilon\,(u-v_{3}+v_{4}-v_{x})\Gamma(\tfrac{1}{2}+\epsilon
v_{4})}k_{4}^{1+2\epsilon
v_{4}}\hat{j}_{[33;3]}^{2u-v_{3}+v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad-\frac{k_{3}}{9(2u-v_{1}-v_{2}+v_{3}-v_{4})}\left[\frac{1}{\epsilon}+2v_{3}(\log
k_{3}+\gamma_{E}+\log 2-\tfrac{7}{6})-\frac{1}{3}(u-v_{4}+3v_{x})\right]$
$\displaystyle\qquad-\frac{k_{4}}{9(2u-v_{1}-v_{2}-v_{3}+v_{4})}\left[\frac{1}{\epsilon}+2v_{4}(\log
k_{4}+\gamma_{E}+\log 2-\tfrac{7}{6})-\frac{1}{3}(u-v_{3}+3v_{x})\right]$
$\displaystyle\qquad+O(\epsilon).$ (370) $\displaystyle
i^{\text{div}}_{[32,32x3]}$ $\displaystyle=s^{-3-2\epsilon
v_{x}}\left[\hat{j}_{[32;3]}(k_{1},k_{2},s)\hat{i}_{[332]}(s,k_{3},k_{4})\right.$
$\displaystyle\qquad+\frac{1}{2\epsilon\,(u-v_{3}-v_{4}-v_{x})}\left(\frac{k_{3}^{2}}{1+2\epsilon
v_{3}}-\frac{k_{4}^{2}}{1-2\epsilon v_{4}}+\frac{s^{2}}{1+2\epsilon
v_{x}}\right)\times$
$\displaystyle\qquad\qquad\qquad\left.\times\hat{j}_{[32;3]}^{2u-v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad-\frac{k_{2}}{2(2u-v_{1}+v_{2}-v_{3}-v_{4})}\left[\frac{1}{\epsilon}+2v_{2}(\log
k_{2}+\gamma_{E}+\log 2-1)-\frac{1}{2}(u-v_{3}-v_{4}+3v_{x})\right]$
$\displaystyle\qquad-\frac{k_{4}}{2(2u-v_{1}-v_{2}-v_{3}+v_{4})}\left[\frac{1}{\epsilon}+2v_{4}(\log
k_{4}+\gamma_{E}+\log 2-1)+\frac{1}{2}(u-v_{3}+v_{4}-3v_{x})\right]$
$\displaystyle\qquad+O(\epsilon).$ (371) $\displaystyle
i^{\text{div}}_{[32,33x3]}$ $\displaystyle=s^{-3-2\epsilon
v_{x}}\hat{j}_{[32;3]}(k_{1},k_{2},s)\hat{i}_{[333]}(s,k_{3},k_{4})$
$\displaystyle\qquad-\frac{s^{-3-2\epsilon
v_{x}}}{8\epsilon}\left[\frac{4^{-\epsilon v_{3}}\Gamma(-\tfrac{3}{2}-\epsilon
v_{3})}{(u+v_{3}-v_{4}-v_{x})\Gamma(\tfrac{3}{2}+\epsilon
v_{3})}k_{3}^{3+2\epsilon
v_{3}}\hat{j}_{[32;3]}^{2u+v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right.$
$\displaystyle\qquad\qquad\qquad\left.+\frac{4^{-\epsilon
v_{4}}\Gamma(-\tfrac{3}{2}-\epsilon
v_{4})}{(u-v_{3}+v_{4}-v_{x})\Gamma(\tfrac{3}{2}+\epsilon
v_{4})}k_{4}^{3+2\epsilon
v_{4}}\hat{j}_{[32;3]}^{2u-v_{3}+v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad+\frac{1}{8\epsilon\,(u-v_{3}-v_{4}+v_{x})}\left[\frac{8}{3+2\epsilon
v_{x}}\hat{i}_{[323]}^{2u-v_{3}-v_{4}+v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right.$
$\displaystyle\qquad\qquad\qquad\left.-\frac{4^{-\epsilon
v_{x}}\Gamma(-\tfrac{3}{2}-\epsilon v_{x})}{\Gamma(\tfrac{3}{2}+\epsilon
v_{x})}\hat{j}_{[32;3]}^{2u-v_{3}-v_{4}+v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad+\frac{1}{36(2u-v_{T})}\left[\frac{-5s^{2}-2k_{1}^{2}+2k_{2}^{2}-9k_{3}^{2}-9k_{4}^{2}}{\epsilon}+\frac{s^{2}}{6}(37(-u+v_{3}+v_{4})+123v_{x})\right.$
$\displaystyle\qquad\qquad\qquad+\frac{2}{3}k_{1}^{2}(-u+6v_{1}+v_{3}+v_{4}+3v_{x})+\frac{2}{3}k_{2}^{2}(u+6v_{2}-v_{3}-v_{4}-3v_{x})$
$\displaystyle\qquad\qquad\qquad\left.+\frac{9}{2}k_{3}^{2}(-u+5v_{3}+v_{4}+3v_{x})+\frac{9}{2}k_{4}^{2}(-u+v_{3}+5v_{4}+3v_{x})\right]+O(\epsilon).$
(372) $\displaystyle i^{\text{div}}_{[33,33x3]}$
$\displaystyle=s^{-3-2\epsilon
v_{x}}\hat{j}_{[33;3]}(k_{1},k_{2},s)\hat{i}_{[333]}(s,k_{3},k_{4})$
$\displaystyle\qquad-\frac{s^{-3-2\epsilon
v_{x}}}{8\epsilon}\left[\frac{4^{-\epsilon v_{3}}\Gamma(-\tfrac{3}{2}-\epsilon
v_{3})}{(u+v_{3}-v_{4}-v_{x})\Gamma(\tfrac{3}{2}+\epsilon
v_{3})}k_{3}^{3+2\epsilon
v_{3}}\hat{j}_{[33;3]}^{2u+v_{3}-v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right.$
$\displaystyle\qquad\qquad\qquad\left.+\frac{4^{-\epsilon
v_{4}}\Gamma(-\tfrac{3}{2}-\epsilon
v_{4})}{(u-v_{3}+v_{4}-v_{x})\Gamma(\tfrac{3}{2}+\epsilon
v_{4})}k_{4}^{3+2\epsilon
v_{4}}\hat{j}_{[33;3]}^{2u-v_{3}+v_{4}-v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad+\frac{1}{8\epsilon\,(u-v_{3}-v_{4}+v_{x})}\left[\frac{8}{3+2\epsilon
v_{x}}\hat{i}_{[333]}^{2u-v_{3}-v_{4}+v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right.$
$\displaystyle\qquad\qquad\qquad\left.-\frac{4^{-\epsilon
v_{x}}\Gamma(-\tfrac{3}{2}-\epsilon v_{x})}{\Gamma(\tfrac{3}{2}+\epsilon
v_{x})}\hat{j}_{[33;3]}^{2u-v_{3}-v_{4}+v_{x}\\{v_{1}v_{2}v_{x}\\}}(k_{1},k_{2},s)\right]$
$\displaystyle\qquad+\frac{k_{1}^{3}}{27(2u+v_{1}-v_{2}-v_{3}-v_{4})}\left[\frac{1}{\epsilon}+2v_{1}(\log
k_{1}+\gamma_{E}+\log
2-\tfrac{7}{3})+\tfrac{1}{3}(u-v_{3}-v_{4}-3v_{x})\right]$
$\displaystyle\qquad+\frac{k_{2}^{3}}{3(2u-v_{1}+v_{2}-v_{3}-v_{4})}\left[\frac{1}{\epsilon}+2v_{2}(\log
k_{2}+\gamma_{E}+\log
2-\tfrac{7}{3})+\tfrac{1}{3}(u-v_{3}-v_{4}-3v_{x})\right]$
$\displaystyle\qquad+\frac{k_{3}^{3}}{3(2u-v_{1}-v_{2}+v_{3}-v_{4})}\left[\frac{1}{\epsilon}+2v_{3}(\log
k_{3}+\gamma_{E}+\log
2-\tfrac{7}{3})-\tfrac{1}{3}(u+v_{3}-v_{4}+3v_{x})\right]$
$\displaystyle\qquad+\frac{k_{4}^{3}}{3(2u-v_{1}-v_{2}-v_{3}+v_{4})}\left[\frac{1}{\epsilon}+2v_{4}(\log
k_{4}+\gamma_{E}+\log
2-\tfrac{7}{3})-\tfrac{1}{3}(u-v_{3}+v_{4}+3v_{x})\right]$
$\displaystyle\qquad+O(\epsilon).$ (373)

##  Weight-shifting operators

In this section, we discuss the application of weight-shifting operators to
relate correlators of different scaling dimensions. In principle, such
operators offer an elegant construction of the various correlators of interest
starting from a smaller set of known ‘seed’ integrals. In the cosmological
context, the operators of Karateev:2017jgd  were applied in Arkani-
Hamed:2018kmz ; Baumann:2019oyu  to construct 4-point correlators of
conformally coupled and massless scalars in four-dimensional de Sitter.
Holographically, these are dual to the three-dimensional CFT correlators of
dimensions $\Delta=2,3$ studied here. However, there are several important
subtleties missing from the analysis of Arkani-Hamed:2018kmz ; Baumann:2019oyu
, which arise quite generally in all cases where renormalization is required:

  * •

While weight-shifting operators can be applied to connect amplitudes in the
dimensionally regulated theory, their action on exchange diagrams generates a
specific linear combination of shifted exchange and shifted contact diagrams.
The precise formula is given in (409) below. If one is to arrive at a pure
shifted exchange diagram, this shifted contact contribution must therefore be
evaluated and subtracted off. In some cases, as we show below, this shifted
contact contribution can itself be constructed through the action of
differential operators and symmetrizations.

  * •

For special values of the operator and spacetime dimensions, the specific
coefficients parametrising the linear combination of shifted exchange and
shifted contact diagrams can themselves vanish. In particular, this must
happen whenever a shift operator connects a finite amplitude to a divergent
amplitude. Clearly the result of acting with a differential operator on any
finite amplitude must also be finite, and so any divergent shifted amplitude
thus produced must be accompanied by a vanishing coefficient. When this
happens, knowledge of the finite amplitude only allows us to obtain the
leading divergence of the shifted amplitude and not its subleading finite
part.

  * •

After renormalization, the application of weight-shifting operators can no
longer reliably connect different renormalized amplitudes. This is because the
weight-shifting operators map only homogeneous solutions of the conformal Ward
identities to shifted homogeneous solutions, whereas the renormalized
correlators obey inhomogeneous Ward identities containing anomalies and beta
functions reflecting the breaking of conformal invariance. As a specific
example, it is impossible to obtain the amplitude $i_{[33,22x2]}$ from
$i_{[22,22x2]}$ via the standard weight-shifting operators. This follows
simply from the fact that $i_{[22,22x2]}$ is anomaly-free whereas
$i_{[33,22x2]}$ is not, and the anomaly cannot be introduced through
differential operators in the external momenta. Thus, weight-shifting
operators can only reliably be applied when working at the level of the
regulated theory.

In this section we present a new momentum-space derivation of the relevant
weight-shifting operator for scalar correlators. We then discuss their
application to construct a reduction scheme providing an alternative route to
the correlators we evaluated earlier by direct calculation of Witten diagrams.
This shows how, with due care, the obstacles above can be avoided and the
regulated amplitudes constructed, although the resulting scheme is not much
simpler than evaluating the diagrams directly.

###  Shift operators from the shadow transform

The weight-shifting operators of principal interest are those shifting two of
the conformal dimensions either up or down by one unit while preserving the
spacetime dimension $d$. Denoting these operators collectively as
$\mathcal{W}_{ij}^{\sigma_{i}\sigma_{j}}$, their action is to send

$\Delta_{i}\rightarrow\Delta_{i}+\sigma_{i},\qquad\Delta_{j}\rightarrow\Delta_{j}+\sigma_{j},$
(374)

where $\sigma_{i},\sigma_{j}\in\pm 1$ represents any independent choice of
signs and $(i,j)$ are any pair of legs. In momentum space, and acting on the
first two legs, they take the form

$\displaystyle\mathcal{W}^{--}_{12}$
$\displaystyle=\frac{1}{2}\,\partial_{12}^{2}$ (375)
$\displaystyle\mathcal{W}^{+-}_{12}$
$\displaystyle=k_{1}^{2(\beta_{1}+1)}\mathcal{W}^{--}_{12}k_{1}^{-2\beta_{1}}$
(376) $\displaystyle\mathcal{W}^{-+}_{12}$
$\displaystyle=k_{2}^{2(\beta_{2}+1)}\mathcal{W}^{--}_{12}k_{2}^{-2\beta_{2}}$
(377) $\displaystyle\mathcal{W}_{12}^{++}$
$\displaystyle=k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}\mathcal{W}_{12}^{--}k_{1}^{-2\beta_{1}}k_{2}^{-2\beta_{2}}$
(378)

where $\partial_{12}^{2}=\partial_{12}^{\mu}\partial_{12\mu}$ with
$\partial_{12\mu}=\partial/\partial k_{1}^{\mu}-\partial/\partial k_{2}^{\mu}$
and $\beta_{i}=\Delta_{i}-d/2$ as usual. The analogous expressions for other
pairs of legs follow trivially through permutations. These operators were
introduced in position space in Karateev:2017jgd , and later applied in
momentum space in Arkani-Hamed:2018kmz ; Baumann:2019oyu . However, the
compact expressions (376) - (378) are new and follow directly from the
momentum-space shadow transform as we explain below. Upon multiplying out, one
can verify that (376) - (378) are equivalent to the expressions used in
Baumann:2019oyu , namely

$\displaystyle\mathcal{W}^{-+}_{12}$
$\displaystyle=k_{2}^{2}\mathcal{W}_{12}^{--}+2\beta_{2}\Big{(}\beta_{2}+1-\frac{d}{2}+k_{2}^{\mu}\partial_{12\mu}\Big{)}$
(379) $\displaystyle\mathcal{W}^{+-}_{12}$
$\displaystyle=k_{1}^{2}\mathcal{W}_{12}^{--}+2\beta_{1}\Big{(}\beta_{1}+1-\frac{d}{2}-k_{1}^{\mu}\partial_{12\mu}\Big{)},$
(380) $\displaystyle\mathcal{W}_{12}^{++}$
$\displaystyle=k_{1}^{2}k_{2}^{2}\mathcal{W}_{12}^{--}+2\beta_{1}\beta_{2}(k_{1}^{2}+k_{2}^{2}-s^{2})$
$\displaystyle\qquad+2\beta_{1}k_{2}^{2}\Big{(}\beta_{1}+1-\frac{d}{2}-k_{1}^{\mu}\partial_{12\mu}\Big{)}+2\beta_{2}k_{1}^{2}\Big{(}\beta_{2}+1-\frac{d}{2}+k_{2}^{\mu}\partial_{12\mu}\Big{)}.$
(381)

The weight-shifting action of these operators has been analyzed from an
embedding space perspective in Karateev:2017jgd ; Baumann:2019oyu . Here, we
offer a simple alternative derivation based primarily in momentum space. Our
analysis is restricted to the regulated theory to avoid divergences.

The action of the lowering operator $\mathcal{W}_{ij}^{--}$ follows
immediately since it is simply the Fourier transform (modulo a factor of one
half) of $x_{ij}^{2}$. The latter acts as a lowering operator in position
space since the general position-space $n$-point function can be parametrised
as

$\langle\mathcal{O}(\bm{x}_{1})\ldots\mathcal{O}(\bm{x}_{n})\rangle=\prod_{1\leq
i<j\leq n}x_{ij}^{2\alpha_{ij}}f(\bm{u}),$ (382)

where the $f$ is an arbitrary function of the independent cross ratios
$\bm{u}$. In particular, inversions constrain the $\alpha_{ij}$ to satisfy

$\Delta_{i}=-\sum_{j=1}^{n}\alpha_{ij},\qquad i=1,2,\ldots,n$ (383)

where $\alpha_{ij}=\alpha_{ji}$ and $\alpha_{ii}=0$. Multiplying by
$x_{ij}^{2}$ thus shifts $\alpha_{ij}\rightarrow\alpha_{ij}+1$ and hence we
obtain a new solution with $\Delta_{i}\rightarrow\Delta_{i}-1$ and
$\Delta_{j}\rightarrow\Delta_{j}-1$ but the same $f(\bm{u})$ and spacetime
dimension $d$.

Returning to momentum space, the action of this operator can also be seen from
its intertwining relation with the conformal Ward identities. By direct
computation

$\displaystyle\big{(}\mathcal{K}_{1}^{\mu}(\Delta_{1}-1)+\mathcal{K}_{2}^{\mu}(\Delta_{2}-1)\big{)}\mathcal{W}^{--}_{12}=\mathcal{W}^{--}_{12}\big{(}\mathcal{K}_{1}^{\mu}(\Delta_{1})+\mathcal{K}_{2}^{\mu}(\Delta_{2})\big{)},$
(384)

where

$\mathcal{K}^{\mu}_{i}(\Delta_{i})=k_{i}^{\mu}\frac{\partial}{\partial
k_{i}^{\nu}}\frac{\partial}{\partial
k_{i\nu}}-2k_{i}^{\nu}\frac{\partial}{\partial
k_{i}^{\nu}}\frac{\partial}{\partial
k_{i\mu}}+2(\Delta_{i}-d)\frac{\partial}{\partial k_{i\mu}}$ (385)

is the special conformal generator in momentum space. One can further check
that $\mathcal{W}_{12}^{--}$ commutes with all remaining generators
$\mathcal{K}^{\mu}_{i}(\Delta_{i})$ for $i\neq 1,2$. Given some solution
$\mathcal{I}_{\\{\Delta_{i}\\}}$ of the special conformal Ward identities,

$0=\sum_{i=1}^{n-1}\mathcal{K}_{i}^{\mu}(\Delta_{i})\mathcal{I}_{\\{\Delta_{i}\\}},$
(386)

by acting with $\mathcal{W}_{12}^{--}$ to the left it then follows that

$0=\sum_{i=1}^{n-1}\mathcal{K}_{i}^{\mu}(\Delta_{i}-\delta_{i1}-\delta_{i2})\mathcal{W}_{12}^{--}\mathcal{I}_{\\{\Delta_{i}\\}}.$
(387)

Thus, $\mathcal{W}_{12}^{--}\mathcal{I}_{\\{\Delta_{i}\\}}$ satisfies the
shifted special conformal Ward identities obtained by replacing
$\Delta_{i}\rightarrow\Delta_{i}-\delta_{i1}-\delta_{i2}$. The corresponding
shifted dilatation Ward identity is also satisfied as can be seen by noting
that $\mathcal{W}_{12}^{--}$ is homogeneous with dimension $-2$.

The remaining operators (377) - (378) can then be constructed by applying the
shadow transform. In momentum space, this takes the form of the operator
identity

$\mathcal{K}_{i}^{\mu}(d-\Delta_{i})k_{i}^{-2\beta_{i}}=k^{-2\beta_{i}}\mathcal{K}_{i}^{\mu}(\Delta_{i}).$
(388)

Multiplying a given solution of the special conformal Ward identities by
$k_{i}^{-2\beta_{i}}$ thus generates a new solution with dimension
$\Delta_{i}\rightarrow d-\Delta_{i}$. The corresponding dilatation Ward
identity is also satisfied as can be verified by power counting. To construct
raising operators we then simply shadow transform a selection of momenta,
apply the lowering operator $\mathcal{W}_{12}^{--}$, and then invert the
shadow transform. For example,

$\displaystyle\mathcal{W}_{12}^{++}=k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}\mathcal{W}_{12}^{--}k_{1}^{-2\beta_{1}}k_{2}^{-2\beta_{2}},$
(389)

since multiplying by $k_{1}^{-2\beta_{1}}k_{2}^{-2\beta_{2}}$ sends
$\Delta_{i}\rightarrow\Delta^{\prime}_{i}=d-\Delta_{i}$ for $i=1,2$, after
which $\mathcal{W}_{12}^{--}$ sends
$\Delta^{\prime}_{i}\rightarrow\Delta_{i}^{\prime\prime}=\Delta^{\prime}_{i}-1=d-1-\Delta_{i}$.
The shadow transform is then inverted by applying
$k_{1}^{-2\beta_{1}^{\prime\prime}}k_{1}^{-2\beta_{2}^{\prime\prime}}=k_{1}^{-2\Delta_{1}^{\prime\prime}+d}k_{2}^{-2\Delta_{2}^{\prime\prime}+d}=k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}$
sending
$\Delta_{i}^{\prime\prime}\rightarrow\Delta_{i}^{\prime\prime\prime}=d-\Delta_{i}^{\prime\prime}=\Delta_{i}+1$.
Thus, the net effect is to shift $\Delta_{i}\rightarrow\Delta_{i}+1$ for
$i=1,2$ leaving all other dimensions intact. If instead we had shadow
transformed with respect to only one of the momenta, say the first, then we
would have obtained $\mathcal{W}_{12}^{+-}$ since the action on $\Delta_{2}$
would remain that of $\mathcal{W}_{12}^{--}$.

As for $\mathcal{W}_{12}^{--}$, we can prove the action of these operators
more formally by verifying the intertwining relations. For example, by direct
calculation

$\big{(}\mathcal{K}_{1}^{\mu}(\Delta_{1}+1)+\mathcal{K}_{2}^{\mu}(\Delta_{2}+1)\big{)}\mathcal{W}^{++}_{12}=\mathcal{W}^{++}_{12}\big{(}\mathcal{K}_{1}^{\mu}(\Delta_{1})+\mathcal{K}_{2}^{\mu}(\Delta_{2})\big{)}$
(390)

while all other $\mathcal{K}_{i}^{\mu}(\Delta_{i})$ with $i\neq 1,2$ commute
with $\mathcal{W}_{12}^{++}$. Thus, given some solution of the special
conformal Ward identities (386), by acting to the left with
$\mathcal{W}_{12}^{++}$ we find

$0=\sum_{i=1}^{n-1}\mathcal{K}_{i}^{\mu}(\Delta_{i}+\delta_{i1}+\delta_{i2})\mathcal{W}_{12}^{++}\mathcal{I}_{\\{\Delta_{i}\\}},$
(391)

and so $\mathcal{W}_{12}^{++}$ indeed generates a new solution with raised
$\Delta_{1}$ and $\Delta_{2}$. Again, the dilatation Ward identity is
automatically satisfied since $\mathcal{W}_{12}^{++}$ is homogeneous of
dimension $+2$. A similar analysis applies for $\mathcal{W}_{12}^{-+}$ and
$\mathcal{W}_{12}^{+-}$.

###  Action on exchange diagrams

To analyze the action of the shift operators (375) - (378) on the various
amplitudes we study, it is convenient to convert to Mandelstam variables. In
fact, it will suffice to consider their action on contact diagrams and
$s$-channel exchanges only. Since these amplitudes are all independent of $t$,
we can then further simplify the shift operators by setting all
$t$-derivatives to zero. The operator $\mathcal{W}^{--}_{12}$ then takes the
form

$\displaystyle\mathcal{W}^{--}_{12}$
$\displaystyle=\frac{1}{2}\Big{[}\partial_{1}^{2}+\partial_{2}^{2}+\frac{(d-1)}{k_{1}}\partial_{1}+\frac{(d-1)}{k_{2}}\partial_{2}+\frac{1}{k_{1}k_{2}}(k_{1}^{2}+k_{2}^{2}-s^{2})\partial_{1}\partial_{2}\Big{]}+O(\partial_{t}),$
(392)

where $\partial_{i}=\partial/\partial k_{i}$ while the remaining operators
follow from (376) - (378).

Let us now apply $\mathcal{W}_{12}^{--}$ to an $s$-channel exchange diagram (•
‣ 2). Acting on a product of bulk-boundary propagators (3), we find the
identity

$\displaystyle\mathcal{W}^{--}_{12}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}$
$\displaystyle\quad=\frac{1}{8(\beta_{1}-1)(\beta_{2}-1)}\Big{(}\Box_{z}-(\beta_{1}+\beta_{2}-2)(\beta_{1}+\beta_{2}+d-2)\Big{)}\mathcal{K}_{d,\Delta_{1}-1}(z,k_{1})\mathcal{K}_{d,\Delta_{2}-1}(z,k_{2})$
(393)

where the box operator for $(d+1)$-dimensional AdS is

$\Box_{z}=z^{2}\partial_{z}^{2}+(1-d)z\partial_{z}-s^{2}z^{2}.$ (394)

Thus, if we act with $\mathcal{W}^{--}_{12}$ on an $s$-channel exchange
diagram, we obtain a box operator which can be integrated by parts in $z$ so
as to act on the bulk-bulk propagator. This produces a delta function and a
constant term, since the bulk-bulk propagator (6) obeys

$(-\Box_{z}+m_{x}^{2})\mathcal{G}_{d,\Delta_{x}}(z,s;\zeta)=z^{d+1}\delta(z-\zeta),\qquad
m_{x}^{2}=\Delta_{x}(\Delta_{x}-d)=\beta_{x}^{2}-\frac{d^{2}}{4}.$ (395)

In the regulated theory then, the action of $\mathcal{W}^{--}_{12}$ on an
exchange diagram yields a linear combination of a shifted exchange and a
shifted contact diagram,

$\mathcal{W}^{--}_{12}\,\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}=\mathcal{N}^{--}_{exch.}\,\hat{i}_{[\Delta_{1}-1,\Delta_{2}-1,\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}+\mathcal{N}^{--}_{cont.}\,\hat{i}_{[\Delta_{1}-1,\Delta_{2}-1,\Delta_{3},\Delta_{4}]}$
(396)

where

$\displaystyle\mathcal{N}^{--}_{exch.}$
$\displaystyle=\frac{\beta_{x}^{2}-d^{2}/4-(\beta_{1}+\beta_{2}-2)(\beta_{1}+\beta_{2}+d-2)}{8(\beta_{1}-1)(\beta_{2}-1)},$
(397) $\displaystyle\mathcal{N}^{--}_{cont.}$
$\displaystyle=-\frac{1}{8(\beta_{1}-1)(\beta_{2}-1)}.$ (398)

If our goal is to obtain just the shifted exchange diagram, we must therefore
evaluate and subtract off this contribution from the shifted contact diagram.
Sometimes this can be achieved through the further use of weight-shifting
operators as discussed in Section 6.3.1.

Let us next consider the action of the raising operator
$\mathcal{W}_{12}^{++}$ on an $s$-channel exchange. This can easily be
understood through use of (393) in combination with the shadow relation for
the bulk-boundary propagator,

$k^{-2\beta}\mathcal{K}_{d,\Delta}(z,k)=\Lambda_{\beta}\,\mathcal{K}_{d,d-\Delta}(z,k),\qquad\Lambda_{\beta}=\frac{\Gamma(-\beta)}{4^{\beta}\Gamma(\beta)}.$
(399)

First, we write

$\displaystyle\mathcal{W}_{12}^{++}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}$
$\displaystyle\quad=k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}\mathcal{W}_{12}^{--}k_{1}^{-2\beta_{1}}k_{2}^{-2\beta_{2}}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}$
$\displaystyle\quad=\Lambda_{\beta_{1}}\Lambda_{\beta_{2}}k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}\mathcal{W}_{12}^{--}\Big{(}\mathcal{K}_{d,d-\Delta_{1}}(z,k_{1})\mathcal{K}_{d,d-\Delta_{2}}(z,k_{2})\Big{)}$
$\displaystyle\quad=\frac{\Lambda_{\beta_{1}}\Lambda_{\beta_{2}}k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}}{8(\beta_{1}+1)(\beta_{2}+1)}$
$\displaystyle\qquad\times\Big{(}\Box_{z}-(\beta_{1}+\beta_{2}+2)(\beta_{1}+\beta_{2}-d+2)\Big{)}\mathcal{K}_{d,d-\Delta_{1}-1}(z,k_{1})\mathcal{K}_{d,d-\Delta_{2}-1}(z,k_{2}).$
(400)

We now commute the factor $k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}$
through the differential operator and use again the shadow identity (399)
(replacing $\Delta\rightarrow d-\Delta-1$) to find

$\displaystyle
k_{1}^{2(\beta_{1}+1)}k_{2}^{2(\beta_{2}+1)}\mathcal{K}_{d,d-\Delta_{1}-1}(z,k_{1})\mathcal{K}_{d,d-\Delta_{2}-1}(z,k_{2})$
$\displaystyle\qquad=\Lambda_{-\beta_{1}-1}\Lambda_{-\beta_{2}-1}\mathcal{K}_{d,\Delta_{1}+1}(z,k_{1})\mathcal{K}_{d,\Delta_{2}+1}(z,k_{2}).$
(401)

After simplifying the prefactor

$\displaystyle\frac{\Lambda_{\beta_{1}}\Lambda_{\beta_{2}}}{8(\beta_{1}+1)(\beta_{2}+1)}\Lambda_{-\beta_{1}-1}\Lambda_{-\beta_{2}-1}=2\beta_{1}\beta_{2},$
(402)

we arrive the desired identity

$\displaystyle\mathcal{W}_{12}^{++}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}$
$\displaystyle\quad=2\beta_{1}\beta_{2}\Big{(}\Box_{z}-(\beta_{1}+\beta_{2}+2)(\beta_{1}+\beta_{2}-d+2)\Big{)}\mathcal{K}_{d,\Delta_{1}+1}(z,k_{1})\mathcal{K}_{d,\Delta_{2}+1}(z,k_{2}).$
(403)

This identity is the counterpart to (393) and can be used in the same way to
evaluate the action of $\mathcal{W}_{12}^{++}$ on an $s$-channel exchange
diagram. After integration by parts in $z$ and use of (395), we obtain the
result

$\mathcal{W}^{++}_{12}\,\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}=\mathcal{N}^{++}_{exch.}\,\hat{i}_{[\Delta_{1}+1,\Delta_{2}+1,\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}+\mathcal{N}^{++}_{cont.}\,\hat{i}_{[\Delta_{1}+1,\Delta_{2}+1,\Delta_{3},\Delta_{4}]}$
(404)

where

$\displaystyle\mathcal{N}^{++}_{exch.}=2\beta_{1}\beta_{2}\Big{(}\beta_{x}^{2}-\frac{d^{2}}{4}-(\beta_{1}+\beta_{2}+2)(\beta_{1}+\beta_{2}-d+2)\Big{)},\qquad\mathcal{N}^{++}_{cont.}$
$\displaystyle=-2\beta_{1}\beta_{2}.$ (405)

The steps above can similarly be repeated for the remaining operators
$\mathcal{W}_{12}^{+-}$ and $\mathcal{W}_{12}^{-+}$. Examining the form of
these results, and those above, we find they can all be collected into a
common form:

$\displaystyle\mathcal{W}_{12}^{\sigma_{1}\sigma_{2}}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}=\mathcal{N}^{\sigma_{1}\sigma_{2}}\Big{(}\Box_{z}-m^{2}_{\sigma_{1}\sigma_{2}}\Big{)}\mathcal{K}_{d,\Delta_{1}+\sigma_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}+\sigma_{2}}(z,k_{2}).$
(406)

where $\sigma_{1},\sigma_{2}\in\pm 1$ and

$\displaystyle\mathcal{N}^{\sigma_{1}\sigma_{2}}$
$\displaystyle=2^{\sigma_{1}+\sigma_{2}-1}(\beta_{1})_{\sigma_{1}}(\beta_{2})_{\sigma_{2}},$
(407) $\displaystyle m^{2}_{\sigma_{1}\sigma_{2}}$
$\displaystyle=(\sigma_{1}\beta_{1}+\sigma_{2}\beta_{2}+2)(\sigma_{1}\beta_{1}+\sigma_{2}\beta_{2}-d+2).$
(408)

Here the Pochhammer symbol
$(\beta)_{\sigma}=\Gamma(\beta+\sigma)/\Gamma(\beta)$ is just a convenient way
of writing $(\beta)_{1}=\beta$ and $(\beta)_{-1}=(\beta-1)^{-1}$. Acting on an
$s$-channel exchange, we find

$\displaystyle\mathcal{W}^{\sigma_{1}\sigma_{2}}_{12}\,\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}=\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}\,\hat{i}_{[\Delta_{1}+\sigma_{1},\Delta_{2}+\sigma_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}+\mathcal{N}^{\sigma_{1}\sigma_{2}}_{cont.}\,\hat{i}_{[\Delta_{1}+\sigma_{1},\Delta_{2}+\sigma_{2},\Delta_{3},\Delta_{4}]}$
(409)

where

$\displaystyle\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}$
$\displaystyle=\mathcal{N}^{\sigma_{1}\sigma_{2}}(m_{x}^{2}-m_{\sigma_{1}\sigma_{2}}^{2}),\qquad\mathcal{N}^{\sigma_{1}\sigma_{2}}_{cont.}=-\mathcal{N}^{\sigma_{1}\sigma_{2}}.$
(410)

Where divergences are present, this identity again applies only in the
regulated theory.

Using the definition of $m_{x}^{2}$ from (395), we can rewrite

$\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}=-\mathcal{N}^{\sigma_{1}\sigma_{2}}\Big{(}\frac{d}{2}-\sigma_{1}\tilde{\beta}_{1}-\sigma_{2}\tilde{\beta}_{2}+\beta_{x}\Big{)}\Big{(}\frac{d}{2}-\sigma_{1}\tilde{\beta}_{1}-\sigma_{2}\tilde{\beta}_{2}-\beta_{x}\Big{)}$
(411)

where the $\tilde{\beta}_{i}=\beta_{i}+\sigma_{i}$ are the shifted values of
the $\beta_{i}$ parameters under $\Delta_{i}\rightarrow\Delta_{i}+\sigma_{i}$.
From this expression, we see that $\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}$
has a zero whenever the condition

$0=\frac{d}{2}-\sigma_{1}\tilde{\beta}_{1}-\sigma_{2}\tilde{\beta}_{2}-\sigma_{x}\beta_{x}$
(412)

is satisfied for some choice of the signs
$(\sigma_{1},\sigma_{2},\sigma_{x})=(\pm,\pm,\pm)$, where each sign can be
chosen independently of the others. This makes sense since whenever this
condition is satisfied the shifted exchange diagram is singular222The
divergence of the shifted exchange diagram derives from the region where the
vertex corresponding to the shifted operators approaches the boundary of AdS:
effectively, this is the lower limit of an integral of three Bessel $K$
functions, two of which come from the bulk-boundary propagators and the third
from the exterior part of the bulk-bulk propagator. The singularities of such
triple-$K$ integrals, which also arise for the momentum-space 3-point function
$\langle\\!\langle\mathcal{O}_{\Delta_{1}+\sigma_{1}}\mathcal{O}_{\Delta_{2}+\sigma_{2}}\mathcal{O}_{\Delta_{x}}\rangle\\!\rangle$,
have been analyzed in Bzowski:2015pba  leading to precisely the condition
(412)., and a differential operator such as
$\mathcal{W}^{\sigma_{1}\sigma_{2}}_{12}$ cannot map a finite exchange diagram
to a singular one. Rather, acting with
$\mathcal{W}^{\sigma_{1}\sigma_{2}}_{12}$ yields a finite result which, in the
regulated theory, derives from the product of a vanishing coefficient
$\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}$ and a divergent shifted exchange
diagram, plus a contact contribution. In such situations where
$\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}$ has a zero, the application of
$\mathcal{W}^{\sigma_{1}\sigma_{2}}_{12}$ to a finite diagram therefore only
allows us to extract the leading divergence of the shifted exchange diagram:
to additionally compute its finite piece requires knowing the finite unshifted
exchange diagram to subleading orders in the regulator $\epsilon$. This is
generally harder to compute since the corresponding Bessel function indices
are no longer half-integer.

For this reason, such divergences and their corresponding zeros in the
exchange coefficients $\mathcal{N}^{\sigma_{i}\sigma_{j}}_{exch.}$ have
important consequences for the construction of reduction schemes relating
different exchange diagrams. Fortunately, as we will see in Section 6.4, it
turns out that for the correlators of interest in this paper, all such zeros
of $\mathcal{N}^{\sigma_{i}\sigma_{j}}_{exch.}$ can be avoided through making
appropriate use of shadow transformations in addition to the
$\mathcal{W}^{\sigma_{i}\sigma_{j}}_{ij}$ operators.

###  Action on contact diagrams

As we saw above, the action of the $\mathcal{W}_{12}^{\sigma_{1}\sigma_{2}}$
on an exchange diagram generates a linear combination of a shifted exchange
diagram and a shifted contact diagram. The action of
$\mathcal{W}_{12}^{\sigma_{1}\sigma_{2}}$ on a contact diagram produces a
linear combination of a shifted contact diagram and a shifted contract diagram
containing an AdS box vertex. In this section we discuss how this box
contribution, which also follows from to the action of the Casimir on the
shifted contact diagram, can in some cases be removed through an appropriate
symmetrization. Further shift relations for contact diagrams are discussed in
appendix D.1. Once all shifted contact diagrams are known, either via weight-
shifting operators or from direct evaluation, their contribution can be
subtracted enabling all shifted exchange diagrams to be constructed.

####  Exchanges to contacts

As an intermediate step, one can always pass from an exchange diagram to the
corresponding contact diagram (i.e., with no shift to the operator dimensions)
through the well-known action of the quadratic Casimir operator

$\displaystyle\mathcal{C}_{12}$
$\displaystyle=\big{(}2k_{1\mu}k_{2\nu}-k_{1}\cdot
k_{2}\,\delta_{\mu\nu}\big{)}\partial_{12}^{\mu}\partial_{12}^{\nu}+2\big{(}(\Delta_{2}-d)k_{1\mu}-(\Delta_{1}-d)k_{2\mu}\big{)}\partial_{12}^{\mu}$
$\displaystyle\quad-(\Delta_{1}+\Delta_{2}-2d)(\Delta_{1}+\Delta_{2}-d).$
(413)

In Mandelstam variables,

$\displaystyle\mathcal{C}_{12}$
$\displaystyle=\frac{1}{2}(s^{2}+k_{1}^{2}-k_{2}^{2})K_{1}+\frac{1}{2}(s^{2}+k_{2}^{2}-k_{1}^{2})K_{2}-\big{(}L_{1}+L_{2}+\frac{3d}{2}\big{)}^{2}+\frac{d^{2}}{4}+O(\partial_{t})$
(414)

where

$K_{i}=\partial_{i}^{2}+\frac{1-2\beta_{i}}{k_{i}}\partial_{i},\qquad
L_{i}=k_{i}\partial_{i}-\Delta_{i}$ (415)

and we have omitted $t$-derivatives since $s$-channel scalar exchanges are
independent of $t$. Indeed, we can simplify this further since for $s$-channel
scalar exchanges

$(K_{1}-K_{2})\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}=0.$
(416)

The action of the Casimir on such diagrams is thus equal to that of the
reduced operator

$\tilde{\mathcal{C}}_{12}=\frac{s^{2}}{2}(K_{1}+K_{2})-\big{(}L_{1}+L_{2}+\frac{3d}{2}\big{)}^{2}+\frac{d^{2}}{4}.$
(417)

We can also use this reduced Casimir operator when acting on scalar contact
diagrams, since these are also independent of $t$ (and $s$) and satisfy
$(K_{1}-K_{2})\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]}=0$.

One can now verify directly the identity

$\displaystyle\tilde{\mathcal{C}}_{12}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}=-\Box_{z}\Big{(}\mathcal{K}_{d,\Delta_{1}}(z,k_{1})\mathcal{K}_{d,\Delta_{2}}(z,k_{2})\Big{)}.$
(418)

Acting on an $s$-channel exchange diagram with
$(\tilde{\mathcal{C}}_{12}+m_{x}^{2})$ and integrating by parts with respect
to $z$, we obtain a delta function from (395). This collapses the exchange
diagram to a contact diagram of the same operator dimensions:

$\displaystyle\big{(}\tilde{\mathcal{C}}_{12}+m_{x}^{2}\big{)}\,\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}=\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]}$
(419)

One can alternatively obtain this result by showing that

$\displaystyle[\mathcal{W}_{12}^{--},\mathcal{W}_{12}^{++}]=2(\beta_{1}+\beta_{2})\Big{(}\mathcal{C}_{12}+(\beta_{1}+\beta_{2})^{2}-\frac{d}{2}(d-2)\Big{)},$
(420)
$\displaystyle[\mathcal{W}_{12}^{-+},\mathcal{W}_{12}^{+-}]=2(\beta_{1}-\beta_{2})\Big{(}\mathcal{C}_{12}+(\beta_{1}-\beta_{2})^{2}-\frac{d}{2}(d-2)\Big{)},$
(421)

then making two applications of (409).

####  Shifted contact diagrams from symmetrization

Let us now consider the action of $\mathcal{W}_{12}^{\sigma_{1},\sigma_{2}}$
on the contact diagram
$\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]}$. From (406), we
obtain a linear combination of the shifted contact diagram and a shifted
contact diagram with an AdS box vertex. Via (418), the latter can be re-
expressed as the Casimir operator acting on the shifted contact diagram giving

$\displaystyle\mathcal{W}_{12}^{\sigma_{1},\sigma_{2}}\hat{i}_{[\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]}$
$\displaystyle=-\mathcal{N}^{\sigma_{1}\sigma_{2}}\Big{(}\tilde{\mathcal{C}}_{12}^{\Delta_{1}+\sigma_{1},\Delta_{2}+\sigma_{2}}+m_{\sigma_{1}\sigma_{2}}^{2}\Big{)}\hat{i}_{[\Delta_{1}+\sigma_{1},\Delta_{2}+\sigma_{2},\Delta_{3},\Delta_{4}]},$
(422)

where $\mathcal{N}^{\sigma_{1}\sigma_{2}}$ and $m_{\sigma_{1}\sigma_{2}}^{2}$
are given in (407) and (408) but the dimensions appearing inside the Casimir
operator are the shifted ones as indicated by the superscript indices.

In the special case where at least three of the shifted operator dimensions
are equal, the Casimir operator on the right-hand side of this relation can
now be eliminated through a simple symmetrization procedure. Here, it is
useful to recall that from an embedding space perspective, the $d$-dimensional
conformal Ward identities originate from invariance under the action of the
$SO(d+1,1)$ Lorentz generators

$0=\sum_{i=1}^{4}L^{AB}_{i}$ (423)

where $L_{i}^{AB}=X_{i}^{A}(\partial/\partial
X_{iB})-X_{i}^{B}(\partial/\partial X_{iA})$. Squaring this relation and using
the identities

$\mathcal{C}_{ij}^{\Delta_{i},\Delta_{j}}=\frac{1}{2}(L_{i}^{AB}+L_{j}^{AB})(L_{iAB}+L_{jAB}),\qquad\frac{1}{2}L_{i}^{AB}L_{iAB}=-\Delta_{i}(\Delta_{i}-d),$
(424)

along with

$\mathcal{C}_{12}^{\Delta_{1},\Delta_{2}}=\mathcal{C}_{34}^{\Delta_{3},\Delta_{4}},\qquad\mathcal{C}_{14}^{\Delta_{1},\Delta_{4}}=\mathcal{C}_{23}^{\Delta_{2},\Delta_{3}},\qquad\mathcal{C}_{13}^{\Delta_{1},\Delta_{3}}=\mathcal{C}_{24}^{\Delta_{2},\Delta_{4}},$
(425)

as follows from squaring, e.g.,
$L_{1}^{AB}+L_{2}^{AB}=-(L_{3}^{AB}+L_{4}^{AB})$, we obtain

$\mathcal{C}_{12}^{\Delta_{1},\Delta_{2}}+\mathcal{C}_{13}^{\Delta_{1},\Delta_{3}}+\mathcal{C}_{23}^{\Delta_{2},\Delta_{3}}=\frac{1}{2}\sum_{i<j}^{4}\mathcal{C}_{ij}^{\Delta_{i},\Delta_{j}}=-\sum_{i=1}^{4}\Delta_{i}(\Delta_{i}-d).$
(426)

If the three operator dimensions on the left-hand side are equal, this reduces
to

$\displaystyle\Big{(}\mathcal{C}_{12}^{\Delta,\Delta}+\mathcal{C}_{13}^{\Delta,\Delta}+\mathcal{C}_{23}^{\Delta,\Delta}\Big{)}\hat{i}_{[\Delta,\Delta,\Delta,\Delta_{4}]}=-\Big{(}3\Delta(\Delta-d)+\Delta_{4}(\Delta_{4}-d)\Big{)}\,\hat{i}_{[\Delta,\Delta,\Delta,\Delta_{4}]}.$
(427)

Moreover, we know from (422) that

$\displaystyle\mathcal{W}_{12}^{\sigma_{1},\sigma_{2}}\hat{i}_{[\Delta-\sigma_{1},\Delta-\sigma_{2},\Delta,\Delta_{4}]}$
$\displaystyle=-\mathcal{N}^{\sigma_{1}\sigma_{2}}\Big{(}\mathcal{C}_{12}^{\Delta,\Delta}+m_{\sigma_{1}\sigma_{2}}^{2}\Big{)}\hat{i}_{[\Delta,\Delta,\Delta,\Delta_{4}]}.$
(428)

By permuting momenta and exploiting the fact that the three shifted dimensions
are equal, we can now construct all the remaining terms on the left-hand side
of (427). For example,

$\displaystyle\Big{(}\mathcal{W}_{12}^{\sigma_{1},\sigma_{2}}\hat{i}_{[\Delta-\sigma_{1},\Delta-\sigma_{2},\Delta,\Delta_{4}]}\Big{)}_{\bm{k}_{2}\leftrightarrow\bm{k}_{3}}$
$\displaystyle=-\mathcal{N}^{\sigma_{1}\sigma_{2}}\Big{(}\mathcal{C}_{13}^{\Delta,\Delta}+m_{\sigma_{1}\sigma_{2}}^{2}\Big{)}\hat{i}_{[\Delta,\Delta,\Delta,\Delta_{4}]}.$
(429)

Summing over permutations, the Casimir contributions can now be eliminated

$\displaystyle\Big{(}\mathcal{W}_{12}^{\sigma_{1},\sigma_{2}}\hat{i}_{[\Delta-\sigma_{1},\Delta-\sigma_{2},\Delta,\Delta_{4}]}\Big{)}+(\bm{k}_{2}\leftrightarrow\bm{k}_{3})+(\bm{k}_{1}\leftrightarrow\bm{k}_{3})$
$\displaystyle\qquad=\mathcal{N}^{\sigma_{1}\sigma_{2}}\Big{(}3\Delta(\Delta-d)+\Delta_{4}(\Delta_{4}-d)-3m_{\sigma_{1}\sigma_{2}}^{2}\Big{)}\hat{i}_{[\Delta,\Delta,\Delta,\Delta_{4}]}$
$\displaystyle\qquad=2^{\sigma_{1}+\sigma_{2}-1}(\beta-\sigma_{1})_{\sigma_{1}}(\beta-\sigma_{2})_{\sigma_{2}}$
$\displaystyle\qquad\qquad\times\Big{(}3\beta^{2}+\beta_{4}^{2}-d^{2}-3\beta^{2}(\sigma_{1}+\sigma_{2})^{2}+3\beta
d(\sigma_{1}+\sigma_{2})\Big{)}\hat{i}_{[\Delta,\Delta,\Delta,\Delta_{4}]}$
(430)

where the Pochammer symbols are simply $(\beta-1)_{1}=\beta-1$ and
$(\beta+1)_{-1}=\beta^{-1}$. This relation (and its permutations) thus enable
the construction of shifted contact diagrams containing three equal dimensions
if the appropriate initial integral is known. Some further shift relations for
contact diagrams can be found in appendix D.1.

###  Reduction scheme

Equation (409) can be used to set up a reduction scheme. Starting from a
smaller set of simple master integrals, one can use the weight-changing
operators to obtain other diagrams. Since the lower dimensions generally yield
simpler expressions, one would like to start with the lowest dimensional
diagrams such as $i_{[22,22x\Delta_{x}]}$ and apply the weight-raising
operators $\mathcal{W}^{++}_{ij}$. To see if it is possible, we begin by
investigating all available relations due to (409) between the amplitudes.

####  Available relations

First, notice that one can either use the weight-changing operators directly
or in a combination with the shadow transform. The shadow transform is an
observation that the bulk-to-boundary propagators for the AdS fields with
$\Delta$ and $d-\Delta$ are related according to,

$\displaystyle\operatorname{\mathcal{K}}_{d,d-\Delta}(z,k)=\frac{\Gamma\left(\Delta-\tfrac{d}{2}\right)}{\Gamma\left(\tfrac{d}{2}-\Delta\right)}\left(\frac{k}{2}\right)^{d-2\Delta}\times\operatorname{\mathcal{K}}_{d,\Delta}(z,k).$
(431)

Thus, we can define the shadow transform $\mathcal{S}_{j}$ acting on the
$j$-th external leg of an amplitude as the multiplication by the above
constant. This yields, for example,

$\displaystyle i_{[d-\Delta_{1},\Delta_{2};\Delta_{3}\Delta_{4}x\Delta_{x}]}$
$\displaystyle=\mathcal{S}_{1}i_{[\Delta_{1},\Delta_{2};\Delta_{3}\Delta_{4}x\Delta_{x}]}$
$\displaystyle=\frac{\Gamma\left(\Delta_{1}-\tfrac{d}{2}\right)}{\Gamma\left(\tfrac{d}{2}-\Delta_{1}\right)}\left(\frac{k_{1}}{2}\right)^{d-2\Delta_{1}}\times
i_{[\Delta_{1},\Delta_{2};\Delta_{3}\Delta_{4}x\Delta_{x}]}.$ (432)

It is important to use regulated dimensions $\hat{d}$ and $\hat{\Delta}_{1}$
when dealing with regulated amplitudes.

While the weight-shifting operators $\mathcal{W}^{++}_{ij}$ and
$\mathcal{W}^{--}_{ij}$ always raise or lower a pair of dimensions, we can
compose them with the shadow transform to raise or lower a single dimension
only. As an example, consider $i_{[22,22x2]}$. By acting with
$\mathcal{W}^{++}_{12}$, according to (409) one should obtain a combination of
$i_{[33,22x2]}$ and $i_{[3322]}$. On the other hand, we can use the shadow
transform to flip $\Delta_{2}=2$ to $d-\Delta_{2}=1$. Thus, the combination
$\mathcal{W}^{++}_{12}\circ\mathcal{S}_{2}$ should map $i_{[22,22x2]}$ to a
combination of $i_{[32,22x2]}$ and $i_{[3222]}$. In general, we can carry out
the following operations on a pair of dimensions:

$(22)$$(33)$$(32)$$\mathcal{W}^{++}$$\mathcal{W}^{--}$$\mathcal{W}^{++}\mathcal{S}_{2}$$\mathcal{S}_{2}\mathcal{W}^{--}$
(433)

It is always important to remember that the action of the weight-shifting
operators on an exchange diagram results in a combination of shifted exchange
and contact diagrams.

####  The scheme

At first glance, all exchange amplitudes with a fixed exchange dimension
$\Delta_{x}$ are related through repeated use of the operations in (433). On
practical level, however, this turns out not to be the case. In order for the
reduction scheme to work, three conditions must be satisfied:

  * 1.

One must know all contact diagrams featuring on the right hand side of (409).
These are easy to calculate directly and thus pose no difficulty.

Alternatively, one can attempt to construct them via the shift relation
(6.3.2) starting from $\hat{i}_{[2323]}$. This contact diagram can itself be
obtained from either $\hat{i}_{[2323x2]}$ or $\hat{i}_{[2323x3]}$ via (419).
First, we evaluate

$\displaystyle(\mathcal{W}_{24}^{--}\hat{i}_{[2323]})+(\bm{k}_{2}\leftrightarrow\bm{k}_{3})+(\bm{k}_{3}\leftrightarrow\bm{k}_{4})$
$\displaystyle=-(2+\epsilon)(5+2\epsilon)\hat{i}_{[2222]},$ (434)
$\displaystyle(\mathcal{W}_{13}^{++}\hat{i}_{[2323]})+(\bm{k}_{2}\leftrightarrow\bm{k}_{3})+(\bm{k}_{3}\leftrightarrow\bm{k}_{4})$
$\displaystyle=\epsilon(3-2\epsilon)\hat{i}_{[3333]}.$ (435)

In this last equation, $\mathcal{W}^{++}_{13}$ projects out the leading
$\epsilon^{-1}$ divergence of $\hat{i}_{[2323]}$ meaning the left-hand side is
of order $\epsilon^{0}$. We can therefore only construct the leading
$\epsilon^{-1}$ divergence of $\hat{i}_{[3333]}$ from this relation. Next, via
the shadow transform (431), we find

$\hat{i}_{[1323]}=\mathcal{S}_{1}\hat{i}_{[2323]}=\frac{4^{\beta_{1}}\Gamma(\beta_{1})}{\Gamma(-\beta_{1})}k_{1}^{-2\beta_{1}}\hat{i}_{[2323]}=-\frac{1}{k_{1}}\hat{i}_{[2323]}.$
(436)

We now construct

$\displaystyle(\mathcal{W}_{12}^{+-}\hat{i}_{[1323]})+(\bm{k}_{2}\leftrightarrow\bm{k}_{3})+(\bm{k}_{1}\leftrightarrow\bm{k}_{3})$
$\displaystyle=(3+6\epsilon+2\epsilon^{2})\hat{i}_{[2223]},$ (437)
$\displaystyle(\mathcal{W}_{23}^{++}\hat{i}_{[2223]})+(\bm{k}_{2}\leftrightarrow\bm{k}_{4})+(\bm{k}_{3}\leftrightarrow\bm{k}_{4})$
$\displaystyle=(-1+3\epsilon-2\epsilon^{2})\hat{i}_{[2333]}.$ (438)

From $\hat{i}_{[2323]}$, we can thus obtain all the contact diagrams of weight
two and three operators to finite order, with the exception of
$\hat{i}_{[3333]}$ which we only know to leading $\epsilon^{-1}$ order. The
finite part of this integral can be found either by direction evaluation or
through other shift operators as discussed in appendix D.1.

As we saw in (409), evaluating and subtracting contributions from contact
diagrams is essential when applying the
$\mathcal{W}_{ij}^{\sigma_{i}\sigma_{j}}$ operators to exchange diagrams. In
Baumann:2019oyu , where such a recursion scheme was advocated, these contact
diagrams were not subtracted and the equivalence of
$i_{[\Delta_{1}+1,\Delta_{2}+1,\Delta_{3}\Delta_{4}x\Delta_{x}]}$ and
$\mathcal{W}^{++}_{12}i_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$
was incorrectly assumed.

  * 2.

Regularization must be taken care of. For example, consider
$\mathcal{W}^{++}_{12}i_{[22,22x2]}$, which we denote by
$i_{[\tilde{3}\tilde{3},22x2]}$. Since $i_{[22,22x2]}$ is finite, so is
$i_{[\tilde{3}\tilde{3},22x2]}$. However, we see that both the exchange
diagram $i_{[33,22x2]}$ in (37) as well as the contact diagram $i_{[3322]}$ in
(29) are divergent. Nevertheless, their combination on the right-hand side of
(409) is finite in any regularization scheme and

$\displaystyle\left[-1+\frac{\epsilon}{2}(3u-7v_{1}-7v_{2}+v_{x})+O(\epsilon^{2})\right]\hat{i}_{[33,22x2]}$
$\displaystyle\qquad\qquad\qquad+\left[-\frac{1}{2}-\epsilon(v_{1}+v_{2})+O(\epsilon^{2})\right]\hat{i}_{[3322]}=i_{[\tilde{3}\tilde{3},22x2]}.$
(439)

Thus, one can derive $\hat{i}_{[33,22x2]}$ regulated in an arbitrary scheme
once the regulated contact diagram $\hat{i}_{[3322]}$ is known in that scheme.
It is important to emphasise that, whenever regularization is required, (409)
holds for regulated amplitudes. If the terms of order $\epsilon$ in the above
expression were dropped, the equality would fail.

Indeed, if we naively apply weight-shifting operators to renormalized
diagrams, due to the presence of counterterm contributions the two sides of
(409) will generally only agree up to some local terms. For example,

$\displaystyle\mathcal{W}^{++}_{12}i^{\text{ren}}_{[22,22x2]}$
$\displaystyle=-i^{\text{ren}}_{[33,22x2]}-\frac{1}{2}i^{\text{ren}}_{[3322]}+\frac{k_{3}+k_{4}}{8}\left(3+2\mathfrak{a}_{[3322]}^{(1)}-2\mathfrak{a}_{[33,22x2]}^{(1)}\right),$
(440) $\displaystyle\mathcal{W}^{++}_{12}i^{\text{ren}}_{[22,33x2]}$
$\displaystyle=-i^{\text{ren}}_{[33,33x2]}-\frac{1}{2}i^{\text{ren}}_{[3333]}+\frac{k_{1}^{3}+k_{2}^{3}}{24}\left(3-2\mathfrak{a}_{[3333]}^{(1)}+2\mathfrak{a}_{[33,33x2]}^{(1)}\right)$
$\displaystyle\qquad\qquad+\frac{k_{3}^{3}+k_{4}^{3}}{24}\left(-3-2\mathfrak{a}_{[3333]}^{(1)}+2\mathfrak{a}_{[33,33x2]}^{(1)}\right).$
(441)

The moral is that (409) holds for regulated diagrams and one should be wary of
extrapolating this relation to renormalized diagrams.

  * 3.

The value of the constant $\mathcal{N}^{\sigma_{1}\sigma_{2}}_{exch.}$ on the
right hand side of (409) cannot vanish when $\epsilon\rightarrow 0$. The
constant vanishes when (412) is satisfied for unregulated parameters. In such
a case the finite part of
$\mathcal{W}^{\sigma_{1}\sigma_{2}}_{12}\hat{i}_{[\Delta_{1}\Delta_{2},\Delta_{3}\Delta_{4}x\Delta_{x}]}$
does not allow for the determination of the finite part of the exchange
diagram
$\hat{i}_{[\Delta_{1}+1,\Delta_{2}+1,\Delta_{3}\Delta_{4}x\Delta_{x}]}$. In
cases of $\Delta_{j}=2,3$, $j=1,2,3,4,x$ considered in this paper and $d=3$
this precludes the use of half of the operations listed in (433). The
operations that still can be used in the scheme, depending on the value of the
exchange dimension $\Delta_{x}$ are presented in the following table:

| $\mathcal{W}_{ij}$ | $\mathcal{W}^{--}$ | $\mathcal{W}^{++}\mathcal{S}$ | $\mathcal{S}\mathcal{W}^{--}$
---|---|---|---|---
$\Delta_{x}=2$ | yes | no | no | yes
$\Delta_{x}=3$ | no | yes | yes | no

This means, for example, that the exchange diagram $\hat{i}_{[33,22x3]}$
cannot be obtained from $\hat{i}_{[22,22x3]}$ by the application of (409).
Indeed, in such a case
$\mathcal{N}^{++}_{exch.}=\tfrac{3\epsilon}{2}(u-v_{1}-v_{2}-v_{x})+O(\epsilon^{2})$
and vanishes in the $\epsilon\rightarrow 0$ limit.

Taking into account the above considerations, a workable reduction scheme
connecting the various amplitudes is presented in Figures 5 and 6. Given the
seed integral $\hat{i}_{[32,32x2]}$ we can construct all the integrals with
$\Delta_{x}=2$, and from $\hat{i}_{[33,33x3]}$ we obtain all the integrals
with $\Delta_{x}=3$.

$i_{[22,22x2]}$$\mathcal{W}^{++}_{34}$$\hat{i}_{[22,33x2]}$$\mathcal{W}^{++}_{12}$$\hat{i}_{[33,33x2]}$$\hat{i}_{[32,22x2]}$$\mathcal{W}^{++}_{34}$$\hat{i}_{[32,33x2]}$$\mathcal{S}_{2}\mathcal{W}^{--}_{12}$$\mathcal{S}_{2}\mathcal{W}^{--}_{12}$$\hat{i}_{[32,32x2]}$$\mathcal{S}_{4}\mathcal{W}^{--}_{34}$
Figure 5: Reduction scheme for the amplitudes with $\Delta_{x}=2$.
$i_{[22,22x3]}$$\mathcal{W}^{--}_{34}$$\hat{i}_{[22,33x3]}$$\mathcal{W}^{--}_{12}$$\hat{i}_{[33,33x3]}$$\hat{i}_{[32,22x3]}$$\mathcal{W}^{--}_{34}$$\hat{i}_{[32,33x3]}$$\mathcal{W}^{++}_{12}\mathcal{S}_{2}$$\mathcal{W}^{++}_{12}\mathcal{S}_{2}$$\hat{i}_{[32,32x3]}$$\mathcal{W}^{++}_{34}\mathcal{S}_{4}$
Figure 6: Reduction scheme for the amplitudes with $\Delta_{x}=3$.

####  Shifting the dimension of the exchanged operator

To complete our analysis, we now seek a weight-shifting operator capable of
connecting the two schemes in Figures 5 and 6, so that only a single exchange
diagram is required as a seed integral. This requires shifting the dimension
$\Delta_{x}$ of the exchanged operator.

For exchange diagrams with all external dimensions equal to two, a shift
operator sending $\Delta_{x}\rightarrow\Delta_{x}+1$ was given in Arkani-
Hamed:2018kmz . A detailed derivation of this operator (549) is given in
appendix D.2. Here, however, this operator is not suitable since the direction
of the arrows in Figure 6 is such that connecting $\hat{i}_{[22,22x2]}$ to
$\hat{i}_{[22,22x3]}$ does not allow us to recover the three ‘upstream’
integrals $\hat{i}_{[22,33x3]}$, $\hat{i}_{[33,33x3]}$ and
$\hat{i}_{[32,33x3]}$. We will therefore follow a different route, deriving a
new shift operator based on the properties of Bessel functions that will allow
us to recover $\hat{i}_{[32,32x2]}$, and hence all the $\Delta_{x}=2$
integrals, starting from the $\Delta_{x}=3$ integrals.

In the analysis to follow, it will be clearer to write the AdS propagators as
functions of the spacetime dimension $d$ and the parameter
$\beta_{i}=\Delta_{i}-d/2$. To this end, we re-define

$\displaystyle\tilde{\mathcal{K}}_{d,\beta_{i}}(z,k_{i})$
$\displaystyle=\frac{z^{d/2}k_{i}^{\beta}K_{\beta_{i}}(k_{i}z)}{2^{\beta_{i}-1}\Gamma(\beta_{i})},$
(442) $\displaystyle\tilde{\mathcal{G}}_{d,\beta_{x}}(z,s;\zeta)$
$\displaystyle=(z\zeta)^{d/2}\Big{(}K_{\beta_{x}}(sz)I_{\beta_{x}}(s\zeta)\Theta(z-\zeta)+I_{\beta_{x}}(sz)K_{\beta_{x}}(s\zeta)\Theta(\zeta-z)\Big{)}$
(443)

where the tildes remind us of the change of argument from $(d,\Delta_{i})$ to
$(d,\beta_{i})$. From the elementary properties of Bessel functions, the bulk-
boundary propagator obeys the relation

$\tilde{\mathcal{K}}_{d-2,\,\frac{1}{2}}(z,k_{i})=\tilde{\mathcal{K}}_{d,\,\frac{3}{2}}(z,k_{i})-k_{i}\tilde{\mathcal{K}}_{d,\,\frac{1}{2}}(z,k_{i}).$
(444)

Writing the exchange diagram as

$\displaystyle\mathcal{I}^{(d)}_{[\beta_{1},\beta_{2};\beta_{3},\beta_{4};x\beta_{x}]}$
$\displaystyle\quad=\int_{0}^{\infty}\frac{\mathrm{d}z}{z^{d+1}}\int_{0}^{\infty}\frac{\mathrm{d}\zeta}{\zeta^{d+1}}\tilde{\mathcal{K}}_{d,\beta_{1}}(z,k_{1})\tilde{\mathcal{K}}_{d,\beta_{2}}(z,k_{2})\tilde{G}_{d,\beta_{x}}(z,s;\zeta)\tilde{\mathcal{K}}_{d,\beta_{3}}(\zeta,k_{3})\tilde{\mathcal{K}}_{d,\beta_{4}}(\zeta,k_{4}),$
(445)

and using this relation twice, we then find that

$\mathcal{I}^{(d-2)}_{[\frac{3}{2},\frac{1}{2};\frac{3}{2},\frac{1}{2};\,x\frac{3}{2}]}=\mathcal{I}^{(d)}_{[\frac{3}{2},\frac{3}{2};\frac{3}{2},\frac{3}{2};\,x\frac{3}{2}]}-k_{2}\mathcal{I}^{(d)}_{[\frac{3}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\,x\frac{3}{2}]}-k_{4}\mathcal{I}^{(d)}_{[\frac{3}{2},\frac{3}{2};\frac{3}{2},\frac{1}{2};\,x\frac{3}{2}]}+k_{2}k_{4}\mathcal{I}^{(d)}_{[\frac{3}{2},\frac{1}{2};\frac{3}{2},\frac{1}{2};\,x\frac{3}{2}]}.$
(446)

Here, we exploited the fact that the factor of $(z\zeta)^{-d}$ from the two
integration measures cancels with the factors of $z^{d/2}$ and $\zeta^{d/2}$
from the first and third bulk-boundary propagators and the factor of
$(z\zeta)^{d/2}$ from the bulk-bulk propagator. A shift $d\rightarrow d-2$ on
the second and fourth bulk-boundary propagators thus extends to the entire
exchange diagram.

Meanwhile, the bulk-bulk propagator obeys the identity

$\frac{1}{2}\Big{(}z^{2}+\zeta^{2}-\partial_{s}^{2}+(2\sigma_{x}\beta_{x}-1)\frac{1}{s}\partial_{s}\Big{)}\tilde{\mathcal{G}}_{d-2,\beta_{x}}(z,s;\zeta)=\tilde{\mathcal{G}}_{d,\beta_{x}+\sigma_{x}}(z,s;\zeta),\quad\sigma_{x}=\pm
1,$ (447)

which shifts the spacetime dimension up by two while increasing or decreasing
$\beta_{x}$ by one. To apply this operator to $s$-channel exchange diagrams,
we further replace the factors of $z^{2}$ and $\zeta^{2}$ with Bessel
operators acting on the external legs,

$\Big{(}\partial_{i}^{2}+\frac{1-2\beta_{i}}{k_{i}}\partial_{i}\Big{)}\tilde{\mathcal{K}}_{d,\beta_{i}}(z,k_{i})=z^{2}\tilde{\mathcal{K}}_{d,\beta_{i}}(z,k_{i}).$
(448)

So long as we choose one external leg corresponding to each vertex the result
is the same. Choosing legs 2 and 4, this gives

$\displaystyle\mathcal{I}^{(d)}_{[\beta_{1},\beta_{2};\beta_{3},\beta_{4};x\beta_{x}+\sigma_{x}]}$
$\displaystyle\quad=\frac{1}{2}\Big{(}\partial_{2}^{2}+\frac{1-2\beta_{2}}{k_{2}}\partial_{2}+\partial_{4}^{2}+\frac{1-2\beta_{4}}{k_{4}}\partial_{4}-\partial_{s}^{2}+(2\sigma_{x}\beta_{x}-1)\frac{1}{s}\partial_{s}\Big{)}\mathcal{I}^{(d-2)}_{[\beta_{1},\beta_{2};\beta_{3},\beta_{4};x\beta_{x}]}$
(449)

Thus, applying this operator with $\sigma_{x}=-1$ to the left-hand side of
(446), we find

$\mathcal{I}^{(d)}_{[\frac{3}{2},\frac{1}{2};\frac{3}{2},\frac{1}{2};\,x\frac{1}{2}]}=\frac{1}{2}\Big{(}\partial_{2}^{2}+\partial_{4}^{2}-\partial_{s}^{2}-\frac{4}{s}\partial_{s}\Big{)}\mathcal{I}^{(d-2)}_{[\frac{3}{2},\frac{1}{2};\frac{3}{2},\frac{1}{2};\,x\frac{3}{2}]}.$
(450)

Using (446) and (450), we can now obtain the $\Delta_{x}=2$ master integral
$\hat{i}_{[32,32x2]}$ starting from the $\Delta_{x}=3$ master integral
$\hat{i}_{[33,33x3]}$. In the first step, we compute all the $\Delta_{x}=3$
integrals using the reduction scheme illustrated in Figure 6. The right-hand
side of (446) can then be constructed in the half-integer scheme:

$\displaystyle\mathcal{I}^{(1+2\epsilon)}_{[\frac{3}{2},\frac{1}{2};\frac{3}{2},\frac{1}{2};\,x\frac{3}{2}]}$
$\displaystyle=\hat{i}_{[33,33x3]}-k_{2}\hat{i}_{[32,33x3]}-k_{4}\hat{i}_{[33,32x3]}+k_{2}k_{4}\hat{i}_{[32,32x3]}.$
(451)

Equation (450) now gives the desired result

$\displaystyle\hat{i}_{[32,32x2]}=\frac{1}{2}\Big{(}\partial_{2}^{2}+\partial_{4}^{2}-\partial_{s}^{2}-\frac{4}{s}\partial_{s}\Big{)}\Big{(}\hat{i}_{[33,33x3]}-k_{2}\hat{i}_{[32,33x3]}-k_{4}\hat{i}_{[33,32x3]}+k_{2}k_{4}\hat{i}_{[32,32x3]}\Big{)}.$
(452)

Using this relation in combination with the reduction schemes in Figures 5 and
6 we can thus obtain all exchange diagrams starting from only the exchange
diagram $\hat{i}_{[33,33x3]}$ and contact diagrams. As the most complicated
integral, $\hat{i}_{[33,33x3]}$ is the natural starting point since the action
of differential operators results in a reduction in complexity (i.e., a
lowering of the degree of transcendality and/or a reduction in the order of
polynomial coefficients).

##  Mathematica notebooks

All our results and their derivations can be found in the Mathematica
notebooks included in the arXiv submission of this paper.333If any
discrepancies exist as a result of typos, the expressions in these notebooks
are definitive. Explicit expressions for the regulated and renormalized
amplitudes are stored in the Mathematica package `HandbooK.wl`, whose contents
are described in Section 7.1 below. The package is accompanied by a number of
notebooks where details of the remaining calculations described in this paper
can be found. We summarize the contents of these notebooks in Section 7.2.

###  The package

The package `HandbooK.wl` provides explicit expressions for both regulated and
renormalized 2-, 3-, and 4-point functions.

  * •

The package does not contain a dedicated installer and is loaded instead
through Mathematica’s `Get` command:

  * •

All the regulated and renormalized expressions listed in the previous sections
can be accessed through the following commands:

Amplitude | Regulated | Renormalized
---|---|---
$i_{[\Delta_{1}\Delta_{2}]}$ |  `iReg2pt[`$\hat{d},\\{\hat{\Delta}_{1},\hat{\Delta}_{2}\\}$`]` |  `iRen2pt[`$d,\\{\Delta_{1},\Delta_{2}\\}$`]`
$i_{[\Delta_{1}\Delta_{2}\Delta_{3}]}$ |  `iReg3pt[`$\hat{d},\\{\hat{\Delta}_{1},\hat{\Delta}_{2},\hat{\Delta}_{3}\\}$`]` |  `iRen3pt[`$d,\\{\Delta_{1},\Delta_{2},\Delta_{3}\\}$`]`
$j_{[\Delta_{1}\Delta_{2};\Delta_{3}]}$ |  `jReg3pt[`$\hat{d},\\{\hat{\Delta}_{1},\hat{\Delta}_{2},\hat{\Delta}_{3}\\}$`]` | —
$i_{[\Delta_{1}\Delta_{2}\Delta_{3}\Delta_{4}]}$ |  `iReg4ptC[`$\hat{d},\\{\hat{\Delta}_{1},\hat{\Delta}_{2},\hat{\Delta}_{3},\hat{\Delta}_{4}\\}$`]` |  `iRen4ptC[`$d,\\{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\\}$`]`