$\displaystyle=-p_{1}^{4}-p_{2}^{4}-p_{3}^{4}+2p_{1}^{2}p_{2}^{2}+2p_{2}^{2}p_{3}^{2}+2p_{3}^{2}p_{1}^{2}.$
(241)

###  5.5 Shift operators preserving the spacetime dimension

The creation operators constructed above decrease the spacetime dimension
according to (230). For many applications, we would prefer an operator capable
of changing the scaling dimensions of a contact diagram while preserving the
spacetime dimension. Thus, we seek an operator
$W_{12}^{\sigma_{1},\sigma_{2}}$ such that

$W_{12}^{\sigma_{1},\sigma_{2}}i_{[d;\,\Delta_{1},\Delta_{2},\,\Delta_{3},\,\ldots,\,\Delta_{n}]}\propto\,i_{[d;\,\Delta_{1}+\sigma_{1},\,\Delta_{2}+\sigma_{2},\,\Delta_{3},\,\ldots,\,\Delta_{n}]}$
(242)

for any independent choice of signs $\sigma_{1}=\pm 1$ and $\sigma_{2}=\pm 1$.
Operators of this type are known at three points Karateev:2017jgd ;
Baumann:2019oyu , but their analogue at four points acts on contact diagrams
to generate shifted contact diagrams with derivative vertices Bzowski:2022rlz
. Instead, our discussion of creation operators above can be modified to
enable operators with the action (242) to be identified.131313The shift
operators that we identify will moreover be of minimal order, unlike the
$d$-preserving combination of an annihilator $\partial_{i}$ or
$\bar{\partial}_{i}$ followed by a creation operator $\mathcal{C}_{j}$ or
$\bar{\mathcal{C}}_{j}$. For example, the combination
$\bar{\mathcal{C}}_{1}\partial_{2}-\bar{\mathcal{C}}_{2}\partial_{1}$ produces
the same shift as $W_{12}^{--}$ but is of seventh order in derivatives for the
4-point function, since each product is eighth order and taking the difference
lowers the order by one. In contrast, the 4-point operator $W_{12}^{--}$ we
find will be of only fourth order. At three points, these will coincide with
the operators of Karateev:2017jgd ; Baumann:2019oyu , but at four points and
higher they are novel. Using these operators we will then construct new shift
operators for exchange diagrams.

Our starting point is the observation that, for the GKZ integral (163)
corresponding to the contact diagram,

$W_{12}^{--}\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}=b_{W}(\bm{\gamma})\partial_{2}\mathcal{I}_{\bm{\gamma}}.$
(243)

Recalling the parameter identifications (165), the action of the operators
here is

$\displaystyle W_{12}^{--}:$ $\displaystyle\gamma_{0}$
$\displaystyle\rightarrow\gamma_{0},$ $\displaystyle\gamma_{1}$
$\displaystyle\rightarrow\gamma_{1}-1,$ $\displaystyle\gamma_{2}$
$\displaystyle\rightarrow\gamma_{2}-1,$ $\displaystyle\bar{\partial}_{1}:$
$\displaystyle\gamma_{0}$ $\displaystyle\rightarrow\gamma_{0}+1,$
$\displaystyle\gamma_{1}$ $\displaystyle\rightarrow\gamma_{1}+1,$
$\displaystyle\gamma_{2}$ $\displaystyle\rightarrow\gamma_{2},$
$\displaystyle\partial_{2}:$ $\displaystyle\gamma_{0}$
$\displaystyle\rightarrow\gamma_{0}+1,$ $\displaystyle\gamma_{1}$
$\displaystyle\rightarrow\gamma_{1},$ $\displaystyle\gamma_{2}$
$\displaystyle\rightarrow\gamma_{2}-1,$ (244)

with all remaining $\gamma_{k}$ for $k=3,\ldots,n$ staying the same. As the
shifts produced by the operators on each side of (243) are the same, both
sides involve the same integral $\mathcal{I}_{\bm{\gamma}}$. As previously,
the $b$-function $b_{W}(\bm{\gamma})$ should be a product of linear factors
that vanishes whenever $W_{12}^{--}$ maps us from a finite to a singular
integral. Taking into account the action (176) of the annihilators in (243),
we have

$\displaystyle W_{12}^{--}\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}$
$\displaystyle=-\gamma_{0}W_{12}^{--}\mathcal{I}_{\bm{\gamma}}\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}+1,\,\gamma_{1}\rightarrow\gamma_{1}+1}$
$\displaystyle=-\gamma_{0}b_{W}(\bm{\gamma})\mathcal{I}_{\bm{\gamma}}\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}+1,\gamma_{2}\rightarrow\gamma_{2}-1}=b_{W}(\bm{\gamma})\partial_{2}\mathcal{I}_{\bm{\gamma}}$
(245)

and so the zeros of $b_{W}(\bm{\gamma})$ must cancel the singularities of
$\mathcal{I}_{\bm{\gamma}}|_{\gamma_{0}\rightarrow\gamma_{0}+1,\gamma_{2}\rightarrow\gamma_{2}-1}$.
From (181), this means

$\displaystyle b_{W}(\bm{\gamma})$ $\displaystyle=\prod_{\sigma_{k}\in\pm
1}\frac{1}{2}\Big{(}-(\gamma_{0}+1)-\gamma_{1}-(\gamma_{2}-1)+\sigma_{3}\gamma_{3}+\ldots\sigma_{n}\gamma_{n}\Big{)}$
$\displaystyle=\prod_{\sigma_{k}\in\pm
1}\frac{1}{2}\Big{(}-\gamma_{0}-\gamma_{1}-\gamma_{2}+\sigma_{3}\gamma_{3}+\ldots\sigma_{n}\gamma_{n}\Big{)}.$
(246)

Only the singularities with $\sigma_{1}=\sigma_{2}=-1$ in (181) appear here
since these are the only cases for which
$\mathcal{I}_{\bm{\gamma}}|_{\gamma_{0}\rightarrow\gamma_{0}+1,\gamma_{2}\rightarrow\gamma_{2}-1}$
is singular but the integral
$\mathcal{I}_{\bm{\gamma}}|_{\gamma_{0}\rightarrow\gamma_{0}+1,\gamma_{1}\rightarrow\gamma_{1}+1}$
on which $W_{12}^{--}$ acts is finite. Every possible independent choice of
$\sigma_{k}\in\pm 1$ for all $k=3,\ldots,n$ is permitted, however, and gives
rise to a corresponding factor in (5.5). Once again, we have also chosen to
include trivial factors of one-half in $b_{W}(\bm{\gamma})$ to simplify the
subsequent form of $W_{12}^{--}$. Replacing the parameters $\bm{\gamma}$ in
$b_{W}(\bm{\gamma})$ using the Euler equations (169) and DWI (170), we find

$\displaystyle
W_{12}^{--}\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}=\partial_{2}\big{(}b_{W}(\bm{\gamma})\mathcal{I}_{\bm{\gamma}}\big{)}=\partial_{2}B_{W}(\theta,\bar{\theta})\mathcal{I}_{\bm{\gamma}}$
(247)

where

$\displaystyle B_{W}(\theta,\bar{\theta})$
$\displaystyle=\prod_{\sigma_{k}\in\pm
1}\frac{1}{2}\Big{(}\sum_{j=1}^{n}(\theta_{j}+\bar{\theta}_{j})-(\theta_{1}-\bar{\theta}_{1})-(\theta_{2}-\bar{\theta}_{2})+\sigma_{3}(\theta_{3}-\bar{\theta}_{3})+\ldots\sigma_{n}(\theta_{n}-\bar{\theta}_{n})\Big{)}$
$\displaystyle=\prod_{\sigma_{k}\in\pm
1}\Big{(}\bar{\theta}_{1}+\bar{\theta}_{2}+(\delta_{\sigma_{3},+1}\theta_{3}+\delta_{\sigma_{3},-1}\bar{\theta}_{3})+\ldots+(\delta_{\sigma_{n},+1}\theta_{n}+\delta_{\sigma_{n},-1}\bar{\theta}_{n})\Big{)}.$
(248)

Since $B_{W}(\theta,\bar{\theta})$ is in fact independent of $\theta_{2}$ the
ordering of $\partial_{2}$ and $B_{W}(\theta,\bar{\theta})$ on the right-hand
side of (247) is in fact immaterial, but had this not been the case the
ordering shown would be the correct one when using the unshifted Euler
equations and DWI to replace the $\bm{\gamma}$ parameters.

To identify $W_{12}^{--}$, all that is needed now is to start with
$\partial_{2}B_{W}(\theta,\bar{\theta})$ and, using the toric equations (171),
pull out a right factor of $\bar{\partial}_{1}$ according to (247). As usual,
the resulting operator can then be projected down to the physical hypersurface
using the Euler equations and DWI. These procedures are illustrated for the 3-
and 4-point function below. Finally, given $W_{12}^{--}$ in physical
variables, all the remaining operators in (242) can be found by shadow
conjugation using (175), namely

$\displaystyle(W_{12}^{+-})_{\mathrm{ph}}$
$\displaystyle=x_{1}^{1+\gamma_{1}}(W_{12}^{--})_{\mathrm{ph}}\,x_{1}^{-\gamma_{1}},$
(249) $\displaystyle(W_{12}^{-+})_{\mathrm{ph}}\,$
$\displaystyle=x_{2}^{1+\gamma_{2}}(W_{12}^{--})_{\mathrm{ph}}\,x_{2}^{-\gamma_{2}},$
(250) $\displaystyle(W_{12}^{++})_{\mathrm{ph}}\,$
$\displaystyle=x_{1}^{1+\gamma_{1}}x_{2}^{1+\gamma_{2}}(W_{12}^{--})_{\mathrm{ph}}\,x_{1}^{-\gamma_{1}}x_{2}^{-\gamma_{2}}.$
(251)

####  5.5.1 3-point function

To illustrate the above discussion, for the 3-point function we have

$b_{W}(\bm{\gamma})=\frac{1}{4}(-\gamma_{0}-\gamma_{1}-\gamma_{2}+\gamma_{3})(-\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3})$
(252)

and

$B_{W}(\theta,\bar{\theta})=(\bar{\theta}_{1}+\bar{\theta}_{2}+\theta_{3})(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}).$
(253)

The operator $W_{12}^{--}$ can now be extracted from

$W_{12}^{--}\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}=\partial_{2}B_{W}(\theta,\bar{\theta})\mathcal{I}_{\bm{\gamma}}.$
(254)

For this, we write

$\displaystyle\partial_{2}(\bar{\theta}_{1}+\bar{\theta}_{2}+\theta_{3})(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3})\mathcal{I}_{\bm{\gamma}}$
$\displaystyle\qquad=\partial_{2}\big{[}(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\theta_{3})(\bar{\theta}_{1}+\bar{\theta}_{2})+\theta_{3}\bar{\theta}_{3}]\mathcal{I}_{\bm{\gamma}}$
$\displaystyle\qquad=\big{[}(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\theta_{3})(\bar{x}_{1}\partial_{2}\bar{\partial}_{1}+\bar{x}_{2}\partial_{2}\bar{\partial}_{2})+x_{3}\bar{x}_{3}\partial_{2}\partial_{3}\bar{\partial}_{3}]\mathcal{I}_{\bm{\gamma}}$
$\displaystyle\qquad=\big{[}(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\theta_{3})(\bar{x}_{1}\partial_{2}+\bar{x}_{2}\partial_{1})+x_{3}\bar{x}_{3}\partial_{2}\partial_{1}]\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}$
(255)

where in the penultimate line we used the toric equations (171). Thus

$\displaystyle W_{12}^{--}$
$\displaystyle=(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\theta_{3})(\bar{x}_{1}\partial_{2}+\bar{x}_{2}\partial_{1})+x_{3}\bar{x}_{3}\partial_{2}\partial_{1}$
$\displaystyle=(\bar{x}_{1}\partial_{2}+\bar{x}_{2}\partial_{1})(1+\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\theta_{3})+x_{3}\bar{x}_{3}\partial_{2}\partial_{1},$
(256)

and using the DWI (170) to project to the physical hypersurface (166), we
obtain

$\displaystyle(W_{12}^{--})_{\mathrm{ph}}$
$\displaystyle=(\partial_{2}+\partial_{1})(1-\gamma_{0}-\theta_{1}-\theta_{2})+x_{3}\partial_{2}\partial_{1}$
$\displaystyle=-(\gamma_{0}+\theta_{1}+\theta_{2})(\partial_{1}+\partial_{2})+x_{3}\partial_{1}\partial_{2}$
(257)

where for the 3-point function $\gamma_{0}=d/2$ from (165). A short
calculation shows that

$(W_{12}^{--})_{\mathrm{ph}}=-\frac{1}{4}\Big{(}\partial_{p_{1}}^{2}+\partial_{p_{2}}^{2}+\frac{(d-1)}{p_{1}}\partial_{p_{1}}+\frac{(d-1)}{p_{2}}\partial_{p_{2}}+(p_{1}^{2}+p_{2}^{2}-p_{3}^{2})\frac{1}{p_{1}p_{2}}\partial_{p_{1}}\partial_{p_{2}}\Big{)}$
(258)

which, up to a factor of $-2$, is the 3-point shift operator studied in
Baumann:2019oyu ; Bzowski:2022rlz .

The action of $W_{12}^{--}$ is

$\displaystyle
W_{12}^{--}\mathcal{I}_{\bm{\gamma}}=b_{W}(\bm{\gamma})\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}-1,\,\gamma_{1}\rightarrow\gamma_{1}-1}\mathcal{I}_{\bm{\gamma}}\Big{|}_{\gamma_{1}\rightarrow\gamma_{1}-1,\,\gamma_{2}\rightarrow\gamma_{2}-1}$
(259)

where the shift on the $b$-function derives from the fact that, in the
projection step going from (5.5.1) to (5.5.1), we have chosen that
$W_{12}^{--}$ acts on the integral $\mathcal{I}_{\bm{\gamma}}$ requiring us to
shift the $\bm{\gamma}$ parameters present in (5.5). Evaluating, this gives

$\displaystyle
b_{W}(\bm{\gamma})\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}-1,\,\gamma_{1}\rightarrow\gamma_{1}-1}$
$\displaystyle=\frac{1}{4}(2-\gamma_{0}-\gamma_{1}-\gamma_{2}+\gamma_{3})(2-\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3})$
$\displaystyle=\frac{1}{4}\Big{(}(\gamma_{0}+\gamma_{1}+\gamma_{2}-2)^{2}-\gamma_{3}^{2}\Big{)}$
(260)

such that (259) is consistent with the action of $\mathcal{W}_{12}^{--}$
obtained in Bzowski:2022rlz . Acting on the holographic contact diagram, from
(164) we have

$W_{12}^{--}i_{[d,\,\Delta_{1},\,\Delta_{2},\,\Delta_{3}]}=\frac{1}{4(\gamma_{1}-1)(\gamma_{2}-1)}b_{W}(\bm{\gamma})\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}-1,\,\gamma_{1}\rightarrow\gamma_{1}-1}i_{[d,\,\Delta_{1}-1,\,\Delta_{2}-1,\,\Delta_{3}]}.$
(261)

####  5.5.2 4-point function

At 4-points, we find

$\displaystyle b_{W}(\bm{\gamma})$
$\displaystyle=\frac{1}{16}(-\gamma_{0}-\gamma_{1}-\gamma_{2}+\gamma_{3}+\gamma_{4})(-\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3}+\gamma_{4})$
$\displaystyle\qquad\times(-\gamma_{0}-\gamma_{1}-\gamma_{2}+\gamma_{3}-\gamma_{4})(-\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3}-\gamma_{4})$
(262)

and hence

$\displaystyle B_{W}(\theta,\bar{\theta})$
$\displaystyle=(\bar{\theta}_{1}+\bar{\theta}_{2}+\theta_{3}+\theta_{4})(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\theta_{4})(\bar{\theta}_{1}+\bar{\theta}_{2}+\theta_{3}+\bar{\theta}_{4})(\bar{\theta}_{1}+\bar{\theta}_{2}+\bar{\theta}_{3}+\bar{\theta}_{4}).$
(263)

Once again, to find $W_{12}^{--}$ we must factorise

$W_{12}^{--}\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}=\partial_{2}B_{W}(\theta,\bar{\theta})\mathcal{I}_{\bm{\gamma}}.$
(264)

As a first step, we expand

$B_{W}(\theta,\bar{\theta})=Q_{0}(\bar{\theta}_{1}+\bar{\theta}_{2})+Q_{3}\theta_{3}\bar{\theta}_{3}+Q_{4}\theta_{4}\bar{\theta}_{4}$
(265)

where the coefficients

$\displaystyle Q_{0}$
$\displaystyle=(u_{3}+u_{4}+\bar{\theta}_{1}+\bar{\theta}_{2})\Big{(}2(v_{3}+v_{4})+(u_{3}+\bar{\theta}_{1}+\bar{\theta}_{2})(u_{4}+\bar{\theta}_{1}+\bar{\theta}_{2})\Big{)},$
$\displaystyle Q_{3}$ $\displaystyle=(u_{3}+u_{4})u_{4}+v_{3}-v_{4},$
$\displaystyle Q_{4}$ $\displaystyle=(u_{3}+u_{4})u_{3}-v_{3}+v_{4},$ (266)

and

$u_{k}=\theta_{k}+\bar{\theta}_{k},\qquad
v_{k}=\theta_{k}\bar{\theta}_{k},\qquad k=3,4.$ (267)

Now, since all coefficients are independent of $\theta_{2}$,

$\displaystyle\partial_{2}B_{W}(\theta,\bar{\theta})\mathcal{I}_{\bm{\gamma}}$
$\displaystyle=\Big{[}Q_{0}(\bar{x}_{1}\partial_{2}\bar{\partial}_{1}+\bar{x}_{2}\partial_{2}\bar{\partial_{2}})+Q_{3}x_{3}\bar{x}_{3}\partial_{2}\bar{\partial}_{3}\partial_{3}+Q_{4}x_{4}\bar{x}_{4}\partial_{2}\bar{\partial}_{4}\partial_{4}\Big{]}\mathcal{I}_{\bm{\gamma}}$
(268)
$\displaystyle=\Big{[}\bar{x}_{1}Q_{0}\Big{|}_{\bar{\theta}_{1}\rightarrow\bar{\theta}_{1}+1}\partial_{2}+\bar{x}_{2}Q_{0}\Big{|}_{\bar{\theta}_{2}\rightarrow\bar{\theta}_{2}+1}\partial_{1}$
$\displaystyle\qquad+x_{3}\bar{x}_{3}Q_{3}\Big{|}_{\theta_{3}\rightarrow\theta_{3}+1,\,\bar{\theta}_{3}\rightarrow\bar{\theta}_{3}+1}\partial_{2}\partial_{1}+x_{4}\bar{x}_{4}Q_{4}\Big{|}_{\theta_{4}\rightarrow\theta_{4}+1,\,\bar{\theta}_{4}\rightarrow\bar{\theta}_{4}+1}\partial_{2}\partial_{1}\Big{]}\bar{\partial}_{1}\mathcal{I}_{\bm{\gamma}}$

where in the second line we used the toric equations (171). We thus have

$\displaystyle W_{12}^{--}$
$\displaystyle=(\bar{x}_{1}\partial_{2}+\bar{x}_{2}\partial_{1})Q_{0}\Big{|}_{\bar{\theta}_{1}\rightarrow\bar{\theta}_{1}+1}$
$\displaystyle\qquad+\partial_{1}\partial_{2}\Big{(}x_{3}\bar{x}_{3}Q_{3}\Big{|}_{\theta_{3}\rightarrow\theta_{3}+1,\,\bar{\theta}_{3}\rightarrow\bar{\theta}_{3}+1}+x_{4}\bar{x}_{4}Q_{4}\Big{|}_{\theta_{4}\rightarrow\theta_{4}+1,\,\bar{\theta}_{4}\rightarrow\bar{\theta}_{4}+1}\Big{)}$
(269)

where in the first line we used the fact that $\bar{\theta}_{1}$ and
$\bar{\theta}_{2}$ enter $Q_{0}$ only in the combination
$\bar{\theta}_{1}+\bar{\theta}_{2}$ and so the replacement
$\bar{\theta}_{1}\rightarrow\bar{\theta}_{1}+1$ produces the same result as
$\bar{\theta}_{2}\rightarrow\bar{\theta}_{2}+1$ allowing us to combine the two
$Q_{0}$ terms. We have in addition moved $Q_{0}$, $Q_{3}$ and $Q_{4}$ to the
right (noting that all coefficients are independent of $\theta_{1}$ and
$\theta_{2}$) so as to be able to use the Euler equations for
$\mathcal{I}_{\bm{\gamma}}$ to project to the physical hypersurface. For this,
we set $\bar{x}_{k}\rightarrow 1$ and
$\bar{\theta}_{k}\rightarrow\theta_{k}-\gamma_{k}$ inside all $Q_{k}$
coefficients giving

$\displaystyle(W_{12}^{--})_{\mathrm{ph}}$
$\displaystyle=(\partial_{1}+\partial_{2})Q_{0}\Big{|}_{\bar{\theta}_{k}\rightarrow\theta_{k}-\gamma_{k}+\delta_{k,1}}$
(270)
$\displaystyle\quad+\partial_{1}\partial_{2}\Big{(}x_{3}Q_{3}\Big{|}_{\theta_{3}\rightarrow\theta_{3}+1,\,\bar{\theta}_{k}\rightarrow\theta_{k}-\gamma_{k}+\delta_{k,3}}+x_{4}Q_{4}\Big{|}_{\theta_{4}\rightarrow\theta_{4}+1,\,\bar{\theta}_{k}\rightarrow\theta_{k}-\gamma_{k}+\delta_{k,4}}\Big{)}.$

In all the replacements here, $\bar{\theta}_{k}$ stands for any index
$k=1,\ldots,4$. Evaluating this formula explicitly using the coefficients in
(5.5.2), we find

$\displaystyle(W_{12}^{--})_{\mathrm{ph}}=(\partial_{1}+\partial_{2})(1-\gamma_{0}-\theta_{1}-\theta_{2})\Big{(}2\theta_{3}(\theta_{3}-\gamma_{3})+2\theta_{4}(\theta_{4}-\gamma_{4})$
(271)
$\displaystyle\qquad+(1+2\theta_{3}-\gamma_{3}+\theta_{1}-\gamma_{1}+\theta_{2}-\gamma_{2})(1+2\theta_{4}-\gamma_{4}+\theta_{1}-\gamma_{1}+\theta_{2}-\gamma_{2})\Big{)}$
$\displaystyle\qquad+\partial_{1}\partial_{2}\Big{[}x_{3}\Big{(}(2+2\theta_{3}-\gamma_{3}+2\theta_{4}-\gamma_{4})(2\theta_{4}-\gamma_{4})+(1+\theta_{3})(1+\theta_{3}-\gamma_{3})-\theta_{4}(\theta_{4}-\gamma_{4})\Big{)}$
$\displaystyle\qquad+x_{4}\Big{(}(2+2\theta_{3}-\gamma_{3}+2\theta_{4}-\gamma_{4})(2\theta_{3}-\gamma_{3})+(1+\theta_{4})(1+\theta_{4}-\gamma_{4})-\theta_{3}(\theta_{3}-\gamma_{3})\Big{)}\Big{]}$

where for the 4-point function $\gamma_{0}=d$ from (165).

Alternatively, we can use the DWI (170) projected to the physical
hypersurface,

$0=\Big{(}\frac{1}{2}(\gamma_{0}-\gamma_{t})+\sum_{k=1}^{4}\theta_{k}\Big{)}\mathcal{I}_{\bm{\gamma}},\qquad\gamma_{t}=\sum_{k=1}^{4}\gamma_{k},$
(272)

to eliminate the factors of $\theta_{1}+\theta_{2}$ on the second line of
(271). After further moving all factors of $\partial_{1}$ and $\partial_{2}$
to the right, this gives the equivalent form

$\displaystyle(W_{12}^{--})_{\mathrm{ph}}$
$\displaystyle=-(\gamma_{0}+\theta_{1}+\theta_{2})\Big{(}(\theta_{3}+\theta_{4})(\theta_{3}+\theta_{4}-\gamma_{3}-\gamma_{4})$
(273)
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{1}{4}(2-\gamma_{0}-\gamma_{t}+2\gamma_{3})(2-\gamma_{0}-\gamma_{t}+2\gamma_{4})\Big{)}(\partial_{1}+\partial_{2})$
$\displaystyle+\Big{[}x_{3}\Big{(}(2+2\theta_{3}-\gamma_{3}+2\theta_{4}-\gamma_{4})(2\theta_{4}-\gamma_{4})+(1+\theta_{3})(1+\theta_{3}-\gamma_{3})-\theta_{4}(\theta_{4}-\gamma_{4})\Big{)}$
$\displaystyle+x_{4}\Big{(}(2+2\theta_{3}-\gamma_{3}+2\theta_{4}-\gamma_{4})(2\theta_{3}-\gamma_{3})+(1+\theta_{4})(1+\theta_{4}-\gamma_{4})-\theta_{3}(\theta_{3}-\gamma_{3})\Big{)}\Big{]}\partial_{1}\partial_{2}.$

The action of $W_{12}^{--}$ is

$\displaystyle
W_{12}^{--}\mathcal{I}_{\bm{\gamma}}=b_{W}(\bm{\gamma})\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}-1,\,\gamma_{1}\rightarrow\gamma_{1}-1}\mathcal{I}_{\bm{\gamma}}\Big{|}_{\gamma_{1}\rightarrow\gamma_{1}-1,\,\gamma_{2}\rightarrow\gamma_{2}-1}$
(274)

where, once again, the shift on the $b$-function derives from the fact that in
projecting from GKZ variables to the physical hypersurface we chose
$W_{12}^{--}$ to act on the unshifted integral $\mathcal{I}_{\bm{\gamma}}$
requiring us to shift the $\bm{\gamma}$ parameters present in (5.5).
Explicitly, this is

$\displaystyle
b_{W}(\bm{\gamma})\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}-1,\,\gamma_{1}\rightarrow\gamma_{1}-1}$
$\displaystyle=\frac{1}{16}\Big{(}\gamma_{3}^{2}+\gamma_{4}^{2}-(\gamma_{0}+\gamma_{1}+\gamma_{2}-2)^{2}\Big{)}^{2}-\frac{1}{4}\gamma_{3}^{2}\gamma_{4}^{2}.$
(275)

Acting on the holographic contact diagram, from (164) we again have

$\displaystyle
W_{12}^{--}i_{[d;\,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]}=\frac{1}{4(\gamma_{1}-1)(\gamma_{2}-1)}b_{W}(\bm{\gamma})\Big{|}_{\gamma_{0}\rightarrow\gamma_{0}-1,\,\gamma_{1}\rightarrow\gamma_{1}-1}i_{[d;\,\Delta_{1}-1,\Delta_{2}-1,\Delta_{3},\Delta_{4}]}.$
(276)

To our knowledge, this is the first time an operator that shifts the 4-point
contact diagram in this fashion has been identified. We emphasise that the
3-point operator (258), when applied to 4-point contact diagrams, generates
shifted contact diagrams but with derivative vertices and hence does not
satisfy this requirement Bzowski:2022rlz .

##### Examples:

Contact diagrams for which the Bessel functions have half-integer indices can
be evaluated directly. This yields many simple examples for which the action
of $W_{12}^{--}$ can be checked. For instance, with
$(d,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4})=(5,3,4,3,4)$, we find

$\displaystyle i_{[5;3,4,3,4]}$
$\displaystyle=\frac{1}{p_{t}^{3}}\Big{(}p_{1}^{2}+p_{3}^{2}+2(p_{2}^{2}+p_{4}^{2})+3(p_{1}+p_{3})(p_{2}+p_{4})+2p_{1}p_{3}+6p_{2}p_{4}\Big{)}$
(277)

where $p_{t}=\sum_{j=1}^{4}p_{j}$, while the shifted integral with
$(d,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4})=(5,2,3,3,4)$ is

$\displaystyle i_{[5;2,3,3,4]}$
$\displaystyle=-\frac{(p_{t}+2p_{4})}{p_{1}p_{t}^{3}}.$ (278)

Evaluating the action of $W_{12}^{--}$ in (273) using (195), we can verify
(276), namely

$W_{12}^{--}i_{[5;3,4,3,4]}=-\frac{63}{2}i_{[5;2,3,3,4]}.$ (279)

####  5.5.3 Combinations of operators

To round up our discussion of shift operators for contact diagrams, we have
identified operators mapping

$\displaystyle\partial_{i}:\quad\gamma_{0}\rightarrow\gamma_{0}+1,\quad\gamma_{i}\rightarrow\gamma_{i}-1\qquad\bar{\partial}_{i}:\quad\gamma_{0}\rightarrow\gamma_{0}+1,\quad\gamma_{i}\rightarrow\gamma_{i}+1$
$\displaystyle\mathcal{C}_{i}:\quad\gamma_{0}\rightarrow\gamma_{0}-1,\quad\gamma_{i}\rightarrow\gamma_{i}+1\qquad\bar{\mathcal{C}}_{i}:\quad\gamma_{0}\rightarrow\gamma_{0}-1,\quad\gamma_{i}\rightarrow\gamma_{i}-1$
$\displaystyle
W_{ij}^{\sigma_{i}\sigma_{j}}:\quad\gamma_{i}\rightarrow\gamma_{i}+\sigma_{i},\quad\gamma_{j}\rightarrow\gamma_{j}+\sigma_{j},\qquad\\{\sigma_{i},\sigma_{j}\\}\in\pm
1.$ (280)

Combining these allows us to construct still further shifts, for example:

$\displaystyle\mathcal{C}_{i}\bar{\mathcal{C}}_{i}:\quad\gamma_{0}\rightarrow\gamma_{0}-2,\qquad\mathcal{C}_{i}\bar{\partial}_{i}:\quad\gamma_{i}\rightarrow\gamma_{i}+2,\qquad\bar{\mathcal{C}}_{i}\partial_{i}:\quad\gamma_{i}\rightarrow\gamma_{i}-2.$
(281)

Acting on the 3-point function specifically,

$\mathcal{C}_{1}W_{23}^{++}:\quad\gamma_{0}\rightarrow\gamma_{0}-1,\quad\gamma_{i}\rightarrow\gamma_{i}+1\quad\forall\,\,i=1,2,3$
(282)

which is equivalent to shifting $d\rightarrow d-2$ while preserving all
operator dimensions $\Delta_{i}$.

Finally, one might wonder why all these operators produce a shift of two
units: why, for example, can one not construct an operator shifting
$\gamma_{0}\rightarrow\gamma_{0}+1$ only, or just
$\gamma_{1}\rightarrow\gamma_{1}+1$? The absence of such operators can be
traced to the spacing of the singular hyperplanes of the contact diagram,
specifically the term $-2m$ appearing in the singularity condition (181). As
$m\in\mathbb{Z}^{+}$, this means that the singularities are effectively spaced
by two units. Any operator that produced a shift of a single unit would
require a $b$-function containing an infinite number of factors, since there
are infinitely many finite integrals that are only one unit away from a
singular integral. (Namely, those for which $m$ is half-integer.) As the
number of factors in the $b$-function corresponds to the order of the
differential operator, there is thus no single-shift operator of finite order.
In contrast, for an operator shifting by two units, the number of finite
integrals that can be mapped to singular integrals is finite, and hence the
$b$-functions and shift operators are also of finite order.

###  5.6 Exchange diagrams

Having analysed contact diagrams, we now turn to the $s$-channel exchange
diagrams (159). Rather than constructing an explicit GKZ representation, here
we simply note that shifts of the form

$i_{[d;\,\Delta_{1},\Delta_{2};\,\Delta_{3},\Delta_{4};\,\Delta_{x}]}\rightarrow
i_{[d;\,\Delta_{1}+\sigma_{1},\Delta_{2}+\sigma_{2};\,\Delta_{3},\Delta_{4};\,\Delta_{x}]}$
(283)

for any $\\{\sigma_{1},\sigma_{2}\\}\in\pm 1$ can be obtained by combining the
3- and 4-point $W_{12}^{--}$ operators given in (5.5.1) and (273) with the
$s$-channel Casimir operator. As with contact diagrams, it is sufficient to
focus on the case $\sigma_{1}=\sigma_{2}=-1$, since all remaining operators
follow by shadow conjugation according to (249). We emphasise however that
both the original and the shifted exchange diagrams we consider have purely
non-derivative vertices. Moreover, any operator and spacetime dimensions are
permitted, provided we work in dimensional regularisation where necessary to
avoid divergences.

For purposes of disambiguation, let us define the operator

$\displaystyle\mathcal{W}^{--}_{12}$
$\displaystyle=(d+2\theta_{1}+2\theta_{2})(\partial_{1}+\partial_{2})-2s^{2}\partial_{1}\partial_{2}$
(284)

where $d$ is the boundary spacetime dimension and
$\partial_{i}=\partial/\partial x_{i}$ with $x_{i}=p_{i}^{2}$ as usual. This
is simply the 3-point operator $-2W_{12}^{--}$ in (258), but with $p_{3}^{2}$
replaced by the Mandelstam variable $s^{2}=(\bm{p}_{1}+\bm{p}_{2})^{2}$ as
appropriate for acting on $s$-channel exchange diagrams. (The factor of $-2$
is included for consistency with the $\mathcal{W}^{--}_{12}$ defined in
Baumann:2019oyu ; Bzowski:2022rlz .) In the following, we will then use
$W_{12}^{--}$ to refer exclusively to the 4-point $W_{12}^{--}$ operator given
in (273).

As shown in Bzowski:2022rlz , the action of $\mathcal{W}^{--}_{12}$ on an
$s$-channel exchange diagram is to produce a linear combination of a shifted
exchange and a shifted contact diagram:

$\displaystyle\mathcal{W}^{--}_{12}\,i_{[d;\,\Delta_{1},\Delta_{2};\Delta_{3},\Delta_{4};\,\Delta_{x}]}$
$\displaystyle=\mathcal{N}_{exch.}\,i_{[d;\,\Delta_{1}-1,\Delta_{2}-1;\Delta_{3},\Delta_{4};\,\Delta_{x}]}$
$\displaystyle\quad+\mathcal{N}_{cont.}\,i_{[d;\,\Delta_{1}-1,\Delta_{2}-1,\Delta_{3},\Delta_{4}]}$
(285)

where the coefficients141414 Where the shifted exchange diagram has a pole (or
double pole) in dimensional regularisation, one (or both) of the factors on
the right-hand side of (286) vanish, see Bzowski:2022rlz .

$\displaystyle\mathcal{N}_{exch.}$
$\displaystyle=\Big{(}\frac{d}{2}-2+\gamma_{1}+\gamma_{2}+\gamma_{x}\Big{)}\Big{(}\frac{d}{2}-2+\gamma_{1}+\gamma_{2}-\gamma_{x}\Big{)}\mathcal{N}_{cont.}$
(286) $\displaystyle\mathcal{N}_{cont.}$
$\displaystyle=-\frac{1}{8(\gamma_{1}-1)(\gamma_{2}-1)}$ (287)

where $\gamma_{i}=\Delta_{i}-d/2$ and $\gamma_{x}=\Delta_{x}-d/2$. Thus, in
order to go from an exchange diagram to shifted exchange diagram only, the
shifted contact contribution in (5.6) must be subtracted.

This can be accomplished in two steps. First, the unshifted contact diagram is
obtained by acting on the original exchange diagram with the reduced Casimir
operator,

$\displaystyle\hat{\mathcal{C}}_{12}\,i_{[d;\,\Delta_{1},\Delta_{2};\Delta_{3},\Delta_{4};\,\Delta_{x}]}=i_{[d;\,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]},$
(288)

where

$\displaystyle\hat{\mathcal{C}}_{12}$
$\displaystyle=2s^{2}\Big{(}(\theta_{1}+1-\gamma_{1})\partial_{1}+(\theta_{2}+1-\gamma_{2})\partial_{2}\Big{)}-\Big{(}2\theta_{1}+2\theta_{2}-\gamma_{1}-\gamma_{2}+\frac{d}{2}\Big{)}^{2}+\gamma_{x}^{2}$
(289)

with $\theta_{i}=x_{i}\partial_{i}$. The action of this operator on an
$s$-channel exchange is equivalent to that of the Casimir operator plus the
square of the exchanged mass Bzowski:2022rlz .151515Specifically,
$\hat{\mathcal{C}}_{12}=\tilde{\mathcal{C}}_{12}+m_{x}^{2}$ with
$\tilde{\mathcal{C}}_{12}$ as defined in (6.44) of Bzowski:2022rlz  and
$m_{x}^{2}=\gamma_{x}^{2}-d^{2}/4$. If desired, $\hat{\mathcal{C}}_{12}$ can
be shortened using the identity

$0=\Big{(}(\theta_{1}+1-\gamma_{1})\partial_{1}-(\theta_{2}+1-\gamma_{2})\partial_{2}\Big{)}i_{[d;\,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}\,x\,\Delta_{x}]}$
(290)

which corresponds to the difference of the Bessel operators acting on legs 1
and 2, i.e., $K_{1}-K_{2}$ where
$K_{i}=\partial_{p_{i}}^{2}+(1-2\gamma_{i})p_{i}^{-1}\partial_{p_{i}}$.
However, (289) is symmetric under $1\leftrightarrow 2$.

For the second step, we now construct the shifted contact diagram using the
4-point $W_{12}^{--}$ operator defined in (273). From (276), this has the
action

$\displaystyle
W_{12}^{--}i_{[d;\,\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}]}=\mathcal{N}_{W}\,\mathcal{N}_{cont.}\,i_{[d;\,\Delta_{1}-1,\Delta_{2}-1,\Delta_{3},\Delta_{4}]}$
(291)

with $\mathcal{N}_{cont.}$ from (287) and

$\displaystyle\mathcal{N}_{W}=-\frac{1}{8}\Big{[}\left(\gamma_{3}^{2}+\gamma_{4}^{2}-(d+\gamma_{1}+\gamma_{2}-2)^{2}\right)^{2}-4\gamma_{3}^{2}\gamma_{4}^{2}\Big{]}.$
(292)

Putting everything together, we find the operator

$\displaystyle\Omega_{12}^{--}=\mathcal{N}_{W}\,\mathcal{W}_{12}^{--}-W_{12}^{--}\hat{\mathcal{C}}_{12}$
(293)

whose action is

$\displaystyle\Omega^{--}_{12}\,i_{[d;\,\Delta_{1},\Delta_{2};\,\Delta_{3},\Delta_{4};\,\Delta_{x}]}$
$\displaystyle=\mathcal{N}_{W}\mathcal{N}_{exch.}\,i_{[d;\,\Delta_{1}-1,\Delta_{2}-1;\,\Delta_{3},\Delta_{4};\,\Delta_{x}]}.$
(294)

This is therefore the desired operator mapping an exchange to a shifted
exchange diagram.

Written out explicitly, with $\gamma_{t}=\sum_{j=1}^{4}\gamma_{j}$, we have

$\displaystyle\Omega^{--}_{12}$
$\displaystyle=-\frac{1}{8}\Big{[}\left(\gamma_{3}^{2}+\gamma_{4}^{2}-(d+\gamma_{1}+\gamma_{2}-2)^{2}\right)^{2}-4\gamma_{3}^{2}\gamma_{4}^{2}\Big{]}\Big{(}(d+2\theta_{1}+2\theta_{2})(\partial_{1}+\partial_{2})-2s^{2}\partial_{1}\partial_{2}\Big{)}$
$\displaystyle\quad-\Big{[}-(d+\theta_{1}+\theta_{2})\Big{(}(\theta_{3}+\theta_{4})(\theta_{3}+\theta_{4}-\gamma_{3}-\gamma_{4})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{1}{4}(2-d-\gamma_{t}+2\gamma_{3})(2-d-\gamma_{t}+2\gamma_{4})\Big{)}(\partial_{1}+\partial_{2})$
$\displaystyle+x_{3}\Big{(}(2+2\theta_{3}-\gamma_{3}+2\theta_{4}-\gamma_{4})(2\theta_{4}-\gamma_{4})+(1+\theta_{3})(1+\theta_{3}-\gamma_{3})-\theta_{4}(\theta_{4}-\gamma_{4})\Big{)}\partial_{1}\partial_{2}$
$\displaystyle+x_{4}\Big{(}(2+2\theta_{3}-\gamma_{3}+2\theta_{4}-\gamma_{4})(2\theta_{3}-\gamma_{3})+(1+\theta_{4})(1+\theta_{4}-\gamma_{4})-\theta_{3}(\theta_{3}-\gamma_{3})\Big{)}\partial_{1}\partial_{2}\Big{]}\times$
$\displaystyle\times\Big{[}2s^{2}\Big{(}(\theta_{1}+1-\gamma_{1})\partial_{1}+(\theta_{2}+1-\gamma_{2})\partial_{2}\Big{)}-\Big{(}2\theta_{1}+2\theta_{2}-\gamma_{1}-\gamma_{2}+\frac{d}{2}\Big{)}^{2}+\gamma_{x}^{2}\Big{]}.$
(295)

##### Examples:

All exchange diagrams involving fields of $\Delta=2,3$ in $d=3$ were computed
recently in Bzowski:2022rlz  and are available in the associated Mathematica
package HandbooK.wl. These results enable many tests of the operator
$\Omega_{12}^{--}$ in (5.6) and its shadow conjugates

$\displaystyle\Omega_{12}^{+-}$
$\displaystyle=x_{1}^{1+\gamma_{1}}\Omega_{12}^{--}\,x_{1}^{-\gamma_{1}},$
(296) $\displaystyle\Omega_{12}^{-+}\,$
$\displaystyle=x_{2}^{1+\gamma_{2}}\Omega_{12}^{--},x_{2}^{-\gamma_{2}},$
(297) $\displaystyle\Omega_{12}^{++}\,$
$\displaystyle=x_{1}^{1+\gamma_{1}}x_{2}^{1+\gamma_{2}}\Omega_{12}^{--}\,x_{1}^{-\gamma_{1}}x_{2}^{-\gamma_{2}}.$
(298)

For this, we work in the dimensionally regulated theory with $d\rightarrow
d+2\epsilon$ and $\Delta_{i}\rightarrow\Delta_{i}+\epsilon$ for all
$i=1,2,3,4,x$. This scheme has the virtue of preserving the half-integer
values of all Bessel function indices $\gamma_{i}=\Delta_{i}-d/2$. The
simplest such example is

$\displaystyle i_{[3;22;22;2]}$
$\displaystyle=-\frac{1}{2s}\mathcal{D}^{(+)},$ (299) $\displaystyle
i_{[3;33;22;2]}$
$\displaystyle=\frac{1}{2}(p_{3}+p_{4})\Gamma(2\epsilon)p_{T}^{-2\epsilon}+\frac{1}{4s}(p_{1}^{2}+p_{2}^{2}-s^{2})\mathcal{D}^{(+)}$
$\displaystyle\quad+\frac{1}{2}(p_{1}+p_{2})\Big{[}\log\Big{(}\frac{l_{34+}}{p_{T}}\Big{)}+1\Big{]}+\frac{7}{8}(p_{3}+p_{4})+O(\epsilon)$
(300)

where

$p_{T}=\sum_{i=1}^{4}p_{i},\qquad l_{ij\pm}=p_{i}+p_{j}\pm s,$ (301)

and

$\mathcal{D}^{(+)}=\operatorname{Li}_{2}\Big{(}\frac{l_{12-}}{p_{T}}\Big{)}+\operatorname{Li}_{2}\Big{(}\frac{l_{34-}}{p_{T}}\Big{)}+\log\Big{(}\frac{l_{12+}}{p_{T}}\Big{)}\log\Big{(}\frac{l_{34+}}{p_{T}}\Big{)}-\frac{\pi^{2}}{6}.$
(302)

By direct differentiation, one then finds

$\Omega_{12}^{--}i_{[3;33;22;2]}=(90+261\epsilon+O(\epsilon^{2}))i_{[3;22;22;2]}+O(\epsilon)$
(303)

consistent with (294). Note $\mathcal{N}_{W}\mathcal{N}_{exch.}$ on the right-
hand side here is expanded to order $\epsilon$ since $i_{[3;22;22;2]}$ has an
$\epsilon^{-1}$ pole. We have performed similar checks for all other values of
the $\Delta_{i}$ and $\Delta_{x}$, and for the shadow conjugated operators.

This ability to shift exchange diagrams directly to other exchange diagrams
means that, instead of computing all the diagrams individually, we can compute
the easiest diagram (namely, $i_{[3;22;22;2]}$) to sufficiently high order in
the regulator $\epsilon$, and then obtain all others by acting with
$\Omega_{12}^{\sigma_{1}\sigma_{2}}$ and $\Omega_{34}^{\sigma_{3}\sigma_{4}}$.

##  6 Creation operators for Feynman diagrams

In this section, we analyse various Feynman integrals presenting their GKZ
representations, their singularities, and the associated creation operators.
Many of the examples we study have appeared in the recent works
de_la_Cruz_2019 ; Klausen:2019hrg ; Feng_2020 ; Chestnov:2022alh . Here, our
focus will be the construction of creation operators and ways to automate
their computation using standard Gröbner basis and convex hulling algorithms.

In all cases, we start with an $L$-loop scalar integral in the momentum
representation

$\displaystyle
I=\Big{(}\prod_{j=1}^{L}\int\frac{\mathrm{d}^{d}\bm{k}_{j}}{(2\pi)^{d}}\Big{)}\frac{1}{P_{1}^{\gamma_{1}}\ldots
P_{N}^{\gamma_{N}}},$ (304)

where the propagators $P_{i}$ for $i=1,\ldots,N$ are raised to generalised
powers $\gamma_{i}$. As shown in appendix A (see also de_la_Cruz_2019 ;
Weinzierl:2022eaz ), the corresponding GKZ integral is

$\displaystyle\mathcal{I}_{\bm{\gamma}}=\Big{(}\prod_{i=1}^{N}\int_{0}^{\infty}\mathrm{d}z_{i}\,z_{i}^{\gamma_{i}-1}\Big{)}\,\mathcal{D}^{-\gamma_{0}},\qquad\gamma_{0}=\frac{d}{2}$
(305)

where the denominator $\mathcal{D}$ is formed from the Lee-Pomeransky
denominator $\mathcal{G}=\mathcal{U}+\mathcal{F}$, the sum of first and second
Symanzik polynomials, by replacing the coefficient of every term with an
independent variable $x_{k}$. The Feynman integral (304) now corresponds to

$I=c_{\bm{\gamma}}\mathcal{I}_{\bm{\gamma}},\qquad
c_{\bm{\gamma}}=\frac{(4\pi)^{-L\gamma_{0}}\Gamma\left(\gamma_{0}\right)}{\Gamma\left((L+1)\gamma_{0}-\gamma_{t}\right)\prod_{i=1}^{N}\Gamma(\gamma_{i})},\qquad\gamma_{t}=\sum_{i=1}^{N}\gamma_{i}$
(306)

with the $x_{k}$ restored to their physical (Lee-Pomeransky) values. Knowing
the coefficient $c_{\bm{\gamma}}$ enables the action of a creation operator on
the GKZ integral $\mathcal{I}_{\bm{\gamma}}$ to be related to its action on
the Feynman integral $I$.

###  6.1 Bubble diagram

First, we consider the 1-loop bubble integral with propagators of mass $m_{1}$
and $m_{2}$. As a warm-up, we begin with the single-mass case
$(m_{1},m_{2})=(0,m)$ before turning to general masses. The fully massless
case $m_{1}=m_{2}=0$ is trivial (evaluating to a simple power of the momentum)
and so will be omitted.

####  6.1.1 1-mass bubble

$1$ $2$ $\bm{p}$ Figure 8: The single-mass bubble integral (307), with
massless and massive propagators represented by dashed and undashed lines
respectively.

The single-mass bubble diagram

$\displaystyle
I=\int\frac{\mathrm{d}^{d}\bm{k}}{(2\pi)^{d}}\frac{1}{\bm{k}^{2\gamma_{1}}\left((\bm{p}-\bm{k})^{2}+m^{2}\right)^{\gamma_{2}}},$
(307)

corresponds via (306) to the GKZ integral de_la_Cruz_2019 ; Klausen:2019hrg

$\displaystyle\mathcal{I}_{\bm{\gamma}}=\int_{\mathbb{R}_{+}^{2}}\mathrm{d}z_{1}\mathrm{d}z_{2}\frac{z_{1}^{\gamma_{1}-1}z_{2}^{\gamma_{2}-1}}{(x_{1}z_{1}+x_{2}z_{2}+x_{3}z_{1}z_{2}+x_{4}z_{2}^{2})^{\gamma_{0}}}$
(308)

evaluated on the physical hypersurface

$(x_{1},x_{2},x_{3},x_{4})=(1,1,p^{2}+m^{2},m^{2}).$ (309)

In this simple case, the GKZ integral can of course be evaluated directly,

$\displaystyle\mathcal{I}_{\bm{\gamma}}$
$\displaystyle=\frac{\Gamma(\gamma_{1})\Gamma\left(\gamma_{0}-\gamma_{1}\right)\Gamma\left(\gamma_{1}+\gamma_{2}-\gamma_{0}\right)\Gamma\left(2\gamma_{0}-\gamma_{1}-\gamma_{2}\right)}{\Gamma(\gamma_{0})^{2}}$
$\displaystyle\qquad\times
m^{2(\gamma_{0}-\gamma_{1}-\gamma_{2})}{}_{2}F_{1}\left(\gamma_{1},\gamma_{1}+\gamma_{2}-\gamma_{0};\gamma_{0};-\frac{p^{2}}{m^{2}}\right),$
(310)

enabling the action of all creation operators to be verified. The
$\mathcal{A}$-matrix is

$\displaystyle\mathcal{A}=\left(\begin{array}[]{cccc}1&1&1&1\\\ 1&0&1&0\\\
0&1&1&2\\\ \end{array}\right),$ (314)

and from its kernel, we find a single toric equation

$0=(\partial_{1}\partial_{4}-\partial_{2}\partial_{3})\mathcal{I}_{\bm{\gamma}}.$
(315)

The Euler equations can be read off from the rows of the $\mathcal{A}$-matrix,

$0=(\gamma_{0}+\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4})\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{1}+\theta_{1}+\theta_{3})\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{2}+\theta_{2}+\theta_{3}+2\theta_{4})\mathcal{I}_{\bm{\gamma}}.$
(316)

The (rescaled) Newton polytope derived from the column vectors of the
$\mathcal{A}$-matrix is the parellelogram shown in figure 9. From (109), the
GKZ integral is then singular for

$2\gamma_{0}-\gamma_{1}-\gamma_{2}=-k_{1},\quad-\gamma_{0}+\gamma_{1}+\gamma_{2}=-k_{2},\quad\gamma_{0}-\gamma_{1}=-k_{3},\quad\gamma_{1}=-k_{4},\quad
k_{i}\in\mathbb{Z}^{+}$ (317)

consistent with the poles of the gamma functions in (6.1.1).

($0,\gamma_{0}$)($\gamma_{0},0$)($0,2\gamma_{0}$)($\gamma_{0},\gamma_{0}$)$\gamma_{1}$$\gamma_{2}$
Figure 9: The rescaled Newton polytope associated to the 1-mass bubble
integral (308).

The annihilation operators $\partial_{j}$ send
$\bm{\gamma}\rightarrow\bm{\gamma}^{\prime}$ while the creation operators
$\mathcal{C}_{j}$ send $\bm{\gamma}^{\prime}\rightarrow\bm{\gamma}$ where, for
each $j$, these parameters are related by

$\displaystyle j$ $\displaystyle=1:$ $\displaystyle\gamma_{0}^{\prime}$
$\displaystyle=\gamma_{0}+1,$ $\displaystyle\gamma_{1}^{\prime}$
$\displaystyle=\gamma_{1}+1,$ $\displaystyle\gamma_{2}^{\prime}$
$\displaystyle=\gamma_{2},$ $\displaystyle j$ $\displaystyle=2:$
$\displaystyle\gamma_{0}^{\prime}$ $\displaystyle=\gamma_{0}+1,$
$\displaystyle\gamma_{1}^{\prime}$ $\displaystyle=\gamma_{1},$
$\displaystyle\gamma_{2}^{\prime}$ $\displaystyle=\gamma_{2}+1,$
$\displaystyle j$ $\displaystyle=3:$ $\displaystyle\gamma_{0}^{\prime}$
$\displaystyle=\gamma_{0}+1,$ $\displaystyle\gamma_{1}^{\prime}$
$\displaystyle=\gamma_{1}+1,$ $\displaystyle\gamma_{2}^{\prime}$
$\displaystyle=\gamma_{2}+1,$ $\displaystyle j$ $\displaystyle=4:$
$\displaystyle\gamma_{0}^{\prime}$ $\displaystyle=\gamma_{0}+1,$
$\displaystyle\gamma_{1}^{\prime}$ $\displaystyle=\gamma_{1},$
$\displaystyle\gamma_{2}^{\prime}$ $\displaystyle=\gamma_{2}+2.$ (318)

Knowing the location of the singular hyperplanes and the shifts generated by
the creation operators, the $b$-functions can be constructed according to
(138),

$\displaystyle b_{1}=\gamma_{1}(2\gamma_{0}-\gamma_{1}-\gamma_{2}),$
$\displaystyle
b_{2}=(\gamma_{0}-\gamma_{1})(2\gamma_{0}-\gamma_{1}-\gamma_{2}),$
$\displaystyle b_{3}=\gamma_{1}(-\gamma_{0}+\gamma_{1}+\gamma_{2}),$
$\displaystyle
b_{4}=(\gamma_{0}-\gamma_{1})(-\gamma_{0}+\gamma_{1}+\gamma_{2}).$ (319)

Their zeros serve to cancel the singularities that arise whenever the action
of a creation operator shifts us from a finite to a singular integral. For
example, $\mathcal{C}_{4}$ shifts $k_{2}\rightarrow k_{2}+1$ and
$k_{3}\rightarrow k_{3}+1$ which, according to (317), generates a singular
integral when acting on finite integrals with either $k_{2}=-1$ or $k_{3}=-1$.
These singularities, however, are cancelled by the zeros of $b_{4}$.

Using the DWI and the Euler equations (316), we can now re-write

$\mathcal{C}_{j}\partial_{j}\mathcal{I}_{\bm{\gamma}}=b_{j}(\bm{\gamma})\mathcal{I}_{\bm{\gamma}}=B_{j}(\theta)\mathcal{I}_{\bm{\gamma}}$
(320)

where

$\displaystyle
B_{1}=(\theta_{1}+\theta_{3})(\theta_{1}+\theta_{2})=(\theta_{1}+\theta_{2}+\theta_{3})\theta_{1}+\theta_{2}\theta_{3},$
$\displaystyle
B_{2}=(\theta_{2}+\theta_{4})(\theta_{1}+\theta_{2})=(\theta_{1}+\theta_{2}+\theta_{4})\theta_{2}+\theta_{1}\theta_{4},$
$\displaystyle
B_{3}=(\theta_{1}+\theta_{3})(\theta_{3}+\theta_{4})=(\theta_{1}+\theta_{3}+\theta_{4})\theta_{3}+\theta_{1}\theta_{4},$
$\displaystyle
B_{4}=(\theta_{2}+\theta_{4})(\theta_{3}+\theta_{4})=(\theta_{4}+\theta_{2}+\theta_{3})\theta_{4}+\theta_{2}\theta_{3}.$
(321)

By inspection, every term in $B_{j}$ either contains an explicit factor of
$\partial_{j}$ already through $\theta_{j}$, or else such a factor can be
introduced using the toric equations. In $B_{1}$ and $B_{4}$, for instance, we
replace $\theta_{2}\theta_{3}=x_{2}x_{3}\partial_{2}\partial_{3}\rightarrow
x_{2}x_{3}\partial_{1}\partial_{4}$. This enables the $B_{j}$ to be factored
(modulo the toric equations) in the form (320) yielding the creation operators

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=x_{1}(1+\theta_{1}+\theta_{2}+\theta_{3})+x_{2}x_{3}\partial_{4},$
$\displaystyle\mathcal{C}_{2}$
$\displaystyle=x_{2}(1+\theta_{1}+\theta_{2}+\theta_{4})+x_{1}x_{4}\partial_{3},$
$\displaystyle\mathcal{C}_{3}$
$\displaystyle=x_{3}(1+\theta_{1}+\theta_{3}+\theta_{4})+x_{1}x_{4}\partial_{2},$
$\displaystyle\mathcal{C}_{4}$
$\displaystyle=x_{4}(1+\theta_{2}+\theta_{3}+\theta_{4})+x_{2}x_{3}\partial_{1}.$
(322)

These creation operators act on the full GKZ integral (308). To obtain their
counterparts acting on the Feynman integral (307), we must project to the
physical hypersurface (309). Given the form of the operators (6.1.1), it is
useful to first simplify using the DWI to

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=x_{1}(1-\gamma_{0}-\theta_{4})+x_{2}x_{3}\partial_{4},$
$\displaystyle\mathcal{C}_{2}$
$\displaystyle=x_{2}(1-\gamma_{0}-\theta_{3})+x_{1}x_{4}\partial_{3},$
$\displaystyle\mathcal{C}_{3}$
$\displaystyle=x_{3}(1-\gamma_{0}-\theta_{2})+x_{1}x_{4}\partial_{2},$
$\displaystyle\mathcal{C}_{4}$
$\displaystyle=x_{4}(1-\gamma_{0}-\theta_{1})+x_{2}x_{3}\partial_{1}.$ (323)

Next, as all factors of $x_{j}$ are placed to the left of all derivatives, we
set

$(x_{1},x_{2},x_{3},x_{4})\rightarrow(1,1,m^{2}+p^{2},m^{2})$ (324)

and replace all derivatives lying in directions off this hypersurface (namely
$\partial_{1}$ and $\partial_{2}$) with those lying along the hypersurface.
This can be accomplished using the Euler equations (316) projected according
to (324), namely

$\partial_{1}\rightarrow-\gamma_{1}-(m^{2}+p^{2})\partial_{3},\qquad\partial_{2}\rightarrow-\gamma_{2}-(m^{2}+p^{2})\partial_{3}-2m^{2}\partial_{4}.$
(325)

In addition, we use the chain rule with $p^{2}=x_{3}-x_{4}$ and $m^{2}=x_{4}$
to replace

$\partial_{3}=\partial_{p^{2}},\qquad\partial_{4}=-\partial_{p^{2}}+\partial_{m^{2}}.$
(326)

This yields

$\displaystyle\mathcal{C}_{1}^{\mathrm{ph}}$
$\displaystyle=1-\gamma_{0}+p^{2}\partial_{m^{2}}-\theta_{p^{2}},$
$\displaystyle\mathcal{C}_{2}^{\mathrm{ph}}$
$\displaystyle=1-\gamma_{0}-\theta_{p^{2}},$
$\displaystyle\mathcal{C}_{3}^{\mathrm{ph}}$
$\displaystyle=(1-\gamma_{0})m^{2}+(1-\gamma_{0}+\gamma_{2})p^{2}+(p^{2}-m^{2})\theta_{p^{2}}+2p^{2}\theta_{m^{2}}$
$\displaystyle\mathcal{C}_{4}^{\mathrm{ph}}$
$\displaystyle=(1-\gamma_{0})m^{2}+\gamma_{1}p^{2}-(p^{2}+m^{2})\theta_{p^{2}}.$
(327)

From (132), the action on the projected GKZ integral is then

$\mathcal{C}_{1}^{\mathrm{ph}}\mathcal{I}_{\gamma_{0}^{\prime},\gamma_{1}^{\prime},\gamma_{2}^{\prime}}(p^{2},m^{2})=-\gamma_{0}^{-1}b_{1}\mathcal{I}_{\gamma_{0},\gamma_{1},\gamma_{2}}=-\gamma_{0}^{-1}\gamma_{1}(2\gamma_{0}-\gamma_{1}-\gamma_{2})\mathcal{I}_{\gamma_{0},\gamma_{1},\gamma_{2}}$
(328)

and similarly for the other operators. When acting the original Feynman
integral, there is an additional factor of
$c_{\bm{\gamma}^{\prime}}/c_{\bm{\gamma}}$ from (306) we must take into
account giving

$\mathcal{C}_{1}^{\mathrm{ph}}I_{\gamma_{0}^{\prime},\gamma_{1}^{\prime},\gamma_{2}^{\prime}}(p^{2},m^{2})=-\frac{1}{4\pi}I_{\gamma_{0},\gamma_{1},\gamma_{2}}.$
(329)

All these results can be checked directly using (6.1.1) and the standard shift
operators for the ${}_{2}F_{1}$ (see e.g., NIST:DLMF ).

####  6.1.2 Massive bubble

Next we consider the full bubble graph with general masses $m_{1}$ and
$m_{2}$,

$\displaystyle
I=\int\frac{\mathrm{d}^{d}\bm{k}}{(2\pi)^{d}}\frac{1}{(\bm{k}^{2}+m_{1}^{2})^{\gamma_{1}}\left((\bm{p}-\bm{k})^{2}+m_{2}^{2}\right)^{\gamma_{2}}}.$
(330)

The corresponding GKZ integral is

$\displaystyle\mathcal{I}_{\bm{\gamma}}=\int_{\mathbb{R}_{+}^{2}}\mathrm{d}z_{1}\mathrm{d}z_{2}\frac{z_{1}^{\gamma_{1}-1}z_{2}^{\gamma_{2}-1}}{(x_{1}z_{1}+x_{2}z_{2}+x_{3}z_{1}^{2}+x_{4}z_{2}^{2}+x_{5}z_{1}z_{2})^{\gamma_{0}}},$
(331)

where $\gamma_{0}=d/2$ and the physical hypersurface is

$(x_{1},x_{2},x_{3},x_{4},x_{5})=(1,1,m_{1}^{2},m_{2}^{2},m_{1}^{2}+m_{2}^{2}+p^{2}).$
(332)

$1$ $2$ $\bm{p}$ Figure 10: The massive bubble integral (331).

From the kernel of the $\mathcal{A}$-matrix

$\displaystyle\mathcal{A}=\left(\begin{array}[]{ccccc}1&1&1&1&1\\\
1&0&2&0&1\\\ 0&1&0&2&1\\\ \end{array}\right)$ (336)

we obtain the toric equations

$0=(\partial_{3}\partial_{4}-\partial_{5}^{2})\mathcal{I}_{\bm{\gamma}},\qquad
0=(\partial_{2}\partial_{3}-\partial_{1}\partial_{5})\mathcal{I}_{\bm{\gamma}},\qquad
0=(\partial_{1}\partial_{4}-\partial_{2}\partial_{5})\mathcal{I}_{\bm{\gamma}},$
(337)

while the DWI and the Euler equations can be read off from the rows:

$0=\Big{(}\gamma_{0}+\sum_{i=1}^{5}\theta_{i}\Big{)}\mathcal{I}_{\bm{\gamma}},\qquad
0=(\gamma_{1}+\theta_{1}+2\theta_{3}+\theta_{5})\mathcal{I}_{\bm{\gamma}},\qquad
0=(\gamma_{2}+\theta_{2}+2\theta_{4}+\theta_{5})\mathcal{I}_{\bm{\gamma}}.$
(338)

The rescaled Newton polytope corresponding to this $\mathcal{A}$-matrix is the
quadrilateral shown in figure 11. The singular hyperplanes lie parallel to and
outside the facets of this polytope:

$\gamma_{1}=-k_{1},\quad\gamma_{2}=-k_{2},\quad
2\gamma_{0}-\gamma_{1}-\gamma_{2}=-k_{3},\quad-\gamma_{0}+\gamma_{1}+\gamma_{2}=-k_{4},\qquad
k_{i}\in\mathbb{Z}^{+}.$ (339)

($0,\gamma_{0}$)($\gamma_{0},0$)($0,2\gamma_{0}$)($2\gamma_{0},0$)$\gamma_{1}$$\gamma_{2}$
Figure 11: The rescaled Newton polytope associated with the massive bubble GKZ
integral (331).

For illustration, let us now discuss the creation operator $\mathcal{C}_{5}$.
All others can be obtained by similar computations. The annihilator
$\partial_{5}$ sends $\bm{\gamma}\rightarrow\bm{\gamma}^{\prime}$ where

$\partial_{5}:\qquad\gamma_{0}^{\prime}=\gamma_{0}+1,\qquad\gamma_{1}^{\prime}=\gamma_{1}+1,\qquad\gamma_{2}^{\prime}=\gamma_{2}+1,$
(340)

while the creation operator $\mathcal{C}_{5}$ acts in the opposite direction
sending $\bm{\gamma}^{\prime}\rightarrow\bm{\gamma}$. Given this shift and the
location of the singular hyperplanes, we identify the $b$-function as

$b_{5}=\gamma_{1}\gamma_{2}(\gamma_{0}-\gamma_{1}-\gamma_{2}).$ (341)

Using DWI and Euler equations, this can be re-written in terms of Euler
operators as

$B_{5}=(\theta_{1}+2\theta_{3}+\theta_{5})(\theta_{2}+2\theta_{4}+\theta_{5})(\theta_{3}+\theta_{4}+\theta_{5}).$
(342)

This expression can now be factorised as $\mathcal{C}_{5}\partial_{5}$ by
expanding out and using the toric equations to replace any terms not involving
$\partial_{5}$ with equivalent terms containing this factor. Stripping off the
factor of $\partial_{5}$ then yields $\mathcal{C}_{5}$ in GKZ variables,

$\displaystyle\mathcal{C}_{5}$
$\displaystyle=x_{5}\big{[}2\theta_{3}^{2}+2\theta_{4}^{2}+8\theta_{3}\theta_{4}+3(\theta_{3}+\theta_{4})(1+\theta_{5})+(1+\theta_{5})^{2}+\theta_{2}(1+3\theta_{3}+\theta_{4}+\theta_{5})$
$\displaystyle\qquad+\theta_{1}(1+\theta_{2}+\theta_{3}+3\theta_{4}+\theta_{5})\big{]}+x_{2}x_{3}\partial_{1}(1+\theta_{1}+2\theta_{3})+x_{1}x_{4}\partial_{2}(1+\theta_{2}+2\theta_{4})$
$\displaystyle\quad+2x_{3}x_{4}\partial_{5}(4+\theta_{1}+\theta_{2}+2\theta_{3}+2\theta_{4}).$
(343)

To project this operator to the physical hypersurface (332), we first use the
Euler equations to replace

$\theta_{1}\rightarrow-\gamma_{1}-2\theta_{3}-\theta_{5},\qquad\theta_{2}\rightarrow-\gamma_{2}-2\theta_{4}-\theta_{5}.$
(344)

The two occurrences of $\partial_{1}$ and $\partial_{2}$ can be dealt with
similarly by writing $\partial_{i}=(x_{i})^{-1}\theta_{i}$ for $i=1,2$ and
using (344). Then, setting
$(x_{1},x_{2},x_{3},x_{4},x_{5})\rightarrow(1,1,x_{3},x_{4},x_{5})$, we obtain

$\displaystyle\mathcal{C}_{5}^{\mathrm{ph}}=$ $\displaystyle
x_{5}\bigl{[}(1-\gamma_{1})(1-\gamma_{2})+(1-\gamma_{1}-\gamma_{2}-\theta_{5})(\theta_{3}+\theta_{4})\bigr{]}$
$\displaystyle+x_{3}(\gamma_{1}+2\theta_{3}+\theta_{5})(\gamma_{1}-1+\theta_{5})+x_{4}(\gamma_{2}+2\theta_{4}+\theta_{5})(\gamma_{2}-1+\theta_{5})$
$\displaystyle\quad-2x_{3}x_{4}\partial_{5}(\gamma_{1}+\gamma_{2}-4+2\theta_{5})$
(345)

The remaining variables here are all physical since

$(x_{3},x_{4},x_{5})=(m_{1}^{2},m_{2}^{2},m_{1}^{2}+m_{2}^{2}+p^{2})$ (346)

and

$\displaystyle\partial_{3}=\partial_{m_{1}^{2}}-\partial_{p^{2}},\qquad\partial_{4}=\partial_{m_{2}^{2}}-\partial_{p^{2}},\qquad\partial_{5}=\partial_{p^{2}}.$
(347)

###  6.2 Massive triangle

Since the massless triangle integral is equivalent Bzowski:2013sza  to the
3-point contact Witten diagram studied in sections 5.3.1 and 5.5.1, let us
examine here the massive triangle integral

$\displaystyle
I=\int\frac{\mathrm{d}^{d}\bm{k}}{(2\pi)^{d}}\frac{1}{(\bm{k}^{2}+m_{3}^{2})^{\gamma_{3}}\left((\bm{k}-\bm{p}_{1})^{2}+m_{2}^{2}\right)^{\gamma_{2}}\left((\bm{k}+\bm{p}_{2})^{2}+m_{1}^{2}\right)^{\gamma_{1}}}.$
(348)

$\bm{p}_{1}$ $\bm{p}_{2}$ $\bm{p}_{3}$ $2$ $1$ $3$ Figure 12: The massive
triangle graph (348).

The corresponding GKZ integral according to (306) is

$\displaystyle\mathcal{I}_{\bm{\gamma}}=\int_{\mathbb{R}_{+}^{3}}\mathrm{d}z_{1}\mathrm{d}z_{2}\mathrm{d}z_{3}\,z_{1}^{\gamma_{1}-1}z_{2}^{\gamma_{2}-1}z_{3}^{\gamma_{3}-1}\mathcal{D}^{-\gamma_{0}},$
(349)

where

$\displaystyle\mathcal{D}=$ $\displaystyle
x_{1}z_{1}+x_{2}z_{2}+x_{3}z_{3}+x_{4}z_{2}z_{3}+x_{5}z_{1}z_{3}+x_{6}z_{1}z_{2}+x_{7}z_{1}^{2}+x_{8}z_{2}^{2}+x_{9}z_{3}^{2}$
(350)

with $\gamma_{0}=d/2$. The physical hypersurface is

$\bm{x}=(1,1,1,p_{1}^{2}+m_{2}^{2}+m_{3}^{2},p_{2}^{2}+m_{1}^{2}+m_{3}^{2},p_{3}^{2}+m_{1}^{2}+m_{2}^{2},m_{1}^{2},m_{2}^{2},m_{3}^{2})$
(351)

and the $\mathcal{A}$-matrix reads

$\displaystyle\mathcal{A}=\left(\begin{array}[]{ccccccccc}1&1&1&1&1&1&1&1&1\\\
1&0&0&0&1&1&2&0&0\\\ 0&1&0&1&0&1&0&2&0\\\ 0&0&1&1&1&0&0&0&2\\\
\end{array}\right).$ (356)

For larger $\mathcal{A}$-matrices such as this one, it is useful to automate
the calculation of creation operators using Gröbner basis algorithms. To this
end, in place of the five independent toric equations spanning the kernel of
the $\mathcal{A}$-matrix, we will use instead the full set of 17 (non-
independent) toric equations forming the toric ideal:161616 These can be
obtained using the Singular code singular :

    LIB "toric.lib";

    ring r=0,(x1,x2,x3,x4,x5,x6,x7,x8,x9),dp;

    intmat A[4][9]=1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,2,0,0,0,1,0,1,0,1,0,2,0,0,0,1,1,1,0,0,0,2;

    ideal I=toric_ideal(A,"du");

    I;

$\displaystyle I_{toric}=\\{$
$\displaystyle\partial_{2}\partial_{6}-\partial_{1}\partial_{8},~{}\partial_{7}\partial_{8}-\partial_{6}^{2},~{}\partial_{1}\partial_{6}-\partial_{2}\partial_{7},~{}\partial_{1}\partial_{5}-\partial_{3}\partial_{7},~{}\partial_{3}^{2}\partial_{9}-\partial_{7},~{}\partial_{1}\partial_{4}-\partial_{3}\partial_{6},$
$\displaystyle~{}\partial_{4}^{2}\partial_{9}-\partial_{8},~{}\partial_{5}\partial_{6}-\partial_{4}\partial_{7},~{}\partial_{4}\partial_{6}-\partial_{5}\partial_{8},~{}\partial_{4}\partial_{5}\partial_{9}-\partial_{6},~{}\partial_{3}^{2}\partial_{8}\partial_{9}-\partial_{2}^{2},~{}\partial_{3}^{2}\partial_{7}\partial_{9}-\partial_{1}^{2},$
$\displaystyle~{}\partial_{3}^{2}\partial_{6}\partial_{9}-\partial_{1}\partial_{2},~{}\partial_{3}\partial_{5}\partial_{9}-\partial_{1},~{}\partial_{3}\partial_{4}\partial_{9}-\partial_{2},~{}\partial_{2}\partial_{5}-\partial_{3}\partial_{6},~{}\partial_{2}\partial_{4}-\partial_{3}\partial_{8}\\}.$
(357)

Each entry here corresponds to a toric equation, for example the first is
$0=(\partial_{2}\partial_{6}-\partial_{1}\partial_{8})\mathcal{I}_{\bm{\gamma}}$
and similarly for the rest. Since all the partial derivatives commute, these
equations can be treated as a system of polynomial equations by mapping
$\partial_{i}$ to an ordinary commutative variable $y_{i}$. As we will show
below, this enables the factorisation step to be handled via ordinary
commutative Gröbner basis methods. (For alternative constructions of creation
operators using non-commutative Gröbner bases over the Weyl algebra, see
saito_sturmfels_takayama_1999 .)

The DWI and Euler equations for this $\mathcal{A}$-matrix are

$\displaystyle
0=\Big{(}\gamma_{0}+\sum_{i=1}^{9}\theta_{i}\Big{)}\mathcal{I}_{\bm{\gamma}},$
$\displaystyle
0=(\gamma_{1}+\theta_{1}+\theta_{5}+\theta_{6}+2\theta_{7})\mathcal{I}_{\bm{\gamma}},$
$\displaystyle
0=(\gamma_{2}+\theta_{2}+\theta_{4}+\theta_{6}+2\theta_{8})\mathcal{I}_{\bm{\gamma}},$
$\displaystyle
0=(\gamma_{3}+\theta_{3}+\theta_{4}+\theta_{5}+2\theta_{9})\mathcal{I}_{\bm{\gamma}}$
(358)

and the corresponding Newton polytope is depicted in figure 13.

{1}$\bullet${2}$\bullet${3}$\bullet${4}$\bullet${5}$\bullet${6}$\bullet${7}$\bullet${8}$\bullet${9}$\bullet$
Figure 13: The Newton polytope associated to the denominator of the massive
triangle integral (350). The label {i} denotes the vector defined by the $i$th
column of the $\mathcal{A}$-matrix.

From its facets, we obtain the singularity conditions

$\displaystyle\begin{array}[]{llll}\\{23489\\}:&\gamma_{1}=-k_{1},&\\{1379\\}:&\gamma_{2}=-k_{2},\\\
\\{1278\\}:&\gamma_{3}=-k_{3},&\\{123\\}:&\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3}=+k_{4},\\\
\\{4789\\}:&\gamma_{1}+\gamma_{2}+\gamma_{3}-2\gamma_{0}=+k_{5},&&\end{array}$
(362)

where all $k_{i}\in\mathbb{Z}^{+}$. For the facet $\\{123\\}$, for example, we
have the outward-pointing normal $\bm{n}=(-1,-1,-1)$ which leads via (108) to
the spacing of singular hyperplanes $\delta^{(J)}=1$.

Let us now compute the creation operator $\mathcal{C}_{4}$ which acts on the
GKZ integral to shift $\bm{\gamma}^{\prime}\rightarrow\bm{\gamma}$, where

$\gamma_{0}^{\prime}=\gamma_{0}+1,\qquad\gamma_{1}^{\prime}=\gamma_{1},\qquad\gamma_{2}^{\prime}=\gamma_{2}+1,\qquad\gamma_{3}^{\prime}=\gamma_{3}+1.$
(363)

From these parameter shifts and the location of the singular hyperplanes, the
corresponding $b$-function is

$b_{4}=\gamma_{2}\gamma_{3}(\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3}).$
(364)

Using the DWI and Euler equations, this can be re-expressed as

$B_{4}=\Big{(}\sum_{i=4}^{9}\theta_{i}\Big{)}(\theta_{2}+\theta_{4}+\theta_{6}+2\theta_{8})(\theta_{3}+\theta_{4}+\theta_{5}+2\theta_{9}).$
(365)

Our goal is now to factorise $B_{4}$ as $\mathcal{C}_{4}\partial_{4}$ using
the toric equations. To achieve this in an automated fashion, we decompose
$B_{4}$ over the Gröbner basis formed from the toric ideal (6.2) and
$\partial_{4}$. Treating the partial derivatives as ordinary commutative
variables and computing this Gröbner basis, we obtain

$\bm{g}=\\{\partial_{4},~{}\partial_{2},~{}\partial_{8},~{}\partial_{3}^{2}\partial_{7}\partial_{9}-\partial_{1}^{2},~{}\partial_{6},~{}\partial_{1}\partial_{5}-\partial_{3}\partial_{7},~{}\partial_{3}\partial_{5}\partial_{9}-\partial_{1},~{}\partial_{5}^{2}\partial_{9}-\partial_{7}\\}.$
(366)

Expanding out $B_{4}$ and rewriting all terms in the form (131) so that all
partial derivatives $\partial_{i}$ lie to the right of all $x_{i}$, we can now
decompose each term of $B_{4}$ in this Gröbner basis. This yields

$B_{4}=\bm{Q}\cdot\bm{g}=Q_{1}\partial_{4}+\sum_{i=2}^{8}Q_{i}g_{i}.$ (367)

where the coefficients $Q_{i}$ are polynomials in the $x_{j}$ and $\theta_{j}$
(with $j=1,..,9$) which can be computed automatically.171717 In Mathematica,
for example, after writing $B_{4}$ in the form (131) with all derivatives to
the right, we replace all $\partial_{i}$ (both in $B_{4}$ and in the toric
ideal) by commutative variables $y[i]$. The code

    v = {y[5], y[6], y[7], y[8], y[9], y[1], y[2], y[3], y[4]};

    toric = {y[2] y[6] - y[1] y[8], y[6]2 - y[7] y[8], y[1] y[6] - y[2] y[7], ...};

    g = GroebnerBasis[Append[toric, y[4]], v]

    Q = PolynomialReduce[B4, g, v][[1]]

then evaluates the $Q_{i}$ coefficients with all derivatives $y[i]$ placed to
the right. These can then be re-expressed in terms of Euler operators by
rewriting $y_{i}^{n}=x_{i}^{-n}\theta_{i}(\theta_{i}-1)\ldots(\theta_{i}-n+1)$
leading to (6.2).  To extract the required overall factor of $\partial_{4}$,
we now re-express those $g_{i}$ ($i=2,..,18$) that are not already complete
toric equations (and hence zero) in terms of $\partial_{4}$. For example,
using the third from last toric equation in (6.2), we can replace

$g_{2}=\partial_{2}~{}\rightarrow~{}\partial_{3}\partial_{4}\partial_{9}.$
(368)

In this fashion, we can replace the basis $\bm{g}$ with the equivalent basis
(modulo the toric equations)

$\bm{\tilde{g}}=\\{\partial_{4},~{}\partial_{3}\partial_{4}\partial_{9},~{}\partial_{4}^{2}\partial_{9},~{}0,~{}\partial_{4}\partial_{5}\partial_{9},~{}0,~{}0\\}.$
(369)

All surviving terms then have an explicit factor of $\partial_{4}$ which can
be removed to obtain the creation operator

$\displaystyle\mathcal{C}_{4}$
$\displaystyle=Q_{1}+Q_{2}\partial_{3}\partial_{9}+Q_{3}\partial_{4}\partial_{9}+Q_{5}\partial_{5}\partial_{9},$
(370)

where the coefficients are

$\displaystyle Q_{1}$
$\displaystyle=x_{4}\big{[}(1+\theta_{4})(\theta_{5}+\theta_{7}+\theta_{9})+\theta_{5}(\theta_{6}+\theta_{7}+3\theta_{8}+\theta_{9})+\theta_{6}(\theta_{7}+\theta_{8}+3\theta_{9})$
$\displaystyle\qquad\quad+(\theta_{8}+\theta_{9})(1+2\theta_{7}+\theta_{8}+\theta_{9})+4\theta_{8}\theta_{9}+(1+\theta_{4}+\theta_{5}+\theta_{6}+\theta_{8}+\theta_{9})^{2}$
$\displaystyle\qquad\quad+\theta_{3}(1+\theta_{4}+\theta_{5}+2\theta_{6}+\theta_{7}+3\theta_{8}+\theta_{9})$
$\displaystyle\qquad\quad+\theta_{2}(1+\theta_{3}+\theta_{4}+2\theta_{5}+\theta_{6}+\theta_{7}+\theta_{8}+3\theta_{9})\big{]},$
$\displaystyle Q_{2}$
$\displaystyle=x_{2}(\theta_{3}+\theta_{5}+2\theta_{9})(\theta_{5}+\theta_{6}+\theta_{7}+\theta_{8}+\theta_{9}),$
$\displaystyle Q_{3}$
$\displaystyle=x_{8}(\theta_{3}+\theta_{5}+2\theta_{9})\bigl{(}2\theta_{5}+3\theta_{6}+2(1+\theta_{7}+\theta_{8}+\theta_{9})\bigr{)},$
$\displaystyle Q_{5}$
$\displaystyle=x_{6}(\theta_{3}+\theta_{5}+2\theta_{9})(1+\theta_{5}+\theta_{6}+\theta_{7}+\theta_{9}).$
(371)

Finally, to project to the physical hypersurface, we use the Euler equations
to eliminate the unphysical variables $\theta_{1},\theta_{2},\theta_{3}$ and
set $x_{1}=x_{2}=x_{3}=1$. This yields the physical creation operator

$\displaystyle\mathcal{C}_{4}^{\mathrm{ph}}=$ $\displaystyle
x_{4}\bigl{[}(1-\gamma_{2})(1-\gamma_{3})+(\theta_{5}+\theta_{6}+\theta_{7}+\theta_{8}+\theta_{9})(1-\gamma_{2}-\gamma_{3}-\theta_{4})\bigl{]}$
$\displaystyle+(\gamma_{3}+\theta_{4})\bigl{[}\partial_{9}(\theta_{5}+\theta_{6}+\theta_{7}+\theta_{8}+\theta_{9})(\gamma_{3}+\gamma_{4}+\gamma_{5}+2\theta_{9})$
$\displaystyle\quad-
x_{6}\partial_{5}(1+\theta_{5}+\theta_{6}+\theta_{7}+3\theta_{8}+\theta_{9})-2x_{8}\partial_{4}\partial_{9}(1+\theta_{5}+\theta_{7}+\theta_{8}+\theta_{9})\bigr{]}$
(372)

where the $x_{i}$ are as given in (351) and

$\displaystyle\partial_{4}$ $\displaystyle=\partial_{p_{1}^{2}},$
$\displaystyle\partial_{5}$ $\displaystyle=\partial_{p_{2}^{2}},$
$\displaystyle\partial_{6}$ $\displaystyle=\partial_{p_{3}^{2}},$
$\displaystyle\partial_{7}$
$\displaystyle=\partial_{m_{1}^{2}}-\partial_{p_{2}^{2}}-\partial_{p_{3}^{2}},$
$\displaystyle\partial_{8}$
$\displaystyle=\partial_{m_{2}^{2}}-\partial_{p_{1}^{2}}-\partial_{p_{3}^{2}},$
$\displaystyle\partial_{9}$
$\displaystyle=\partial_{m_{3}^{2}}-\partial_{p_{1}^{2}}-\partial_{p_{2}^{2}}.$
(373)

The automated approach outlined here can be applied similarly to other
examples.

###  6.3 Massless on-shell box

Next we consider the massless box integral

$\displaystyle I$
$\displaystyle=\int\frac{\mathrm{d}^{d}\bm{q}}{(2\pi)^{d}}\frac{1}{|\bm{q}|^{2\gamma_{1}}|\bm{q}+\bm{P}_{1}|^{2\gamma_{2}}|\bm{q}+\bm{P}_{2}|^{2\gamma_{3}}|\bm{q}+\bm{P}_{3}|^{2\gamma_{4}}},$
(374)

where

$\bm{P}_{k}=\sum_{j=1}^{k}\bm{p}_{j},\quad\mathrm{for}\quad
k=1,2,3,\qquad\sum_{i=1}^{4}\bm{p}_{i}=0.$ (375)

$1$ $2$ $3$ $4$ $\bm{p}_{1}$ $\bm{p}_{2}$ $\bm{p}_{3}$ $\bm{p}_{4}$ Figure 14:
The on-shell massless box integral (374).

For simplicity, we will restrict to the on-shell case181818 Creation operators
for the off-shell box are also computable but the results are rather long.
where all $p_{i}^{2}=0$ for $i=1,..,4$. According to (306), the corresponding
GKZ integral is

$\displaystyle\mathcal{I}_{\bm{\gamma}}=\prod_{i=1}^{4}\left(\int_{0}^{\infty}\mathrm{d}z_{i}\>z_{i}^{\gamma_{i}-1}\right)(x_{1}z_{1}+x_{2}z_{2}+x_{3}z_{3}+x_{4}z_{4}+x_{5}z_{1}z_{3}+x_{6}z_{2}z_{4})^{-\gamma_{0}},$
(376)

where the physical hypersurface

$\bm{x}=(1,1,1,1,s^{2},t^{2})$ (377)

with $s^{2}=(\bm{p}_{1}+\bm{p}_{2})^{2}$ and
$t^{2}=(\bm{p}_{2}+\bm{p}_{3})^{2}$ the Mandelstam invariants. The integral
can be evaluated as a linear combination of the hypergeometric function
${}_{3}F_{2}$ Tarasov:2017jen .

The $\mathcal{A}$-matrix

$\displaystyle\mathcal{A}=\left(\begin{array}[]{cccccc}1&1&1&1&1&1\\\
1&0&0&0&1&0\\\ 0&1&0&0&0&1\\\ 0&0&1&0&1&0\\\ 0&0&0&1&0&1\\\
\end{array}\right)$ (383)

yields a single toric equation

$0=(\partial_{1}\partial_{3}\partial_{6}-\partial_{2}\partial_{4}\partial_{5})\mathcal{I}_{\bm{\gamma}},$
(384)

along with the DWI and Euler equations

$\displaystyle
0=\Big{(}\gamma_{0}+\sum_{i=1}^{6}\theta_{i}\Big{)}\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{1}+\theta_{1}+\theta_{5})\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{2}+\theta_{2}+\theta_{6})\mathcal{I}_{\bm{\gamma}},$ $\displaystyle
0=(\gamma_{3}+\theta_{3}+\theta_{5})\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{4}+\theta_{4}+\theta_{6})\mathcal{I}_{\bm{\gamma}}.$ (385)

To determine the singularities of the integral, we need to find the equations
of the facets of the rescaled Newton polytope corresponding to the GKZ
denominator in (376). As this polytope lives in four dimensions, it is
convenient to use an automated hulling algorithm. Using the Mathematica
package Nhull , for example, we can enter the vertices $\bm{a}_{j}$ of the
non-rescaled Newton polytope (where $\bm{a}_{j}$ is the $j$th column of the
$\mathcal{A}$-matrix without the top row) as row vectors:

       verts = {{1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1},{1,0,1,0},{0,1,0,1}};

The command CHNQuickHull[verts] then returns a list of the vertex vectors that
make up the convex hull (labelled according to the numbering specified in the
input), followed by a list of the facets. The latter are specified by the
vertex vectors they contain. Thus, in this example, we obtain

          {{1,2,3,4,5,6}, {{1,2,3,5}, {1,2,4,3}, {1,2,5,6}, {1,2,6,4},
                {1,3,4,5}, {1,4,6,5}, {2,3,5,6}, {2,3,6,4}, {3,4,5,6}}}

where the first set indicates that all six vertices belong to the convex hull,
while the remainder ($\\{1,2,3,5\\}$, $\\{1,2,4,3\\}$, etc.) list the facets.
Here, each $\\{ijkl\\}$ is a co-dimension one facet containing the points
$(\bm{a}_{i},\bm{a}_{j},\bm{a}_{k},\bm{a}_{l})$.

The equations for the facets of the rescaled Newton polytope with vertices
$\gamma_{0}\bm{a}_{i}$ can now be computed through a determinant such as
(113). For the facet $\\{1,2,3,4\\}$, for example, we have

$0=\bm{\gamma}\cdot\bm{N}=\det\,(\,\bm{\gamma}\,|\,\bm{\mathcal{A}}_{1}\,|\,\bm{\mathcal{A}}_{2}\,|\,\bm{\mathcal{A}}_{3}\,|\,\bm{\mathcal{A}}_{4}\,)=\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3}-\gamma_{4}$
(386)

and hence $\bm{N}=(n_{0},\bm{n})=(1,-1,-1,-1,-1)$. The fact that $\bm{n}$ is
outwards-pointing can be verified by showing
$d_{i}^{(J)}=-\bm{\mathcal{A}}_{i}\cdot\bm{N}>0$ for any vertex $i=5,6$ not
lying in the facet. The spacing of the set of singular hyperplanes parallel to
this facet is then $\delta^{(J)}=1$ using (108) and (99), with the singular
hyperplanes themselves then following from (109). Automating this procedure
and applying it to the other facets, the singularities for the GKZ integral
(376) are

$\displaystyle\gamma_{i}=-k_{i},\quad i=1,2,3,4,$
$\displaystyle\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}-\gamma_{0}=-k_{5},$
$\displaystyle\gamma_{1}+\gamma_{2}-\gamma_{0}=+k_{6},$
$\displaystyle\gamma_{2}+\gamma_{3}-\gamma_{0}=+k_{7},$ (387)
$\displaystyle\gamma_{3}+\gamma_{4}-\gamma_{0}=+k_{8},$
$\displaystyle\gamma_{4}+\gamma_{1}-\gamma_{0}=+k_{9},$

where all $k_{i}\in\mathbb{Z}^{+}$.

We are now in a position to compute the creation operators. Let us choose
$\mathcal{C}_{1}$, which acts on the GKZ integral to shift
$\bm{\gamma}^{\prime}\rightarrow\bm{\gamma}$ where

$\gamma_{0}^{\prime}=\gamma_{0}+1,\qquad\qquad\gamma_{1}^{\prime}=\gamma_{1}+1.$
(388)

From the singularities (387), the corresponding $b$-function is

$b_{1}=-\gamma_{1}(\gamma_{2}+\gamma_{3}-\gamma_{0})(\gamma_{3}+\gamma_{4}-\gamma_{0}),$
(389)

which in terms of the Euler operators reads

$\displaystyle B_{1}$
$\displaystyle=(\theta_{1}+\theta_{2})(\theta_{1}+\theta_{4})(\theta_{1}+\theta_{5})$
$\displaystyle=\big{[}(\theta_{1}+\theta_{2})(\theta_{1}+\theta_{4})+(\theta_{1}+\theta_{2}+\theta_{4})\theta_{5}\big{]}\theta_{1}+\theta_{2}\theta_{4}\theta_{5}$
(390)

Applying the toric equation (384) to the final term now enables us to
factorise $B_{1}$ as $\mathcal{C}_{1}\partial_{1}$ giving the creation
operator

$\mathcal{C}_{1}=x_{1}\left[(1+\theta_{1}+\theta_{2})(1+\theta_{1}+\theta_{4})+(1+\theta_{1}+\theta_{2}+\theta_{4})\theta_{5}\right]+x_{2}x_{4}x_{5}\partial_{3}\partial_{6},$
(391)

where we shifted the factor of $x_{1}$ to the left sending each
$\theta_{1}\rightarrow\theta_{1}+1$. Finally, to obtain the creation operator
acting on the physical variables, we use the Euler equations (6.3) to replace
$\theta_{i}$ for $i=1,2,3,4$ with $\theta_{5}$ and $\theta_{6}$ and project to
(377). This yields the operator

$\mathcal{C}_{1}^{\mathrm{ph}}=(1-\gamma_{1}-\gamma_{2}-\theta_{t^{2}})(1-\gamma_{1}-\gamma_{4}-\theta_{t^{2}})+(\gamma_{1}-1)\theta_{s^{2}}-s^{2}(\gamma_{3}+\theta_{s^{2}})\partial_{t^{2}}.$
(392)

Using the automated determination of the convex hull in this example, and the
factorisation of the $b$-function via Gröbner basis methods in the previous
example, the calculation of any creation operator can be fully automated.

##  7 Discussion

The GKZ formalism enables the construction of highly non-trivial shift
operators known as creation operators. As we have shown, the calculation is
very systematic. First, a Feynman or Witten diagram is represented as a GKZ or
$\mathcal{A}$-hypergeometric function. Second, the $b$-function is identified
by examining the parameter shifts produced by the creation operator in
conjunction with the location of all singular hyperplanes of the integral. The
$b$-function is a function of parameters that multiplies the shifted integral,
and its zeros serve physically to cancel the singularities that would
otherwise arise when the creation operator maps a finite to a singular
integral. Such singularities cannot arise under the action of a finite
differential operator on a finite integral. Next, using the Euler equations
and DWI, the $b$-function is expressed as a function of Euler operators and
factorised into a product of a creation and an annihilation operator with the
aid of the toric equations. The creation operator thus extracted is then re-
expressed in terms of physical variables (i.e., the momenta and masses) using
once again the Euler equations and DWI.

This algorithm has a number of interesting features. First, the parametric
singularities of the integral all lie on hyperplanes parallel to the facets of
the Newton polytope associated with the integral’s denominator. We derived a
precise formula for the spacing of these hyperplanes in (109). The
$b$-function therefore has a geometrical character, as originally shown by
Saito in Saito_param_shift . Second, the algorithm makes heavy use of the
higher-dimensional GKZ space obtained by promoting the coefficient of every
term in the Lee-Pomeransky denominator to an independent variable. This
systematises the set of PDEs obeyed by the integral into two distinct classes:
the Euler equations and DWI, and the toric equations. Using the former, we can
uplift to GKZ space by exchanging all dependence on the parameters
$\bm{\gamma}$ for dependence on the additional unphysical coordinates.
Conversely, we can project back to physical variables by using the Euler
equations and DWI to exchange derivatives with respect to the unphysical
variables for derivatives with respect to the physical variables and
dependence on the parameters $\bm{\gamma}$.

This last step is however a potential weakness of the algorithm. To project a
creation operator from GKZ space back to the physical hypersurface, the total
number of Euler equations (including the DWI) must be equal to, or greater
than, the number of unphysical coordinates. This enables every derivative in
unphysical variables to be replaced by an equivalent expression in purely
physical variables. For higher-loop Feynman integrals, however, the number of
terms in the Lee-Pomeransky denominator, and hence the dimension of the full
GKZ space, typically grows more rapidly than the number of propagators and
hence Euler equations. Thus, while a full set of creation operators can be
constructed in GKZ space, in general we lack sufficient Euler equations to
project back to the physical hypersurface. For this reason, we have focused
initially on 1-loop Feynman integrals.

One possible workaround for this issue is to construct an alternative
projectible GKZ system based on some representation other than the Lee-
Pomeransky. For example, for higher-loop massive sunset (aka melon or banana)
diagrams, one can construct a GKZ representation based on their position-space
formulation as a product of Bessel functions Klemm_2020 ; Zhang:2023fil . In
this manner, these diagrams can be related to (analytic continuations of) the
momentum-space contact Witten diagrams for which we have already constructed
creation operators. For more general classes of diagrams, projectible GKZ
representations can also be obtained from Mellin-Barnes representations as
shown in Feng_2020 ; Feng:2022ude ; Zhang:2023fil . A further possibility
might be to develop a GKZ representation starting from the Baikov
representation.

Nevertheless, using the simplest formulation based on the Lee-Pomeransky
representation, we have already identified a number of useful new shift
operators. In particular, for computations in AdS/CFT, we have found:

  * •

The creation operators (208) and (226), along with their permutations and
shadow conjugates, connecting 3- and 4-point momentum-space contact Witten
diagrams of different operator and spacetime dimensions. These new operators
are the inverse of the simple annihilators first identified in Bzowski:2013sza
; Bzowski:2015yxv . The corresponding creation operators for position-space
contact diagrams are given in appendix B.

  * •

The creation operators (5.5.1) and (273), plus their permutations and shadow
conjugates, relating 3- and 4-point momentum-space contact Witten diagrams of
different operator dimensions but the same spacetime dimension. While the
3-point operator (5.5.1) is known Karateev:2017jgd ; Baumann:2019oyu , the
4-point operator (273) is new.

  * •

Using (273), we obtained a further new operator (5.6) connecting exchange
Witten diagrams of different external operator dimensions but the same
spacetime dimension. Unlike any previous construction, this operator connects
exchange diagrams with purely non-derivative vertices. Working in dimensional
regularisation where necessary to avoid divergences Bzowski:2022rlz , it also
applies for arbitrary operator dimensions.

There is ample scope for building on this first application of creation
operators to Witten diagrams. In particular, our results for exchange Witten
diagrams were obtained from the analysis of contact diagrams. It would be
preferable to develop a GKZ representation for the exchange diagram directly,
both in momentum and in position space. This in turn may enable more compact
expressions to be found, as well as operators acting to shift the dimension of
the exchanged leg. At present, operators achieving this latter goal are known
only for a very restricted set of external scaling dimensions Baumann:2019oyu
; Bzowski:2022rlz . The application of the creation operator formalism to
cosmological correlators in de Sitter spacetime is also worthy of exploration.
We hope to return to these matters in future.

##### Acknowledgments:

FC thanks the School of Maths, Statistics & Physics for support. PM is
supported in part by the UKRI consolidated grant ST/T000708/1.

##  Appendix A GKZ representation of Feynman integrals

In this appendix we relate a generic $L$-loop Feynman integral of the form
(304) to the corresponding GKZ integral (305). Related discussions can be
found in, e.g., de_la_Cruz_2019 ; Weinzierl:2022eaz .

After exponentiating the propagators and integrating out the loop momenta,
(304) has the Schwinger parametrisation

$I=(4\pi)^{-\gamma_{0}L}\Big{(}\prod_{i=1}^{N}\frac{1}{\Gamma(\gamma_{i})}\int_{0}^{\infty}\mathrm{d}t_{i}\,t_{i}^{\gamma_{i}-1}\Big{)}\,\mathcal{U}[t]^{-\gamma_{0}}\exp\Big{(}-\frac{\mathcal{F}[t]}{\mathcal{U}[t]}\Big{)},\qquad\gamma_{0}=\frac{d}{2},$
(393)

where $\mathcal{U}[t]$ and $\mathcal{F}[t]$ are the first and second Symanzik
polynomials respectively, which are homogeneous of weights $L$ and $L+1$ in
the Schwinger parameters $t_{i}$. The prefactor of $(4\pi)^{-\gamma_{0}L}$ is
simply that in (304) multiplied by $L$ factors of $\pi^{d/2}$ from integrating
out the loop momenta. The corresponding Feynman representation is obtained by
reparametrising

$t_{i}=\sigma y_{i},\qquad y_{t}=\sum_{i=1}^{N}y_{i}=1$ (394)

and integrating out the variable $\sigma$. Using the Jacobian191919See, e.g.,
Appendix B of Bzowski:2020kfw .

$\prod_{i=1}^{N}\mathrm{d}t_{i}=\sigma^{N-1}\mathrm{d}\sigma\prod_{i=1}^{N}\mathrm{d}y_{i}\,\delta(1-y_{t}),$
(395)

as well as the homogeneity of the Symanzik polynomials, we find

$\displaystyle I$
$\displaystyle=(4\pi)^{-\gamma_{0}L}\Big{(}\prod_{i=1}^{N}\frac{1}{\Gamma(\gamma_{i})}\int_{0}^{1}\mathrm{d}y_{i}\,y_{i}^{\gamma_{i}-1}\Big{)}\delta(1-y_{t})\mathcal{U}[y]^{-\gamma_{0}}\int_{0}^{\infty}\mathrm{d}\sigma\,\sigma^{\gamma_{t}-\gamma_{0}L-1}\exp\Big{(}-\sigma\frac{\mathcal{F}[y]}{\mathcal{U}[y]}\Big{)}$
$\displaystyle=(4\pi)^{-\gamma_{0}L}\Gamma(\gamma_{t}-\gamma_{0}L)\Big{(}\prod_{i=1}^{N}\frac{1}{\Gamma(\gamma_{i})}\int_{0}^{1}\mathrm{d}y_{i}\,y_{i}^{\gamma_{i}-1}\Big{)}\delta(1-y_{t})\mathcal{U}[y]^{\gamma_{t}-\gamma_{0}(L+1)}\mathcal{F}[y]^{-\gamma_{t}+\gamma_{0}L}.$
(396)

In special cases where $\gamma_{t}-\gamma_{0}(L+1)$ vanishes (e.g., $d=2$
multi-loop sunsets with standard propagators) one can use the $\mathcal{F}$
polynomial alone to construct a GKZ representation Vanhove:2018mto . More
generally, one can use the Lee-Pomeransky representation Lee:2013hzt ;
de_la_Cruz_2019  obtained by combining the two Symanzik polynomial factors
using the Euler beta identity

$\mathcal{U}[y]^{-a}\mathcal{F}[y]^{a-b}=\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_{0}^{\infty}\mathrm{d}s\,s^{a-1}(\mathcal{F}[y]+s\,\mathcal{U}[y])^{-b}$
(397)

with $a=\gamma_{0}(L+1)-\gamma_{t}$ and $b=\gamma_{0}$ giving

$\displaystyle I$
$\displaystyle=c_{\bm{\gamma}}\,\Big{(}\prod_{i=1}^{N}\int_{0}^{1}\mathrm{d}y_{i}\,y_{i}^{\gamma_{i}-1}\Big{)}\delta(1-y_{t})\int_{0}^{\infty}\mathrm{d}s\,s^{\gamma_{0}(L+1)-\gamma_{t}-1}(\mathcal{F}[y]+s\,\mathcal{U}[y])^{-\gamma_{0}}$
(398)

where

$c_{\bm{\gamma}}=\frac{(4\pi)^{-L\gamma_{0}}\Gamma\left(\gamma_{0}\right)}{\Gamma\left((L+1)\gamma_{0}-\gamma_{t}\right)\prod_{i=1}^{N}\Gamma(\gamma_{i})}.$
(399)

Setting $y_{i}=sz_{i}$ and using once again the homogeneity of the Symanzik
polynomials, we can eliminate the $s$ integral since

$\int_{0}^{\infty}\frac{\mathrm{d}s}{s}\,\delta(1-sz_{t})=\int_{0}^{\infty}\frac{\mathrm{d}s}{sz_{t}}\delta(z_{t}^{-1}-s)=1$
(400)

after which

$\displaystyle I$
$\displaystyle=c_{\bm{\gamma}}\,\Big{(}\prod_{i=1}^{N}\int_{0}^{\infty}\mathrm{d}z_{i}\,z_{i}^{\gamma_{i}-1}\Big{)}(\mathcal{F}[z]+\mathcal{U}[z])^{-\gamma_{0}}.$
(401)

Finally, this Lee-Pomeransky representation is upgraded to the GKZ
representation by replacing the coefficient of every term in the denominator
$\mathcal{F}[z]+\mathcal{U}[z]$ with an independent variable $x_{k}$. For the
massless triangle integral, for example,

$\mathcal{U}[z]=z_{1}+z_{2}+z_{3},\qquad\mathcal{F}[z]=p_{1}^{2}z_{2}z_{3}+p_{2}^{2}z_{3}z_{1}+p_{3}^{2}z_{1}z_{2}$
(402)

and so we replace the Lee-Pomeransky denominator

$\mathcal{G}=\mathcal{F}[z]+\mathcal{U}[z]=p_{1}^{2}z_{2}z_{3}+p_{2}^{2}z_{3}z_{1}+p_{3}^{2}z_{1}z_{2}+z_{1}+z_{2}+z_{3}$
(403)

with the GKZ denominator

$\mathcal{D}=x_{1}z_{2}z_{3}+x_{2}z_{3}z_{1}+x_{3}z_{1}z_{2}+x_{4}z_{1}+x_{5}z_{2}+x_{6}z_{3}.$
(404)

The GKZ integral

$\mathcal{I}_{\bm{\gamma}}=\Big{(}\prod_{i=1}^{N}\int_{0}^{\infty}\mathrm{d}z_{i}\,z_{i}^{\gamma_{i}-1}\Big{)}\mathcal{D}^{-\gamma_{0}}$
(405)

is then related to the massless triangle integral by

$I=c_{\bm{\gamma}}\mathcal{I}_{\bm{\gamma}}$ (406)

evaluated on the physical hypersurface

$x_{i}=p_{i}^{2},\qquad x_{i+3}=1,\qquad i=1,2,3.$ (407)

##  Appendix B Creation operators for the position-space contact Witten
diagram

In position space, the $n$-point AdS contact Witten diagram

$I_{n}=\int_{0}^{\infty}\frac{\mathrm{d}z}{z^{d+1}}\int\mathrm{d}^{d}\bm{x}_{0}\prod_{i=1}^{n}C_{\Delta_{i}}\Big{(}\frac{z}{z^{2}+x_{i0}^{2}}\Big{)}^{\Delta_{i}},\qquad
C_{\Delta_{i}}=\frac{\Gamma(\Delta_{i})}{\pi^{d/2}\Gamma(\Delta_{i}-\frac{d}{2})},$
(408)

has the parametric representation202020See, e.g., equations (5.46)–(5.51) and
(B.1)–(B.11) of Bzowski:2020kfw .

$\displaystyle I_{n}$
$\displaystyle=C_{n}\,\Big{(}\prod_{i=1}^{n}\int_{0}^{\infty}\mathrm{d}z_{i}\,z_{i}^{\Delta_{i}-1}\Big{)}\delta\big{(}1-\sum_{i=1}^{n}\kappa_{i}z_{i}\big{)}\Big{(}\sum_{i<j}z_{i}z_{j}x_{ij}^{2}\Big{)}^{-\Delta_{t}/2}.$
(409)

where

$C_{n}=\frac{\pi^{d/2}}{2}\Gamma\Big{(}\frac{\Delta_{t}}{2}\Big{)}\Gamma\Big{(}\frac{\Delta_{t}-d}{2}\Big{)}\prod_{i=1}^{n}\frac{C_{\Delta_{i}}}{\Gamma(\Delta_{i})},\qquad\Delta_{t}=\sum_{i=1}^{n}\Delta_{i},\qquad\bm{x}_{ij}=\bm{x}_{i}-\bm{x}_{j}.$
(410)

The parameters $\kappa_{i}\geq 0$ can be chosen arbitrarily provided they are
not all zero. For the 4-point function specifically, choosing
$\kappa_{i}=\delta_{i4}$ and eliminating $y_{4}$ using the delta function
leads to the GKZ representation

$\displaystyle I_{4}$
$\displaystyle=C_{4}\mathcal{I}_{\bm{\gamma}},\qquad\mathcal{I}_{\bm{\gamma}}=\Big{(}\prod_{i=1}^{3}\int_{0}^{\infty}\mathrm{d}z_{i}\,z_{i}^{\gamma_{i}-1}\Big{)}\mathcal{D}^{-\gamma_{0}}$
(411)

where

$\mathcal{D}=x_{1}z_{2}z_{3}+x_{2}z_{1}z_{3}+x_{3}z_{1}z_{2}+x_{4}z_{1}+x_{5}z_{2}+x_{6}z_{3},$
(412)

the parameters

$\gamma_{1}=\Delta_{1},\qquad\gamma_{2}=\Delta_{2},\qquad\gamma_{3}=\Delta_{3},\qquad\gamma_{0}=\frac{1}{2}(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}),$
(413)

and the GKZ variables are related to the physical coordinate separations by

$(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=(x_{23}^{2},x_{13}^{2},x_{12}^{2},x_{14}^{2},x_{24}^{2},x_{34}^{2}).$
(414)

Comparing with (8), the position-space 4-point contact diagram, also known as
the holographic $D$-function DHoker:1999kzh , is thus equivalent to the
massless triangle integral (see also Bzowski:2020kfw ). As shown on page 2.4,
the massless triangle integral is itself equivalent to the triple-$K$ integral
(or momentum-space 3-point contact diagram) under affine reparametrisation of
the GKZ integral. The creation operators for the position-space contact
diagram are thus those analysed in section 5.3.1 and 5.5.1, except that no
final projection to the physical hypersurface is required as all the GKZ
variables in (414) are physical.

Concretely, the $\mathcal{A}$-matrix (22) leads to the Euler equations (23)
and DWI (24), and the toric equations (27). The Newton polytope corresponds to
the right-hand panel in figure 2. From its facets we obtain the singularity
conditions

$\displaystyle\gamma_{i}=-n_{i},\qquad\qquad\gamma_{0}-\gamma_{i}=-m_{i},\qquad\qquad
i=1,2,3,$ $\displaystyle\gamma_{1}+\gamma_{2}+\gamma_{3}-\gamma_{0}=-n,\qquad
2\gamma_{0}-\gamma_{1}-\gamma_{2}-\gamma_{3}=-m,$ (415)

where $n_{i},m_{i},n,m\in\mathbb{Z}^{+}$. The action of the annihilator
$\partial_{1}=\partial/\partial x_{23}^{2}$ is to raise $\gamma_{0}$,
$\gamma_{2}$ and $\gamma_{3}$ by one which corresponds to raising $\Delta_{2}$
and $\Delta_{3}$ by one, and the action of the creation operator
$\mathcal{C}_{1}$ is the reverse of this. The corresponding $b$-function

$b_{1}=\gamma_{2}\gamma_{3}(\gamma_{0}-\gamma_{1})(\gamma_{1}+\gamma_{2}+\gamma_{3}-\gamma_{0}),$
(416)

when re-expressed in terms of Euler operator is

$B_{1}=(\theta_{1}+\theta_{3}+\theta_{5})(\theta_{1}+\theta_{2}+\theta_{6})(\theta_{1}+\theta_{5}+\theta_{6})(\theta_{1}+\theta_{2}+\theta_{3}).$
(417)

As expected, this is simply (185) under the mapping
$\bar{\theta}_{i}=\theta_{i+3}$ since the affine reparametrisation from the
$\mathcal{A}$-matrix (22) to (56) leaves the creation operators unchanged.
Expanding out and using the toric equations to factorise
$B_{1}=\mathcal{C}_{1}\partial_{1}$, we recover the creation operator (5.3.1)
in GKZ variables. In our present variables (414), this is

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=x_{1}(\theta_{1}+1+u_{2}+u_{3})\big{(}(\theta_{1}+1+u_{2})(\theta_{1}+1+u_{3})+2(v_{2}+v_{3})\big{)}$
$\displaystyle\quad+x_{2}x_{5}\partial_{4}\bigl{(}1+u_{2}+v_{2}-v_{3}+(u_{2}+u_{3}+2)u_{3}\bigr{)}$
$\displaystyle\quad+x_{3}x_{6}\partial_{4}\bigl{(}1+u_{3}+v_{3}-v_{2}+(u_{2}+u_{3}+2)u_{2}\bigr{)}$
(418)

where $u_{i}=\theta_{i}+\theta_{i+3}$ and $v_{i}=\theta_{i}\theta_{i+3}$. One
likewise obtains the operator (5.5.1), namely

$\displaystyle W_{12}^{--}$
$\displaystyle=(\theta_{4}+\theta_{5}+\theta_{6}+\theta_{3})(x_{4}\partial_{2}+x_{5}\partial_{1})+x_{3}x_{6}\partial_{2}\partial_{1}.$
(419)

Both these operators can be rewritten in various equivalent forms using the
DWI and Euler equations. Their action on the position-space contact diagram
follows from (59), namely

$\mathcal{C}_{1}:\,\,\Delta_{2}\rightarrow\Delta_{2}-1,\quad\Delta_{3}\rightarrow\Delta_{3}-1,\qquad
W_{12}^{--}:\,\,\Delta_{3}\rightarrow\Delta_{3}+1,\quad\Delta_{4}\rightarrow\Delta_{4}-1.$
(420)

##  Appendix C Non-minimal b-functions

As we have seen, creation operators are constructed starting from a polynomial
$b(\bm{\gamma})$ in the spectral parameters known as the $b$-function. In
section 4.4, we argued that $b(\bm{\gamma})$ must possess a certain minimal
set of zeros, namely, those required to cancel the singularities arising when
a creation operator shifts us from a finite to a singular integral. Notice
however that this argument does not preclude the existence of additional zeros
besides this minimal set. For all the Feynman and Witten diagram examples in
the main text, the minimal $b$-functions were sufficient for the construction
of all creation operators. As these $b$-functions contain the fewest factors,
the resulting creation operators were moreover of lowest possible order in
derivatives. Nevertheless, there are instances where the minimal $b$-function
is not sufficient: a simple example, which we analyse in this appendix, is the
GKZ integral (71). As we will show, additional factors must be appended to the
minimal $b$-functions in order to be able to apply the toric equations and
factorise into a product of creation and annihilation operators. The zeros of
these additional factors are all parallel to the facets of the rescaled Newton
polytope, and in most (though not all) cases correspond to additional singular
hyperplanes of the GKZ integral.

Let us recall the necessary analysis of section 3. The integral (71), namely

$\mathcal{I}_{\bm{\gamma}}=\int_{0}^{\infty}\mathrm{d}z_{1}\int_{0}^{\infty}\mathrm{d}z_{2}\,z_{1}^{\gamma_{1}-1}z_{2}^{\gamma_{2}-1}(x_{1}+x_{2}z_{2}+x_{3}z_{1}^{2}+x_{4}z_{1}z_{2}^{2})^{-\gamma_{0}},$
(421)

corresponds to the $\mathcal{A}$-matrix

$\mathcal{A}=\left(\begin{matrix}1&1&1&1\\\ 0&0&2&1\\\
0&1&0&2\end{matrix}\right)$ (422)

with DWI and Euler equations

$0=(\gamma_{0}+\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4})\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{1}+2\theta_{3}+\theta_{4})\mathcal{I}_{\bm{\gamma}},\quad
0=(\gamma_{2}+\theta_{2}+2\theta_{4})\mathcal{I}_{\bm{\gamma}},$ (423)

and a single toric equation

$0=(\partial_{1}^{3}\partial_{4}^{2}-\partial_{2}^{4}\partial_{3})\mathcal{I}_{\bm{\gamma}}.$
(424)

The singularities of this integral, derived in (93), are

$\displaystyle\gamma_{1}=-m_{1},\quad\gamma_{2}=-m_{2},\quad\gamma_{0}+\gamma_{1}-\gamma_{2}=-m_{3},\quad
4\gamma_{0}-2\gamma_{1}-\gamma_{2}=-3m_{4},$ (425)

for all $m_{j}\in\mathbb{Z}^{+}$. The annihilation operators $\partial_{j}$
send $\bm{\gamma}\rightarrow\bm{\gamma}^{\prime}$ while the creation operators
$\mathcal{C}_{j}$ send $\bm{\gamma}^{\prime}\rightarrow\bm{\gamma}$, where for
each $j$ these parameters are related by

$\displaystyle j$ $\displaystyle=1:$ $\displaystyle\gamma_{0}^{\prime}$
$\displaystyle=\gamma_{0}+1,$ $\displaystyle\gamma_{1}^{\prime}$
$\displaystyle=\gamma_{1},$ $\displaystyle\gamma_{2}^{\prime}$
$\displaystyle=\gamma_{2}$ $\displaystyle j$ $\displaystyle=2:$
$\displaystyle\gamma_{0}^{\prime}$ $\displaystyle=\gamma_{0}+1,$
$\displaystyle\gamma_{1}^{\prime}$ $\displaystyle=\gamma_{1},$
$\displaystyle\gamma_{2}^{\prime}$ $\displaystyle=\gamma_{2}+1$ $\displaystyle
j$ $\displaystyle=3:$ $\displaystyle\gamma_{0}^{\prime}$
$\displaystyle=\gamma_{0}+1,$ $\displaystyle\gamma_{1}^{\prime}$
$\displaystyle=\gamma_{1}+2,$ $\displaystyle\gamma_{2}^{\prime}$
$\displaystyle=\gamma_{2},$ $\displaystyle j$ $\displaystyle=4:$
$\displaystyle\gamma_{0}^{\prime}$ $\displaystyle=\gamma_{0}+1,$
$\displaystyle\gamma_{1}^{\prime}$ $\displaystyle=\gamma_{1}+1,$
$\displaystyle\gamma_{2}^{\prime}$ $\displaystyle=\gamma_{2}+2.$ (426)

According to (138), the minimal $b$-functions containing only the zeros
necessary to cancel the singularities produced by the action of the
$\mathcal{C}_{j}$ are

$\displaystyle b_{1}^{\mathrm{min}}$
$\displaystyle=(\gamma_{0}+\gamma_{1}-\gamma_{2})\prod_{m_{4}=0}^{1}(4\gamma_{0}-2\gamma_{1}-\gamma_{2}+3m_{4}),$
$\displaystyle b_{2}^{\mathrm{min}}$
$\displaystyle=\gamma_{2}(4\gamma_{0}-2\gamma_{1}-\gamma_{2}),$ $\displaystyle
b_{3}^{\mathrm{min}}$
$\displaystyle=\prod_{m_{1}=0}^{1}(\gamma_{1}+m_{1})\prod_{m_{3}=0}^{2}(\gamma_{0}+\gamma_{1}-\gamma_{2}+m_{3}),$
$\displaystyle b_{4}^{\mathrm{min}}$
$\displaystyle=\gamma_{1}\prod_{m_{2}=0}^{1}(\gamma_{2}+m_{2}).$ (427)

For example, $\mathcal{C}_{3}$ shifts $m_{1}\rightarrow m_{1}+2$ and
$m_{3}\rightarrow m_{3}+3$, and so the five singular integrals with
$m_{1}=0,1$ and $m_{3}=0,1,2$ in (425) are accessible starting from finite
integrals. This means $b_{3}^{\mathrm{min}}$ has the five zeros shown, which
act to cancel these singularities. The operator $\mathcal{C}_{1}$ is however a
special case: this sends $m_{3}\rightarrow m_{3}+1$ and $m_{4}\rightarrow
m_{4}+4/3$, corresponding to a non-integer $F_{1}^{(4)}=4/3$ in (139). Only
the singularities with $m_{3}=0$ and $m_{4}=0,1$ are then accessible starting
from finite integrals for which all $m_{j}<0$. (In other words, integrals for
which the GKZ representation (421) converges without meromorphic
continuation.)

Using the DWI and Euler equations to rewrite these $b$-functions in terms of
Euler operators, we then find

$\displaystyle B_{1}^{\mathrm{min}}$
$\displaystyle=-(\theta_{1}+3\theta_{3})\prod_{m_{4}=0}^{1}(4\theta_{1}+3\theta_{2}-3m_{4}),$
$\displaystyle B_{2}^{\mathrm{min}}$
$\displaystyle=(\theta_{2}+2\theta_{4})(4\theta_{1}+3\theta_{2}),$
$\displaystyle B_{3}^{\mathrm{min}}$
$\displaystyle=-(2\theta_{3}+\theta_{4})(2\theta_{3}+\theta_{4}-1)\prod_{m_{3}=0}^{2}(\theta_{1}+3\theta_{3}-m_{3}),$
$\displaystyle B_{4}^{\mathrm{min}}$
$\displaystyle=-(2\theta_{3}+\theta_{4})(\theta_{2}+2\theta_{4})(\theta_{2}+2\theta_{4}-1).$
(428)

At this point a problem appears: to extract a creation operator requires
factorising

$B_{j}\mathcal{I}_{\bm{\gamma}}=\mathcal{C}_{j}\partial_{j}\mathcal{I}_{\bm{\gamma}},$
(429)

however the only toric equation we have available for this purpose, (424), is
of fifth order in derivatives. While $B_{3}^{\mathrm{min}}$ is indeed of fifth
order, the remaining $B_{j}^{\mathrm{min}}$ are of at most third order. Upon
expanding out and ordering terms according to (131), we find

$\displaystyle B_{1}^{\mathrm{min}}$
$\displaystyle=(\ldots)\partial_{1}-27x_{2}^{2}x_{3}\partial_{2}^{2}\partial_{3},$
$\displaystyle B_{2}^{\mathrm{min}}$
$\displaystyle=(\ldots)\partial_{2}+8x_{1}x_{4}\partial_{1}\partial_{4},$
$\displaystyle B_{3}^{\mathrm{min}}$
$\displaystyle=(\ldots)\partial_{3}-x_{1}^{3}x_{4}^{2}\partial_{1}^{3}\partial_{4}^{2},$
$\displaystyle B_{4}^{\mathrm{min}}$
$\displaystyle=(\ldots)\partial_{4}-2x_{2}^{2}x_{3}\partial_{2}^{2}\partial_{3}.$
(430)

For $B_{3}^{\mathrm{min}}$, we obtain the necessary factorisation (429) upon
using (424) allowing a successful construction of $\mathcal{C}_{3}$. For the
others, the order in derivatives is too low to apply (424).

To find $\mathcal{C}_{1}$, $\mathcal{C}_{2}$ and $\mathcal{C}_{4}$, therefore,
we look for new (non-minimal) $B_{j}$ of the form:

$\displaystyle B_{1}$
$\displaystyle=(\ldots)\partial_{1}+(\ldots)x_{2}^{4}x_{3}\partial_{2}^{4}\partial_{3},$
$\displaystyle B_{2}$
$\displaystyle=(\ldots)\partial_{2}+(\ldots)x_{1}^{3}x_{4}^{2}\partial_{1}^{3}\partial_{4}^{2},$
$\displaystyle B_{4}$
$\displaystyle=(\ldots)\partial_{4}+(\ldots)x_{2}^{4}x_{3}\partial_{2}^{4}\partial_{3}.$
(431)

By construction, these are all of fifth order and can be factorised into the
desired form (429) using (424). Since the $B_{j}$ must be functions of the
Euler operators, and

$x_{i}^{n}\partial_{i}^{n}=\theta_{i}(\theta_{i}-1)\ldots(\theta_{i}-n+1),$
(432)

this is equivalent to seeking

$\displaystyle B_{1}$
$\displaystyle=(\ldots)\theta_{1}+(\ldots)\theta_{2}(\theta_{2}-1)(\theta_{2}-2)(\theta_{2}-3)\theta_{3},$
$\displaystyle B_{2}$
$\displaystyle=(\ldots)\theta_{2}+(\ldots)\theta_{1}(\theta_{1}-1)(\theta_{1}-2)\theta_{4}(\theta_{4}-1),$
$\displaystyle B_{4}$
$\displaystyle=(\ldots)\theta_{4}+(\ldots)\theta_{2}(\theta_{2}-1)(\theta_{2}-2)(\theta_{2}-3)\theta_{3}.$
(433)

For comparison, the singular hyperplanes in (425), when translated to Euler
operators via (423), correspond to the zeros of

$\displaystyle(2\theta_{3}+\theta_{4}-m_{1}),\quad(\theta_{2}+2\theta_{4}-m_{2}),\quad(\theta_{1}+3\theta_{3}-m_{3}),\quad(4\theta_{1}+3\theta_{2}-3m_{4}).$
(434)

As the non-minimal $B_{j}$ in (C) must still contain the factors present in
the minimal $B_{j}^{\mathrm{min}}$ in (C), we see that for $B_{1}$, and
$B_{4}$ it suffices simply to append factors corresponding to additional
singular hyperplanes:

$\displaystyle B_{1}$
$\displaystyle=-(\theta_{1}+3\theta_{3})\prod_{m_{4}=0}^{3}(4\theta_{1}+3\theta_{2}-3m_{4}),$
$\displaystyle B_{4}$
$\displaystyle=-(2\theta_{3}+\theta_{4})\prod_{m_{2}=0}^{3}(\theta_{2}+2\theta_{4}-m_{2}).$
(435)

Each of these non-minimal $B_{j}$ contain the factors already present in the
minimal $B_{j}^{\mathrm{min}}$. Moreover, they are of the form (C) since they
correspond to performing a linear shift in $\theta_{j}$ on each of the factors
present in the second term of each $B_{j}$ in (C). (Equivalently, setting
$\theta_{j}$ to zero in each of the $B_{j}$ in (C) yields the second term of
each $B_{j}$ in (C).) This also shows that they are of the smallest order in
derivatives consistent with (C).

For $B_{2}$, the additional factors we must append to $B_{2}^{\mathrm{min}}$
are parallel to the singular hyperplanes in (434) but have different spacing.
Explicitly, we require

$B_{2}=-\prod_{m_{1}=0}^{1}(\theta_{2}+2\theta_{4}-2m_{1})\prod_{m_{2}=0}^{2}(4\theta_{1}+3\theta_{2}-4m_{2})$
(436)

so that, when expanded in $\theta_{2}$, we obtain an expression of the form
given in (C). Note this is not possible using the spacings in (434).212121The
zeros of (436), and of the corresponding of $b_{2}$ in (C), do however
coincide with the singular hyperplanes of the integral obtained by deleting
the second column of the $\mathcal{A}$-matrix. This removes a vertex from the
Newton polytope changing the spacings of the singular hyperplanes; a procedure
consistent with confining all $\theta_{2}$ dependence in $B_{2}$ to the first
factor in (C).

By construction, the non-minimal $B_{j}$ in (C) and (436) all derive from
corresponding non-minimal $b$-functions which are polynomials in the spectral
parameters,

$\displaystyle b_{1}$
$\displaystyle=(\gamma_{0}+\gamma_{1}-\gamma_{2})\prod_{m_{4}=0}^{3}(4\gamma_{0}-2\gamma_{1}-\gamma_{2}+3m_{4}),$
$\displaystyle b_{2}$
$\displaystyle=\prod_{m_{1}=0}^{1}(\gamma_{2}+2m_{1})\prod_{m_{2}=0}^{2}(4\gamma_{0}-2\gamma_{1}-\gamma_{2}+4m_{2}),$
$\displaystyle b_{4}$
$\displaystyle=\gamma_{1}\prod_{m_{2}=0}^{3}(\gamma_{2}+m_{2}).$ (437)

and all lead to valid creation operators via (429). From $B_{2}$, for example,
we find

$\displaystyle\mathcal{C}_{2}$
$\displaystyle=x_{2}\Bigl{[}12\theta_{4}^{2}\Big{(}48\theta_{1}^{2}+12\theta_{1}(3\theta_{2}-5)+9\theta_{2}(\theta_{2}-2)+5\Big{)}$
$\displaystyle\qquad+4\theta_{4}\Big{(}64\theta_{1}^{3}+48\theta_{1}^{2}(3\theta_{2}-4)+4\theta_{1}\Big{(}9\theta_{2}(3\theta_{2}-5)+32\Big{)}+3\theta_{2}\Big{(}9\theta_{2}(\theta_{2}-2)+5\Big{)}\Big{)}$
$\displaystyle\qquad+(\theta_{2}-1)(4\theta_{1}+3\theta_{2}-5)(4\theta_{1}+3\theta_{2}-1)(4\theta_{1}+3\theta_{2}+3)\Big{]}+256x_{1}^{3}x_{4}^{2}\partial_{2}^{3}\partial_{3}.$
(438)

$\gamma_{1}$$\gamma_{2}$ Figure 15: For purposes of illustration, a two-
dimensional example of a non-normal lattice can be obtained by projecting to
the $(\gamma_{1},\gamma_{2})$ plane. The points $(2k+1,0)$ and $(2k+1,1)$ for
$k\in\mathbb{Z}^{+}$ (shown in red) lie within the positive cone
$\sum_{j}\mathbb{R}^{+}\bm{a}_{j}$ (shaded region) spanned by the vertex
vectors $\bm{a}_{j}$ of the Newton polytope (shown in blue). These points also
belong to $\sum_{j}\mathbb{Z}\bm{a}_{j}$ since $(2k+1,0)=k(2,0)+(1,2)-2(0,1)$
and $(2k+1,1)=k(2,0)+(1,2)-(0,1)$. However, these points cannot be expressed
in the form $\sum_{j}\mathbb{Z}^{+}\bm{a}_{j}$ and hence the lattice generated
by the $\bm{a}_{j}$ is non-normal.

Having solved this example, let us note that the failure of the minimal
$b$-functions in (C) can also be understood geometrically. For
$B_{1}^{\mathrm{min}}$, $B_{2}^{\mathrm{min}}$ and $B_{4}^{\mathrm{min}}$ in
(C) to be factorisable as $\mathcal{C}_{j}\partial_{j}$, we would need each of
$\partial_{2}^{2}\partial_{3}\partial_{1}^{-1}$,
$\partial_{1}\partial_{4}\partial_{2}^{-1}$ and
$\partial_{2}^{2}\partial_{3}\partial_{4}^{-1}$ to be expressible as
$\prod_{k=1}^{N}\partial_{k}^{c_{k}}$ for some set of
$c_{k}\in\mathbb{Z}^{+}$. (The inverses here are purely formal: we mean that
$\partial_{2}^{2}\partial_{3}=\partial_{1}\prod_{k=1}^{N}\partial_{k}^{c_{k}}$,
etc.) In terms of the shifts produced by these operators on the spectral
parameters $\bm{\gamma}=(\gamma_{0},\gamma_{1},\gamma_{2})$, this is
equivalent to requiring that

$\displaystyle
2\bm{\mathcal{A}}_{2}+\bm{\mathcal{A}}_{3}-\bm{\mathcal{A}}_{1}$
$\displaystyle=\left(\begin{matrix}2\\\ 2\\\
2\end{matrix}\right),\quad\bm{\mathcal{A}}_{1}+\bm{\mathcal{A}}_{4}-\bm{\mathcal{A}}_{2}=\left(\begin{matrix}1\\\
1\\\ 1\end{matrix}\right),\quad
2\bm{\mathcal{A}}_{2}+\bm{\mathcal{A}}_{3}-\bm{\mathcal{A}}_{4}=\left(\begin{matrix}2\\\
1\\\ 0\end{matrix}\right)$ (439)

are all expressible as $\sum_{k=1}^{N}c_{k}\bm{\mathcal{A}}_{k}$ for some set
of $c_{k}\in\mathbb{Z}^{+}$, where $\bm{\mathcal{A}}_{k}$ denotes the $k$th
column of the $\mathcal{A}$-matrix including the top row of ones. Clearly this
is not possible, although these vectors do all lie in the positive cone
corresponding to solutions with $c_{k}\in\mathbb{R}^{+}$ since

$\displaystyle
2\bm{\mathcal{A}}_{2}+\bm{\mathcal{A}}_{3}-\bm{\mathcal{A}}_{1}$
$\displaystyle=2(\bm{\mathcal{A}}_{1}+\bm{\mathcal{A}}_{4}-\bm{\mathcal{A}}_{2})=\frac{2}{3}(\bm{\mathcal{A}}_{2}+\bm{\mathcal{A}}_{3}+\bm{\mathcal{A}}_{4}),$
$\displaystyle
2\bm{\mathcal{A}}_{2}+\bm{\mathcal{A}}_{3}-\bm{\mathcal{A}}_{4}$
$\displaystyle=\frac{1}{2}(3\bm{\mathcal{A}}_{1}+\bm{\mathcal{A}}_{3}).$ (440)

Mathematically, this is precisely the condition that lattice generated by the
$\bm{\mathcal{A}}_{k}$ (or equivalently, the toric ideal associated with the
$\mathcal{A}$-matrix) is non-normal. Conversely, when the normality condition

$\Big{(}\sum_{k}\mathbb{R}^{+}\bm{\mathcal{A}}_{k}\Big{)}\cap\Big{(}\sum_{k}\mathbb{Z}\bm{\mathcal{A}}_{k}\Big{)}=\Big{(}\sum_{k}\mathbb{Z}^{+}\bm{\mathcal{A}}_{k}\Big{)}$
(441)

is satisfied, it can be shown that the minimal $b$-functions (138) generate
valid creation operators Saito_param_shift ; Saito_restrictions . This
condition is non-trivial as can be seen from the two-dimensional example of a
non-normal lattice illustrated in figure 15.

## References

  * (1) T. Regge, _Algebraic topology methods in the theory of Feynman relativistic amplitudes_ , in _Battelle Recontres: Lectures in Mathematics and Physics_ , pp. 433–458, 1967.
  * (2) M. Kashiwara and T. Kawai, _Holonomic Systems of Linear Differential Equations and Feynman Integrals_ , _Publ. Res. Inst. Math. Sci. Kyoto_ 12 (1977) 131.
  * (3) P. Vanhove, _Feynman integrals, toric geometry and mirror symmetry_ , in _KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory_ , pp. 415–458, 2019 [1807.11466].
  * (4) L. de la Cruz, _Feynman integrals as A-hypergeometric functions_ , _JHEP_ 12 (2019) 123 [1907.00507].
  * (5) R.P. Klausen, _Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems_ , _JHEP_ 04 (2020) 121 [1910.08651].
  * (6) R.P. Klausen, _Kinematic singularities of Feynman integrals and principal A-determinants_ , _JHEP_ 02 (2022) 004 [2109.07584].
  * (7) R.P. Klausen, _Hypergeometric Feynman integrals_ , Ph.D. thesis, Mainz U., 2023.  2302.13184.
  * (8) A. Klemm, C. Nega and R. Safari, _The l-loop banana amplitude from GKZ systems and relative Calabi-Yau periods_ , _Journal of High Energy Physics_ 2020 (2020)  [1912.06201].
  * (9) T.-F. Feng, C.-H. Chang, J.-B. Chen and H.-B. Zhang, _GKZ-hypergeometric systems for Feynman integrals_ , _Nuclear Physics B_ 953 (2020) 114952 [arXiv:1912.01726].
  * (10) V. Chestnov, F. Gasparotto, M.K. Mandal, P. Mastrolia, S.J. Matsubara-Heo, H.J. Munch et al., _Macaulay matrix for Feynman integrals: linear relations and intersection numbers_ , _JHEP_ 09 (2022) 187 [2204.12983].
  * (11) B. Ananthanarayan, S. Banik, S. Bera and S. Datta, _FeynGKZ: A Mathematica package for solving Feynman integrals using GKZ hypergeometric systems_ , _Comput. Phys. Commun._ 287 (2023) 108699 [2211.01285].
  * (12) H.-B. Zhang and T.-F. Feng, _GKZ hypergeometric systems of the three-loop vacuum Feynman integrals_ , _JHEP_ 05 (2023) 075 [2303.02795].
  * (13) I. Gel’fand, M. Kapranov and A. Zelevinsky, _Generalized Euler integrals and A-hypergeometric functions_ , _Advances in Mathematics_ 84 (1990) 255.
  * (14) I. Gel’fand, A. Zelevinsky and M. Kapranov, _Hypergeometric functions and toral manifolds_ , _Functional Analysis and its Applications_ 23 (1989) 94.
  * (15) I. Gel’fand, M. Kapranov and A. Zelevinsky, _Hypergeometric functions, toric varieties and Newton polyhedra_ , in _Special Functions, ICM-90 Satellite Conference Proceedings_ , Springer, 1991.
  * (16) I. Gel’fand, M. Kapranov and A. Zelevinsky, _Discriminants, Resultants, and Multidimensional Determinants_ , Mathematics (Birkhäuser), Springer (1994).
  * (17) R.N. Lee and A.A. Pomeransky, _Critical points and number of master integrals_ , _JHEP_ 11 (2013) 165 [1308.6676].
  * (18) J. Stienstra, _GKZ hypergeometric structures_ , in _Istanbul 2005: CIMPA Summer School on Arithmetic and Geometry Around Hypergeometric Functions_ , 11, 2005 [math/0511351].
  * (19) F. Beukers, _Notes on A-hypergeometric functions, in Arithmetic and Galois theories of differential equations_ , _Séminaires et Congrès, Socièté mathématique de France_ 23 (2011) 25.
  * (20) E. Cattani, “Three lectures on hypergeometric functions.” http://people.math.umass.edu/~cattani/hypergeom_lectures.pdf, 2006.
  * (21) N. Takayama, _A-Hypergeometric Functions_ , in _Encyclopedia of Special Functions: The Askey-Bateman Project_ , vol. 2, Cambridge University Press, 2020.
  * (22) M. Saito, B. Sturmfels and N. Takayama, _Gröbner deformations of hypergeometric differential equations_ , vol. 6, Algorithms & Computations in Mathematics, Springer (2013).
  * (23) T. Reichelt, M. Schulze, C. Sevenheck and U. Walther, _Algebraic aspects of hypergeometric differential equations_ , 2004.07262.
  * (24) O.V. Tarasov, _Connection between Feynman integrals having different values of the space-time dimension_ , _Phys. Rev. D_ 54 (1996) 6479 [hep-th/9606018].
  * (25) R.N. Lee, _Calculating multiloop integrals using dimensional recurrence relation and $D$-analyticity_, _Nucl. Phys. B Proc. Suppl._ 205-206 (2010) 135 [1007.2256].
  * (26) V.V. Bytev, M.Y. Kalmykov and B.A. Kniehl, _Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case_ , _Nucl. Phys. B_ 836 (2010) 129 [0904.0214].
  * (27) V.V. Bytev, M.Y. Kalmykov and S.-O. Moch, _HYPERgeometric functions DIfferential REduction (HYPERDIRE): MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: $F_{D}$ and $F_{S}$ Horn-type hypergeometric functions of three variables_, _Comput. Phys. Commun._ 185 (2014) 3041 [1312.5777].
  * (28) T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer, _Feynman integral relations from parametric annihilators_ , _Lett. Math. Phys._ 109 (2019) 497 [1712.09215].
  * (29) F.A. Dolan and H. Osborn, _Conformal four point functions and the operator product expansion_ , _Nucl. Phys._ B599 (2001) 459 [hep-th/0011040].
  * (30) A. Bzowski, P. McFadden and K. Skenderis, _Evaluation of conformal integrals_ , _JHEP_ 02 (2016) 068 [1511.02357].
  * (31) D. Karateev, P. Kravchuk and D. Simmons-Duffin, _Weight Shifting Operators and Conformal Blocks_ , _JHEP_ 02 (2018) 081 [1706.07813].
  * (32) M.S. Costa and T. Hansen, _AdS Weight Shifting Operators_ , _JHEP_ 09 (2018) 040 [1805.01492].
  * (33) D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee and G.L. Pimentel, _The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds_ , 1910.14051.
  * (34) K.C. Rigatos and X. Zhou, _Yangian Symmetry in Holographic Correlators_ , _Phys. Rev. Lett._ 129 (2022) 101601 [2206.07924].
  * (35) A. Bzowski, P. McFadden and K. Skenderis, _A handbook of holographic 4-point functions_ , _JHEP_ 12 (2022) 039 [2207.02872].
  * (36) M. Saito, _Parameter shift in normal generalized hypergeometric systems_ , _Tohoku Mathematical Journal_ (1992) 523–534.
  * (37) M. Saito and N. Takayama, _Restrictions of A-hypergeometric systems and connection formulas on the $\Delta_{1}\times\Delta_{n-1}$-hypergeometric function_, _International Journal of Mathematics_ (1994) 537.
  * (38) M. Saito, _Contiguity relations for the lauricella functions_ , _Funkcialaj Ekvacioj_ 38 (1995) 37.
  * (39) M. Saito, B. Sturmfels and N. Takayama, _Hypergeometric Polynomials and Integer Programming_ , _Compositio Mathematica_ 115 (1999) 231–240.
  * (40) L. Nilsson and M. Passare, _Mellin transforms of multivariate rational functions_ , _Journal of Geometric Analysis_ 23 (2013) 24 [1010.5060].
  * (41) C. Berkesch, J. Forsgrd and M. Passare, _Euler-Mellin integrals and A-hypergeometric functions_ , _Michigan Mathematical Journal_ 63 (2014) 101 [1103.6273].
  * (42) W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, “Singular 4-3-0 — A computer algebra system for polynomial computations.” http://www.singular.uni-kl.de/Manual/4-0-3/sing_1375.htm, 2022.
  * (43) A. Bzowski, P. McFadden and K. Skenderis, _Implications of conformal invariance in momentum space_ , _JHEP_ 03 (2014) 111 [1304.7760].
  * (44) A. Bzowski, P. McFadden and K. Skenderis, _Conformal correlators as simplex integrals in momentum space_ , _JHEP_ 01 (2021) 192 [2008.07543].
  * (45) N. Arkani-Hamed, A. Hillman and S. Mizera, _Feynman polytopes and the tropical geometry of UV and IR divergences_ , _Phys. Rev. D_ 105 (2022) 125013 [2202.12296].
  * (46) S.-J. Matsubara-Heo, S. Mizera and S. Telen, _Four Lectures on Euler Integrals_ , 2306.13578.
  * (47) T. Kaneko and T. Ueda, _A geometric method of sector decomposition_ , _Comput. Phys. Commun._ 181 (2010) 1352 [0908.2897].
  * (48) K. Schultka, _Toric geometry and regularization of Feynman integrals_ , 1806.01086.
  * (49) A. Bzowski, P. McFadden and K. Skenderis, _Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies_ , _JHEP_ 03 (2016) 066 [1510.08442].
  * (50) L. Petrich, “Computational geometry.” http://lpetrich.org/Science/#CHDV.
  * (51) W. Miller, _Lie Theory and Special Functions_ , vol. 43, Mathematics in Science and Engineering, Academic Press, New York (1968).
  * (52) E. Kalanins, H. Manocha and W. Miller, _The Lie theory of two-variable hypergeometric functions_ , _Studies in Applied Mathematics_ 62 (1980) 143.
  * (53) M.J.E. Smeets, _Parameter shift in GKZ-hypergeometric systems_ , Ph.D. thesis, Eindhoven University of Technology, 2000.  https://doi.org/10.6100/IR532761.
  * (54) “NIST Digital Library of Mathematical Functions.” https://dlmf.nist.gov/.
  * (55) A. Prudnikov, I. Brychkov and O. Marichev, _Integrals and Series, Vol. 2: Special functions, p.404_ , Gordon and Breach Science Publishers (1986).
  * (56) C. Corianò and M.M. Maglio, _On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space_ , _JHEP_ 09 (2019) 107 [1903.05047].
  * (57) A. Bzowski, P. McFadden and K. Skenderis, _Renormalised 3-point functions of stress tensors and conserved currents in CFT_ , _JHEP_ 11 (2018) 153 [1711.09105].
  * (58) A. Bzowski, P. McFadden and K. Skenderis, _Renormalised CFT 3-point functions of scalars, currents and stress tensors_ , _JHEP_ 11 (2018) 159 [1805.12100].
  * (59) S. Weinzierl, _Feynman Integrals_ , Springer (2022), 10.1007/978-3-030-99558-4, [2201.03593].
  * (60) O.V. Tarasov, _Massless on-shell box integral with arbitrary powers of propagators_ , _J. Phys. A_ 51 (2018) 275401 [1709.07526].
  * (61) T.-F. Feng, H.-B. Zhang, Y.-Q. Dong and Y. Zhou, _GKZ-system of the 2-loop self energy with 4 propagators_ , _Eur. Phys. J. C_ 83 (2023) 314 [2209.15194].
  * (62) E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, _Graviton exchange and complete four point functions in the AdS / CFT correspondence_ , _Nucl. Phys. B_ 562 (1999) 353 [hep-th/9903196].