Shift operators from the simplex representation in momentum-space CFT

We derive parametric integral representations for the general $n$-point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman integral with the topology of an $(n-1)$-simplex, featuring an arbitrary function of momentum-space cross ratios. Here, we show all graph polynomials for this integral can be expressed in terms of the first and second minors of the Laplacian matrix for the simplex. Computing the effective resistance between nodes of the corresponding electrical network, an inverse parametrisation is found in terms of the determinant and first minors of the Cayley-Menger matrix. These parametrisations reveal new families of weight-shifting operators, expressible as determinants, that connect $n$-point functions in spacetime dimensions differing by two. Moreover, the action of all previously known weight-shifting operators preserving the spacetime dimension is manifest. Finally, the new parametric representations enable the validity of the conformal Ward identities to be established directly, without recourse to recursion in the number of points.

In position space, the structure of general scalar n-point functions has been understood for over fifty years [27]. A correspondingly general solution in momentum space was proposed only recently in [28,29]. This takes the form of a generalised Feynman integral with the topology of an (n − 1)-simplex, where the integration is taken over the internal momenta q ij running between vertices of the simplex. Here q ij = −q ji runs from vertex i to j, while the external momenta p i enter only via momentum conservation as imposed by the delta function inserted at each vertex. Each propagator corresponds to an edge of the simplex, as illustrated in figure 1, and is raised to a power specified by the parameter α ij . Together, these satisfy the constraints where ∆ i is the scaling dimension of the operator O i . To simplify the writing of such sums we define α ii = 0 and α ji = α ij . Euclidean signature will be assumed throughout. The distinguishing feature of the simplex representation (1.1) is the presence of an arbitrary function f (q) of the independent momentum-space cross ratioŝ denoted collectively by the vectorq. As the simplex representation can be derived from the general position-space solution [28,29], the number of independent cross ratios is the same in both cases, i.e., n(n−3)/2 for n ≤ d+2 and nd−(d+2)(d+1)/2 for n > d+2. For n ≥ 4, the solution of the constraints (1.2) for the α ij is not unique, but making a different choice simply multiplies f (q) by a product of powers of the cross ratios (1.3). Since f (q) is arbitrary, the solution of (1.2) chosen is therefore immaterial.
In this paper, we explore scalar parametric representations of the simplex integral (1.1) obtained by integrating out the internal momenta. This offers several advantages: • The original integral (1.1) features n(n − 1)/2 d-dimensional loop integrations and we have (n − 1) delta functions to help us, with one remaining behind to enforce overall momentum conservation. This leaves the equivalent of (n − 1)(n − 2)d/2 scalar integrals to perform. In contrast, the parametrisations we derive feature fewer integrals: only n(n−1)/2 scalar parametric integrals, one for each edge of the simplex.
• By inverting the graph polynomials that arise, we construct novel weight-shifting operators connecting solutions of the conformal Ward identities in spacetime dimension d to new solutions in dimension d + 2. Remarkably, these operators have a determinantal structure based on the Cayley-Menger matrix familiar from distance geometry. In contrast, the well-known weight-shifting operators introduced in [30] preserve the spacetime dimension. Operators mapping d → d + 2 are we believe known only for 3-point functions, where their existence can be seen from the triple-K representation in momentum space [31] 1 , and for 4-point conformal blocks in position space (the operator E + in [33]). The new d → d + 2 operators we obtain can be viewed as a natural generalisation of the 3-point operators of [31] to arbitrary n-point correlators.
The plan of this paper is as follows. In section 2, we show that all graph polynomials for the simplex integral (1.1) can be constructed from the corresponding Gram matrix. The standard parametric representations for Feynman integrals then follow. Alternatively, by regarding the Schwinger parameters as resistances in an electrical network, we can compute the effective resistances between all vertices of the simplex. This latter set of variables dramatically simplifies the structure of the Schwinger exponential. In section 3, we use these effective resistances to construct new d → d + 2 shift operators for the general n-point function. The cases n = 3, 4 are discussed in detail, and we verify the action of all operators independently through computation of their intertwining relations with the conformal Ward identities. The actions of the d-preserving weight-shifting operators of [30] are also demonstrated from this scalar parametric perspective. In section 4, we prove that the new parametric representations indeed solve the conformal Ward identities. In contrast to the vectorial representation (1.1) (for which the Ward identities are analysed in [28,29]), for the new scalar parametric representations the Ward identities can be verified directly without use of recursive arguments in the number of points n. As we show in section 5, the validity of the conformal Ward identities, as well as the action of the d-preserving weightshifting operators, can also be seen from the position-space counterpart of the simplex. Section 6 concludes with a summary of results and open directions.
This section investigates scalar parametric representations for the simplex integral (1.1). In the following, we identify the necessary graph polynomials (section 2.1), standard parametric representations (section 2.2), and introduce new variables analogous to the effective resistances between nodes of the simplex (section 2.3). To re-formulate the simplex integral in these variables, we solve the inverse problem to express the original Schwinger parameters in terms of the effective resistances (section 2.4). The re-parametrised integral, which will be the basis of our new shift operators, then follows (section 2.5).

Graph polynomials
Exponentiating all propagators via Schwinger parametrisation, the internal momenta can be integrated out reducing the simplex integral to various scalar parametrisations. The structure of the resulting Symanzik polynomials is clearest however when expressed in terms of the inverse of the usual variables. For this reason, we use the inverse Schwinger parametrisation 1 q 2α ij +d ij The resulting polynomials U and F are then related to the standard Symanzik polynomials U and F by For the simplex, the structure of U and F can be expressed in terms of two matrices. The first is the (n − 1) × (n − 1) Gram matrix G ij = p i · p j . For our purposes, the most convenient parametrisation is Here the V ij provide a full set of n(n − 1)/2 symmetric and independent Lorentz invariants.
To write the diagonal entries in the Gram matrix, we used momentum conservation to express p 2 i = − n k =i p i · p k . The second matrix is simply the image of the Gram matrix under the mapping V ij → v ij , namely Since the v ij correspond to the edges of the simplex we define, as we did for the V ij , As shown in appendix A, the graph polynomials are now where |g| = det g, adj g = |g| g −1 is the adjugate matrix and g −1 the inverse matrix. The derivation proceeds by expressing the delta functions of (1.1) in Fourier form and integrating out all internal momenta. Only after this step has been performed are the Fourier integrals for the delta functions then evaluated. As the Gram determinant |G| is proportional to the squared volume of the simplex spanned by the independent momenta, the polynomial U describes the image of this squared volume under the mapping V ij → v ij . Alternatively, by the matrix tree theorem (see e.g., [34]), U is the Kirchhoff polynomial encoding the sum of spanning trees on the simplex. A second useful expression for F can be derived from Jacobi's identity, in combination with the relation This last relation follows from the linearity of the G ij in the V kl , as we saw in (2.3), and the mapping of G ij → g ij under V kl → v kl . The sums run over all k and l such that k < l, corresponding to all edges of the simplex. Substituting (2.9) into (2.7) then using (2.8), 10) or in terms of the raw momenta,

Parametric representations of the n-point correlator
To express correlators compactly, we extract the overall delta function of momentum conservation as We also define an arbitrary function f (v) whose arguments, denoted collectively by the vectorv, are the independent inverse Schwinger parameter cross ratios The simplex integral (1.1) can now be written in a variety of standard forms using the polynomials U and F defined in (2.7) or (2.11). Among the most useful are: 1. Schwinger parametrisation: Here, the v −d/2 ij factors in (2.1) cancel with those associated with U −d/2 via (2.2).

2.
Lee-Pomeransky parametrisation [35]: 3. Feynman parametrisation: 16) where ω = (n − 1)d/2 + n i<j α ij = (−∆ t + (n − 1)d)/2 and the constants κ ij ≥ 0 can be chosen arbitrarily provided they are not all zero. 2 If we choose all κ ij = 1 then the integration region is a simplex in the space spanned by the v ij . Alternatively, we can set a single κ ij to unity and the rest to zero which trivialises one of the integrations at the cost of obscuring permutation invariance.
These representations are all equivalent up to numerical factors; for clarity, we have re-absorbed these into the arbitrary functions. For analysing the action of weight-shifting operators and verifying the conformal Ward identities, we will focus exclusively on the Schwinger parametrisation (2.14). Nevertheless, the Lee-Pomeransky representation (2.15) is well suited for studying the Landau singularities, as discussed in appendix C, and the Feynman parametrisation (2.16) has the virtue that one integral can be performed using the delta function. 2 The Feynman parametrisation follows from the Schwinger parametrisation by setting vij = yij/σ subject to the constraint n i<j κijyij = 1. The U and F are homogeneous polynomials of weights (n − 1) and (n − 2) respectively, meaning that F(vij)/U(vij) = σF(yij)/U(yij) while the Jacobian can be evaluated as per appendix B of [29]. We then perform the scale integral over σ and relabel the yij → vij.
Example: As a quick illustration, the 4-point function in Schwinger parametrisation is where G ij = p i · p j is the 3 × 3 Gram matrix and g is its image The determinant is (2.19) and the equivalence of (2.7) and (2.11) can be verified directly.

The effective resistances
Thus far, we have expressed the Kirchhoff polynomial U as the determinant of g, the image under V ij → v ij of the Gram matrix, where p n is eliminated using momentum conservation. However, since all vertices of the simplex are equivalent, U ought also to be expressible in terms of the n × n matrixg corresponding to the image of the extended Gram matrix G ij = p i · p j for i, j = 1, . . . , n. This is simply the Laplacian matrix for the simplex: (2.20) As every row and column sum of the Laplacian matrix is zero its determinant vanishes identically, but its cofactors (i.e., signed first minors) are in fact all equal to U. To see this, consider the diagonal minor |g (n,n) | formed by deleting row n and column n then taking the determinant. Comparing with (2.5), we then see that |g (n,n) | = |g| = U. As any diagonal minor is equal to its cofactor, U is likewise the (n, n) cofactor. However, by elementary row and column operations one can show that all cofactors of the Laplacian matrix are equal. 3 3 For example, add one to every element ofgij then add all rows to the first row, and all columns to the first column. The top left entry is now n 2 while all remaining entries of the first row and column are n.
Taking the determinant, we first extract an overall factor of n from the top row, then subtract the new top row (whose leftmost entry is now n with all other entries one) from all the other rows. The resulting matrix has zeros in all entries of the first column apart from the top one which is n, and all entries other than those in the first row and column aregij (since we added one then subtracted one). The determinant ofgij plus the all ones matrix is therefore n 2 times the (1, 1) cofactor ofgij. Repeating the exercise for any other choice of row and column yields the same result with the corresponding cofactor, hence all cofactors are equal. Note this also shows that U is n −2 times the determinant of the Laplacian plus the all-ones matrix.
Thus, every cofactor (and every diagonal minor) is equal to U. Note this also confirms that our choice of eliminating p n in section 2.1 was immaterial. Let us now turn to an electrical analogy involving a simplicial network of resistors. Here, the Laplacian matrix naturally encodes the external current I i flowing into node i, since where v ij is the conductivity of the edge connecting nodes i and j and V j is the voltage of node j. Given this identification of the v ij with the conductivities, a natural question to ask is what are the corresponding effective resistances between the nodes? From Kirchhoff, the effective resistance s ij between nodes i and j is given by the ratio of minors [36,37] where |g (I,J) | indicates the minor formed by deleting the set of rows I and columns J then taking the determinant. Thus, |g (ij,ij) | is the second minor formed by deleting rows i and j as well as columns i and j, while |g (j,j) | is the first minor corresponding to deleting row and column j. From (2.20), the element v ij appears only in the row and columns (i, i), (i, j), (j, i) and (j, j) ofg. Forming the first minor |g (j,j) | by deleting row and column j, v ij then appears only once in the (i, i) position. The derivative ∂|g (j,j) |/∂v ij is thus equal to the second minor |g (ij,ij) | formed by additionally deleting row and column i in |g (j,j) |.
Since |g (j,j) | = |g| as above, we have where the second result follows immediately from (2.11). The Schwinger exponent in (2.14) thus encodes the effective resistances s ij between all vertices. Moreover, both U and F have been related to minors of the Laplacian: U is any diagonal first minor (or cofactor), while the coefficients of the F polynomial correspond to the second minors: from (2.11), the coefficient of Earlier, we noted that U = |g| is proportional to the squared volume of the (n − 1)simplex formed by the independent momenta under the map V ij → v ij . By the same token, each coefficient |g (ij,ij) | of the F polynomial thus corresponds to the image of |G (ij,ij) |, the second minor of the extended Gram matrix. However, this minor is simply the determinant of the reduced Gram matrix formed from all the momenta apart from p i and p j . Thus, the coefficient of V ij in the F polynomial is proportional to the squared volume of the (n − 2)simplex, formed from all the momenta except for p i and p j , under the map V ij → v ij . Similarly, the effective resistance s ij is proportional to the ratio of the squared volume of this (n − 2)-simplex to the squared volume of the full (n − 1)-simplex.

Re-parametrising the simplex
The original Schwinger parametrisation (2.14) is complicated by the non-linear dependence of the exponent on the v ij . As we saw in (2.23), however, the coefficients of the V ij = −p i ·p j in F/U are simply the effective resistances s ij between nodes. The next step is thus to invert the relation (2.23) to find the v ij in terms of the s ij , i.e., to express the conductivities in terms of the effective resistances. The simplex integral can then be fully re-parametrised in terms of the s ij , with the linearity of the Schwinger exponent giving a Fourier-style duality between the V ij and the s ij . This duality means that all momentum derivatives acting on the simplex, and all momenta, can be trivially exchanged for operators constructed from the s ij and derivatives ∂/∂s ij . The latter can then be integrated by parts. This strategy will repeatedly prove useful to us later.
We start by applying Jacobi's relation to further evaluate (2.23), where the matrices ∂g/∂v ij are easily evaluated from (2.5). Defining s ii = 0 for convenience (as we similarly defined v ii = 0) and re-arranging, we find where the diagonal entries reduce to (g −1 ) ii = s in . Inverting this matrix will now give us back the matrix g, as defined in (2.5), but re-expressed in terms of the s ij . The desired expressions for the v ij in terms of the s ij can then be read off from the appropriate entries. In fact, it is sufficient simply to know the determinant |g −1 |. For i < j < n, the (i, j) minor formed by deleting row i and column j of g −1 is |(g −1 ) (i,j) | = −(−1) i+j ∂|g −1 |/∂s ij , since from (2.25) s ij appears (with coefficient minus one-half) only in the positions (i, j) and (j, i) of the symmetric matrix g −1 . The off-diagonal entries of the adjugate matrix are thus Similarly, s in appears in every entry of the i th row of g −1 , and in every entry of the i th column. The coefficients for the off-diagonal entries are all one-half, while that for the diagonal entry is one. The derivative ∂|g −1 |/∂s in then corresponds to summing one-half times the signed minors both along the i th row and down the i th column such that the diagonal entry is counted twice. As g −1 is symmetric, however, these two sums are equal so we can simply sum along the i th row only with coefficient one. This gives where in the final step we used (2.5) to identify the sum of the first n − 1 entries along the i th row of the Laplacian as v in . The relation (2.27) thus holds for all i < j ≤ n.
To simplify this formula further, we observe that |g −1 | can be re-expressed in terms of the determinant of the (n + 1) × (n + 1) Cayley-Menger matrix, When evaluating the determinant, if we subtract the n th column from the first (n − 1) columns, and then the n th row from the first (n − 1) rows, we find This is our desired result expressing all the v ij in terms of the s ij , inverting (2.23). A few additional relations also follow. Jacobi's relation allows us to write since ∂m kl /∂s ij = 2δ i(k δ l)j from (2.29). As the off-diagonal entries of the Laplacian matrix areg ij = −v ij , this means that In fact, as indicated, this equation also holds for the diagonal elements with i = j ≤ n, since if we multiply the (n + 1) th row of m by column i of m −1 we find and since all row and column sums of the Laplacian matrix vanish, Thus, the n×n upper-left submatrix of the inverse Cayley-Menger matrix is minus one-half the Laplacian matrix, using either (2.23) or (2.32) to convert between the s ij and v ij . 4 The appearance of the Cayley-Menger matrix in our analysis is not a total surprise: in Euclidean distance geometry, the Cayley-Menger determinant is proportional to the squared volume of the simplex whose squared side lengths are given by the s ij . Here, the The determinant |g −1 | is thus proportional to the squared volume of the dual (n − 1)-simplex spanned by the independentp i , and by (2.31), the s ij are then the squared side lengths of this dual simplex. This provides an alternative (dual) geometrical interpretation for the s ij , besides the volume ratio discussed at the end of section 2.3.
Example: All the relations above are easily checked for small values of n, and the s ij are always rational functions of the v ij and vice versa. For the 4-point function, we find, e.g., (2.39)

Cayley-Menger parametrisation of the n-point correlator
Using the results above, we can re-express the various parametrisations of the simplex integral in terms of the effective resistances s ij . If we write the external momenta in Cayley-Menger form, the Schwinger exponent can be written as where the constant term n just produces an overall scaling which can be re-absorbed into the arbitrary function of cross-ratios. Moreover, as shown in appendix B.1, the determinant of the Jacobian is where the constant of proportionality can again be absorbed into the arbitrary function. The Schwinger form (2.14) of the simplex integral now becomes where the cross-ratiosv are rational functions of the s ij as defined via (2.13) and (2.32). An alternative expression can be given in terms of the Cayley-Menger minors, since from Jacobi's relation ∂|m| where |m (i,j) | is the minor formed by taking the determinant after deleting row i and column j. After absorbing numerical factors into the arbitrary function, this gives Analogous expressions can be obtained for the Lee-Pomeransky and Feynman representations (2.15) and (2.16), but the Schwinger parametrisations (2.43) and (2.45) are particularly convenient. As noted, the diagonal Schwinger exponent means differential operators in the momenta can easily be traded for equivalent differential operators in the s ij acting on the exponential, whose action can be further evaluated through integration by parts.

Weight-shifting operators
New weight-shifting operators now follow from the Cayley-Menger parametrisation (2.45). Acting on the Schwinger exponent (2.41) with an appropriate polynomial differential operator in the momenta pulls down a corresponding polynomial in the s ij . Choosing these polynomials to be the Cayley-Menger determinant and its minors, we obtain shift operators either increasing α or decreasing one of the α ij by integer units. We discuss these new operators in section 3.1, showing their effect is to increase the spacetime dimension by two while performing assorted shifts of the operator dimensions. Further weight-shifting operators can then be constructed by conjugating these operators with shadow transforms as shown in section 3.2. Explicit examples are given for the 3-and 4-point functions in section 3.3. Then, in section 3.4, we turn to analyse the weight-shifting operators proposed in [30]. These preserve the spacetime dimension but their action can nevertheless be understood using our parametric representations.

New operators sending d → d + 2
Let us begin with the V ij defined in (2.4) as our independent momentum variables. Acting on the Schwinger exponent (2.41), for any i < j allowing differential operators in the momenta to be traded for equivalent operators in the integration variables s ij . The shift operators then serve to pull down factors of |m (i,j) | and |m| respectively, thus their action is to send From (1.2) and (2.46), this is equivalent to shifting and so the superscript on S ++ ij is chosen to indicate its action of raising ∆ i and ∆ j by one. While the Cayley-Menger structure of S ++ ij and S is manifest in the variables V ij , where convenient these operators can easily be rewritten in terms of other scalar invariants (e.g., Mandelstam variables) via the chain rule. We will discuss this for 3-and 4-point functions shortly in section 3.3.
Alternatively, we can express S ++ ij and S in terms of vectorial derivatives with respect to independent momentum p i for i = 1, . . . n − 1. For S, we find where |G| = |p i · p j | is the Gram determinant and the µ i are Lorentz indices. (We leave all Lorentz indices upstairs to avoid confusion with the momentum labels, given we are working on a flat Euclidean metric.) The equivalence of (3.6) to (3.2) can established either by direct calculation for specific n, or else by considering its action on the Schwinger exponential of the representation (2.14). This representation is the appropriate one since, from (2.7), it involves only dot products of the independent momenta. Evaluating, we find where in the last step we used the Levi-Civita identity (n−1)! δ ..k n−1 to generate a product of determinants |g −1 ||G|, with the |G| then cancelling. Referring back to (2.14), since U −d/2 = |g| −d/2 we see the action of S is thus indeed to raise d → d + 2.
Through similar manipulations, we find (3.8) Relative to (3.6), the derivative ∂/∂p µ i i has been replaced by the dependent momentum since, relative to our previous calculation, the matrix element (g −1 ) ij i is missing in the product on the middle line. Instead of obtaining the full determinant |g −1 |, we then get the derivative of this with respect to the missing element. As in (2.28), we can now rewrite using (2.44) in the last step. The action of S ++ in in (3.8) on the exponential is thus to pull down a factor of v ij |g| −1 . From the representation (2.14), this has precisely the required action of sending α ij → α ij − 1 and d → d + 2.
Finally, since the choice of dependent momentum is immaterial, (3.8) generalises to where the hatspμ j j indicates that this factor and index are omitted in the antisymmetrised product, and we take p k j as the dependent momentum. In principle these last few derivations allow use of the s ij variables to be avoided entirely, although in practice the form of the operators (3.6) and (3.11) would be hard to anticipate.

Further shift operators from shadow conjugation
Additional d → d + 2 shift operators can now be constructed -at no expense -by conjugating S ++ ij and S by a pair of shadow transforms. This idea was discussed recently for d-preserving weight-shifting operators in [23].
In momentum space, the shadow transform ∆ i → d − ∆ i (leaving d invariant) simply corresponds to multiplying by p d−2∆ i i . First, notice that attempting to conjugate S ++ ij by shadow transforms on either of ∆ i or ∆ j has no effect: for example, the action of the operator corresponds to the successive parameter shifts which is equivalent to the action of S ++ ij alone. Further computations confirm that the shadow transform on ∆ i or ∆ j commutes with S ++ ij . However, we do obtain new operators if we shadow conjugate S ++ ij on any index k = i, j. For example, the action of corresponds to the successive parameter shifts Thus, in addition to the shifts produced by S ++ ij alone, we have also shifted ∆ k up by two. Shadow conjugating on further variables has the same effect, for example, We can also apply similar considerations to S. The action of corresponds to the shifts Shadow conjugating on further momenta p k leads similarly to shifting ∆ k → ∆ k + 2.
With all these operators obtained by shadow conjugation, notice we can always obtain an equivalent differential operator with purely polynomial coefficients (i.e., an operator in the Weyl algebra) by commuting the inner p d−2∆ k k shadow factors through the differential operator S or S ++ ij , whereupon all non-integer powers cancel with those from the outer shadow transform.

Examples at three and four points
To illustrate the general discussion in the two preceding subsections, let us now compute the explicit form of these d → d + 2 shift operators for 3-and 4-point functions.

3-point shift operators
For the 3-point function, it is convenient to use the three squared momentum magnitudes as variables. Defining via momentum conservation we have From (3.2), writing D i D j = D ij for short, we then find The various signs on the second line reflect our choice to use the Cayley minors in (2.45) and (3.2): had we used instead the cofactors or ∂|m|/∂s ij as per (2.44) then all signs would be the same. Generally, any overall coefficient in S or the S ++ ij can be eliminated by rescaling the corresponding prefactor in the definition of the simplex integral.
As noted in the introduction, these 3-point operators (and their shadow conjugates) are already known from the triple-K representation of the 3-point function. In [31,32], the Bessel shift operators where shown to act on the 3-point function by sending 22) or equivalently, and similarly under permutations. This is consistent with our analysis here, since and S ++ ij augments ∆ i and ∆ j by one and d by two. The R i operators are then their shadow conjugates as defined in (3.13), producing the expected shifts (3.14). Finally, does not appear explicitly in [31], but can be derived as follows. Writing the 3-point function as the triple-K integral where the final line follows by eliminating the sum of R i operators using the dilatation Ward identity. The effect of S is thus to increase d → d + 2 and all β i → β i − 1. All dimensions ∆ i = β i + d/2 are then preserved, consistent with (3.4).

4-point shift operators
The 3-point calculations above provide a first consistency check, but to obtain genuinely new shift operators we now turn to the 4-point function.
To write our results, we introduce the Mandelstam variables, where s 2 = (p 1 + p 2 ) 2 and t 2 = (p 2 + p 3 ) 2 , and define the derivative operators As per (3.4), the S ++ ij increase ∆ i and ∆ j by one and d by two, while S increases d by two. Following section 3.2, we can obtain further shift operators by shadow conjugation. As noted earlier, shadow conjugating each S ++ ij on either of the (i, j) indices has no effect: from (3.30), S ++ ij contains neither D i or D j hence these shadow factors commute through the operator. Instead, we must shadow conjugate each S ++ ij with respect to indices other than (i, j). At four points, once a pair of insertions (i, j) is specified, the remaining set also form a pair (k, l) = (i, j). Shadow conjugating each S ++ ij on the opposite pair (k, l) then definesS Expressed in terms of the variables (3.28), we find The action of each operatorS ++ ij is to shift d → d + 2, ∆ i,j → ∆ i,j + 1 and ∆ k,l → ∆ k,l + 2. This leaves β i and β j invariant while raising β k and β l by one. Heuristically, theseS ++ ij are then the 4-point generalisation of the 3-point R i operators in (3.21). Likewise, the S ++ ij in (3.30) leave β i and β j invariant but lower β k and β l by one, and represent the 4-point generalisation of the 3-point L i operators.
Besides shadow conjugating S ++ ij with respect to the pair (k, l), one can of course also conjugate with respect to only a single index k to find operators sending d → d + 2, ∆ i,j → ∆ i,j + 1 and ∆ k → ∆ k + 2 only. One can also apply the shadow conjugation procedure to the d → d + 2 operator S. All these operators can be evaluated similarly to theS ++ ij above and we will not write them explicitly. One case of particular interest, however, corresponds to acting withS ++ ij followed by S ++ ij , which produces an overall shift of d → d + 4 while increasing all operator dimensions by two. The same shift is produced when acting with these operators in the opposite order (remembering to shift β k,l → β k,l −1 inS ++ ij to account for the prior action of S ++ ij ). By subtracting, we then obtain a shift operator of only second order in derivatives, rather than fourth. For example, and so D 56 shifts d → d + 4 while sending all ∆ i → ∆ i + 2 and preserving the β i . Finally, let us emphasise that the action of all these shift operators is general and not in any way tied to the simplex representation: any solution of the 4-point conformal Ward identities is mapped to an appropriately shifted solution. 5 We have confirmed this explicitly by computing all the relevant intertwining relations between the shift operators in this section and the conformal Ward identities, whose form in Mandelstam variables can be found in e.g., [11,23]. Thus, for example, where K({∆ i }, d) represents schematically any of the special conformal or dilatation Ward identities with the operator and spacetime dimensions as indicated. Applying this relation to any CFT correlator with dimensions ({∆ i }, d), the right-hand side vanishes and the lefthand side then indicates that the action of S ++ 12 produces a solution of the shifted Ward identities. Intertwining relations such as these 6 allow the shift action of operators to be established independently of any integral representation for the correlator.

Operators preserving d
A different class of weight-shifting operators that preserve the spacetime dimension d while shifting the ∆ i was identified in [30]. In momentum space, these operators have been 5 Up to a technical caveat (common to all shift operators) that where divergences occur, one must work in a suitable dimensional regularisation scheme. In some cases the shift operator then only yields the leading divergences of the shifted correlator, see the discussion in [23]. 6 More generally, the right-hand side of (3.35) could feature any operator in the left ideal of the conformal Ward identities, since all that matters is that it vanishes when acting on a solution with dimensions ({∆i}, d).
applied to de Sitter correlators in [11,12]. With the aid of shadow conjugation, we can write them in the compact form [23] where β i = ∆ i − d/2 and 1 ≤ i < j ≤ n − 1 so p n is taken as the dependent momentum.
Their action is to shift In this section, our goal is to understand the action of the simplest of these operators, W −− ij , from the simplex perspective. The action of the others then follows via shadow conjugation, or else can be shown explicitly: for example, we analyse W −+ ij in section 5.2. We begin by writing the Schwinger exponential (2.7) in the form As only the independent momenta feature in this last expression, the action of W −− ij on the Schwinger exponential can be rewritten as a differential operator in the s kl . We will do this in several steps. First, notice that where in the second line s ik vanishes for i = k. This gives and hence To rewrite these momentum dot products as derivatives with respect to the s kl , we now rearrange this sum as follows. Using momentum conservation p 2 k = − n l =k p k · p l , for any generic coefficient A k such that A n = 0, we have where in the final line the sum runs up to n. Setting A k = s ik − s jk − s in + s jn , we find In the second line here, we exchanged p k · p l for ∂ s kl using the first expression in (3.38). The change of variables from ∂ s kl to ∂ v ij in the third line then comes from the Jacobian evaluated in appendix B.2, and in the final line we used (2.23). The action of W −− ij on the full simplex integral (2.14) now follows. First, the outer factor of |g| d/2 in (3.43) cancels with the factor U −d/2 = |g| −d/2 in (2.14). Integrating by parts with respect to v ij , assuming the boundary terms vanish, 7 the derivative then acts on the prefactors as Here, the terms coming from ∂ v ij hitting the cross-ratios (2.13) inside the arbitrary function f (v), as well as those from hitting v −α ij −1 ij , have been repackaged in the form v −1 ijf (v) for some new function of cross-ratiosf (v). Thus, overall, we find The action of W −− ij on the simplex is therefore to send α ij → α ij + 1, up to changes of the arbitrary function. The latter is of no account as far as mapping one solution of the conformal Ward identities to another is concerned. 8 From (1.2), we now confirm that sending α ij → α ij + 1 while keeping the remaining α kl fixed is equivalent to sending ∆ i → ∆ i − 1 and ∆ j → ∆ j − 1 while preserving d, in perfect agreement with (3.37).

Verifying the conformal Ward identities
In this section, we prove that the parametric representation of the simplex integral (2.14) satisfies the conformal Ward identities for any arbitrary function of cross-ratios. The corresponding result for the vectorial simplex integral (1.1) was established in [28,29]. Working purely in momentum space, our approach is to show that the action of the Ward identities on the simplex integral reduces to a total derivative. With a degree of hindsight, the structure of this total derivative, obtained in (4.24), can also be understood from somewhat simpler position-space arguments. We will return to these in section 5.1.
As the dilatation Ward identity can be verified by power counting, we focus on the special conformal Ward identities treating p n as the dependent momentum. As a first step, we rewrite the action of each individual term in (4.1) on the Schwinger exponential as an equivalent differential operator in v ij . From (2.7), we have maps us from a finite correlator to a singular one, corresponding to a solution of the conditions d + n i=1 σi(∆i − d/2) = −2k for some non-negative integer k and a choice of signs {σi} ∈ ±1, see [28]. In such cases, the arbitrary functionf (v) vanishes. In dimensional regularisation, this zero then cancels the pole coming from the divergent correlator such that the result is finite, see [23].

Using (2.25) for the inverse metric and the manipulation (3.42), this last expression can be rewritten analogously to (3.43):
Next, we must deal with In the first line here, notice we extended the sum over k to run up to n, which is possible since the additional term with k = n vanishes as s nn = 0. To get the second line, we then re-expressed the terms for which k > l by swapping k ↔ l. For convenience, it is useful to defineĝ effectively extending the (n − 1) × (n − 1) matrix g −1 ij to an n × n matrixĝ −1 ij by adding a final row and column of zeros. This allows us to compactly rewrite (4.5) and (4.6) as (4.8) Here, the sum over l for the p k · p l terms in (4.5) has similarly been extended to run up to n, noting the additional l = n term vanishes. We then replaced p k · p l by a derivative with respect to s kl using (3.38). The result now simplifies further upon exchanging whereg ab is the Laplacian matrix (2.20) and the Jacobian is evaluated in appendix B.2. First, we write where the sum over k < l of (k, l)-symmetric terms has been rewritten as half the sum over all k and l, noting the terms with k = l explicitly cancel. The final two terms now vanish since all row and column sums of the Laplacian matrixg are zero: To derive this, note the sum over k restricts to k ≤ n − 1 from (4.7), then for i, j ≤ n − 1 we haveg ikĝ −1 kj = g ik g −1 kj . For i = n, j ≤ n − 1 we useg nkĝ −1 kj = − n−1 l g lk g −1 kj and for j = n and any i the sum vanishes from (4.7). With the aid of this identity, we then find where we usedg ab = −v ab for a < b to obtain the Euler operator θ v ab = v ab ∂ v ab .
Assembling the pieces above, the action of the conformal Ward identity is now using (1.2) in the last line. Finally, we need two further identities:  then use the chain rule, which for i, j, k, l ≤ n − 1 gives Inserting these into (4.17), the sum over a = b can be extended to run over all a, b since the term with a = b vanishes. The only non-cancelling term is then −g −1 jj = −s jn as required. For the second identity in (4.16), we use (2.25) to rewrite The sum over a = b can then be extended to run over all a, b as the term with a = b cancels, after which the first and the last terms cancel and the result follows.
With the aid of the identities (4.16), we find that (4.15) becomes where Ω = n k<l v −α kl −1 kl . Recalling that the simplex representation (2.14) is since whenever the index a appears in a cross ratiov [acde] = v ac v de /v ad v ce it enters with equal weight in the numerator and the denominator producing a cancellation. Acting with the Ward identity thus yields a total derivative: The boundary terms vanish under reasonable assumptions: for generic momentum configurations with non-vanishing Gram determinant, the upper limit is suppressed by the decay of the Schwinger exponential; the lower limit is zero provided v −α ab ab f (v) vanishes as v ab → 0, which is satisfied whenever the simplex representation itself converges. The simplex integral thus solves the special conformal Ward identity.

Insight from position space
Thus far, our analysis has been entirely in momentum space. However, as noted above, the form of the total derivative produced by the action of the special conformal Ward identity in (4.24) can also be understood through independent position-space arguments. We present these in section 5.1. Then, in section 5.2, we show how similar position-space arguments can be applied to verify the action of d-preserving shift operators such as W −+ 12 .

The conformal Ward identities
To Fourier transform the simplex representation (2.14) to position space, we compute where x ij = x i − x j , and for the Gaussian integral over momenta we completed the square: The numerical factor from the integration can then be re-absorbed into the arbitrary function by setting (4π and hence the simplex representation in position space is If the arbitrary functionf (v) is a product of powers, this expression reduces to the conformal correlator n i<j x 2α ij ij where theα ij satisfy j =iα ij = −∆ i . More generally, whereverf (v) admits a Mellin-Barnes representation, we recover n i<j x 2α ij ij times a function of position-space cross ratios as shown in [29]. However, the most straightforward way to check that (5.4) solves the conformal Ward identities is to note that, when acting on a function F = F ({x 2 kl }) of the squared coordinate separations, It then follows that where in the last line we integrated by parts 10 then used (1.2). The middle line here accounts for the form of the total derivative we found earlier in (4.24). Multiplying by 9 Recall the analogous relation in a resistor network of simplex topology, namely, that the power dissipated is n i<j vij(Vi − Vj) 2 = n i,jg ij ViVj, where vij is the conductivity and Vi the voltage at node i. 10 As previously, the boundary terms vanish provided v −α kl klf (v) as v kl → 0. −i and Fourier transforming, the first line yields the momentum-space conformal Ward identity acting on the momentum-space simplex representation (i.e., the left-hand side of (4.24)), while the middle line yields where in the second line we evaluated the momentum derivative of the exponential and pushed the factors of Ω = k<l v −α kl −1 kl , f (v) and v ij inside the v ij -derivative which cancels the ∆ i term via (1.2). In the final line, we extended the sum over i to run up to n by replacing g −1 ia withĝ −1 ia and combined it with the sum over j. Up to a relabelling of indices, this final line is now the total derivative appearing on the right-hand side of (4.24).
The manipulations above illustrate a general theme: given the simplicity of the positionspace simplex representation (5.4), it is often profitable to work with the position-space equivalents of differential operators in order to evaluate their action in terms of the v ij variables. Both sides can then be Fourier transformed back to momentum space in order to deduce the action of the corresponding momentum-space operator on the momentumspace simplex in terms of the v ij variables. In many cases this is more straightforward than working in momentum space throughout.

12
As a further illustration of this approach, let us evaluate the action of the shift operator W −+ 12 defined in (3.36). After expanding out the derivative, this operator can easily be Fourier transformed to position space where it reads Acting on a function F = F ({x 2 kl }) of the squared coordinate separations, we find via the chain rule Acting on the Schwinger exponent appearing in the position-space simplex representation (5.4), this can be translated into v ij -derivatives as Integrating by parts, we find We now rewrite the first part of the last line as where in the final step we rewrote ∂ v 12 θ v 12 = (θ v 12 + 1)∂ v 12 and used n i =2 θ v 2if (v) = 0, as follows from (4.23), along with (1.2) with ∆ 2 = β 2 + d/2 to replace (5.13) Substituting (5.12) into (5.11) and making further use of (5.13), we find the result (5.14) Equivalently, acting on the position-space simplex with W −+ 12 corresponds to acting on the arbitrary functionf (v) with the operator The same remains true when we Fourier transform back to momentum space, giving Finally, it remains to check that the action ofW −+ 12 on the arbitrary function produces the required shift in dimensions ∆ 1 → ∆ 1 − 1 and ∆ 2 → ∆ 2 + 1. Since where h(v) is also function of the cross ratios, we see that where h i (v) and h ij (v) are specific functions of the cross ratios. Each term in the first sum then corresponds to a simplex integral with the shifts while each term in the second sum corresponds to a simplex integral with the shifts From (1.2), both (5.19) and (5.20) correspond to shifting ∆ 1 → ∆ 1 − 1 and ∆ 2 → ∆ 2 + 1 leaving all other operator dimensions fixed. The action of W −+ 12 on the simplex thus produces an appropriately shifted simplex integral, whose function of cross ratios is obtained through the action of the operator (5.15).

Discussion
Our analysis has furnished useful parametric representations for the general momentumspace conformal n-point function. Starting from the generalised simplex Feynman integral of [28,29], we showed how all graph polynomials can be obtained from the corresponding Laplacian matrix, or the Gram matrix to which it reduces once momentum conservation has been enforced. With the graph polynomials to hand, all the usual scalar parametrisations of Feynman integrals can be adapted to represent the simplex solution. Only n(n − 1)/2 integrals over Schwinger parameters remain to be performed -one for each leg of the simplex -in contrast to the (n − 1)(n − 2)d/2 scalar integrals we started with.
Building on the analogy between Feynman graph polynomials and those of electrical circuits, we then formulated a second class of parametric representations. For these, the integration variables represent the effective resistances between vertices of the simplex, rather than the conductivities (i.e., the inverse Schwinger parameters) used previously. This change of variables immediately diagonalises the Schwinger exponential, expressing the n-point function as a standard Laplace transform of a product of polynomials raised to generalised powers. These polynomials correspond to the determinant and first minors of the Cayley-Menger matrix for the simplex, which plays an analogous role to the Gram matrix for this second class of parametrisations. From the form of these polynomials, new weight-shifting operators can immediately be constructed to raise the power of these polynomials, with further shift operators following by shadow conjugation. Besides shifting the scaling dimensions of external operators, these new weight-shifting operators raise the spacetime dimension by two. They therefore generalise the 3-point shift operators of [31,32] to n-points, and constitute a distinct class of operators to those identified in [30].
Our results suggest several interesting directions for further pursuit: • Given we now have weight-shifting operators that both preserve and raise the spacetime dimension, is it also possible to construct operators that lower the spacetime dimension? One approach we have explored, explained in appendix D, is to find so-called Bernstein-Sato operators which act to lower the powers to which the various polynomials of interest are raised. In this case, the relevant polynomials are the Cayley-Menger determinant and its minors appearing in the parametrisation (2.45). We found, for example, that replacing v ij → ∂ s ij in the Kirchhoff polynomial U = |g| yields an operator which lowers by one the power to which the Cayley-Menger determinant is raised: For the simplex representation (2.45), a is the parameter α given in (2.46) and so lowering α by one corresponds to sending d → d − 2 if all the operator dimensions are kept fixed. In principle, one would then integrate by parts to obtain an operator acting solely on the Schwinger exponential, which, due to its diagonal structure, could be translated into a differential operator in the external momenta. In practice, however, this approach is complicated by the presence of all the remaining powers of Cayley-Menger minors present in (2.45).
• In sections 4 and 5.1, we saw how the action of the special conformal Ward identity on the simplex reduces to a total derivative. This followed directly from the scalar parametric representation, without any recourse to the recursive arguments developed in [28,29]. Nevertheless, these arguments, and the recursion relation between n-and (n + 1)-point simplices on which they are based, are of considerable interest in their own right and could be reformulated in the scalar-parametric language used here. The deletion/contraction relations of graph polynomials (see, e.g., [34]) and Kron reduction, corresponding to taking the Schur complement of a subset of vertices in the simplex Laplacian (see e.g., [40]), may also yield relevant identities.
• Starting from the general simplex solution, the arbitrary function of momentumspace cross ratios can be restricted by imposing additional conditions of interest: for example, dual conformal invariance [20,[41][42][43], or the Casimir equation for conformal blocks. For such investigations, the connection with position-space developed in section 5 provides a very simple link between the action of a given differential operator in the external momenta or coordinates, and its corresponding action on the arbitrary function of the simplex representation.
• For holographic n-point functions, bulk scalar Witten diagrams have the interesting property that their form is invariant under the action of a shadow transform on any of the external legs. In momentum space, shadow transforming the operator O i corresponds to multiplying the correlator by p −2β i i , where β i = ∆ i − d/2, which has the effect of replacing β i → −β i in the bulk-boundary propagator z d/2 p β i i K β i (p i z). It would be interesting to understand the restriction this condition places on the function of cross-ratios appearing in the simplex representation.
• Finally, the parametric representations we have developed may provide a useful starting point for the construction of general spinning n-point correlators via the action of spin-raising operators [11,12,30], and for bootstrapping cosmological correlators in de Sitter spacetime. [44]. Labelling the vertices of the simplex by i = 1, . . . , n, and the (directed) legs by a = 1, . . . , N where N = n(n − 1)/2, we introduce the incidence matrix Next, exponentiating all propagators of internal momenta (labelled by their leg indices) using the Schwinger representation (2.1), we find 11 Evaluating the q a integrals by completing the square and using (A.4) now gives Since the Laplacian matrix has no inverse, to compute the y k integrals we must first shift This transformation has unit Jacobian, but moreover greatly simplifies the exponent. Since all row and column sums of the Laplacian matrix vanish, and using these identities we then find In the final line here, all the z k · z n and z 2 n terms cancel while the Laplacian matrixg kl reduces to g kl for k, l = 1, . . . , n. The z n integral now gives the overall delta function of momentum conservation which we strip off to obtain the reduced correlator (2.12). The remaining z k integrals can be evaluated by completing the square, given that the inverse g −1 kl exists. This yields our desired result, where the constant can simply be re-absorbed into the arbitrary function f (v). Rewriting the product of legs a as a product over vertices i < j and replacing p k · p l with the Gram matrix G kl , we recover precisely (2.14) with graph polynomials (2.7).

B Jacobian matrix
In this appendix, we compute the Jacobian matrix for the change of variables from v ij to s ij . In section B.1 we evaluate the Jacobian determinant, then in section B.2 we give expressions for its matrix elements enabling conversion between partial derivatives.

B.1 Jacobian determinant
Our first goal is to derive the relation (2.42) for the Jacobian determinant, namely where the constant of proportionality is not required since it can be re-absorbed into the arbitrary function f (v). For small values of n this result can be verified by direct calculation, and the exponent is simply fixed by power counting, but our aim is nevertheless to prove this relation for general n.
We start by noting can be re-expressed as a product of three square matrices of dimension n(n − 1)/2. Each of the index pairs (p, q) and (r, s) is replaced by a single index running over the n(n − 1)/2 independent entries of the (n − 1) × (n − 1) symmetric matrix g, while (i, j) and (k, l) are each replaced by a single index running over the n(n − 1)/2 edges of the simplex. Noting the elements of g are linear in the v, the matrix determinant |∂g/∂v| evaluates to a nonzero constant. On taking the determinant of (B.2), we find hence it suffices to show that ∂ 2 ln |g| ∂g ∂g ∝ |g| −n .

(B.4)
This relation in fact holds for any invertible symmetric square matrix g of dimension n − 1.
To see this, from Jacobi's relation we have Diagonalising g via an orthogonal matrix O, where the last factor is just ∂Λ/∂g. Regarding this as a matrix product, the first and last matrices depend only on O and are inverses of each other. On taking the determinant of the right-hand side, their contributions therefore cancel giving We thus only need to evaluate the latter determinant for the diagonal matrix Λ. From (B.5), the Hessian is nonzero only when the index pairs are equal (p, q) = (r, s), and is thus diagonal when regarded as a square matrix of dimension n(n − 1)/2: The determinant is now since each eigenvalue Λ pp appears a total of n times along the diagonal: for example, Λ 11 appears quadratically in the position (1, 1) and then linearly in each of the (n − 2) entries indexed by (1, q) for q = 2, . . . n − 1. We have thus established (B.4), and hence (B.1).

B.2 Matrix elements
We now compute the elements of the Jacobian matrix required to establish the relation which we used in (3.43). Starting with (2.24), where since g ab is linear in the v ij its second derivative vanishes. Using then gives (B.14) For i, j, k, l = n, we can evaluate this as where we used the symmetry of the inverse matrix g −1 ij , and in the last line we used (2.25). For j = n but i, k, l = n, which is equivalent to (B.15) setting j = n. The same also holds for l = n but i, j, k = n due to the symmetry of (B.14). Finally For completeness, we can also calculate the inverse Jacobian by similar means: Apart from the final (n + 1) th row and column, the inverse Cayley-Menger matrix is minus one half the Laplacian matrixg ij as we showed in (2.34) and (2.36). This gives and if i = k and j = l,

C Landau singularities
The Landau singularities of the simplex integral are best studied in the Lee-Pomeransky representation (2.15). They follow from solving simultaneously for all v ij the conditions Here, the first Landau equation stipulates the vanishing of the Lee-Pomeransky denominator, while the second requires that this vanishing is either a double zero (for v ij = 0), corresponding to a pinching of the v ij integration contour between two converging singularities of the integrand, or else a pinch of the integration contour between a singularity and the end-point of the integration (v ij = 0). The second condition thus ensures the singularity generated by the vanishing denominator cannot be avoided by a deformation of the integration contour. Where the Landau conditions have more than one solution, the solution with the greatest number of v ij = 0 is referred to as the leading singularity. An important feature of the U polynomial (2.7) is that it is multilinear in the v kl : from the determinant structure one sees that all the quadratic v 2 kl terms cancel, and that no higher powers can appear since v kl enters only in the row/columns (k, k), (k, l), (l, k) and (l, l). Alternatively, this result follows from the matrix tree theorem where the Kirchhoff polynomial U is the generator of spanning trees on the simplex. Since U is also homogeneous of degree (n − 1), it follows that k<l ∂U ∂v kl v kl = (n − 1)U. (C.2) We now find and so a solution of the first Landau condition for all k < l is v kl = λ V kl = −λ p k · p l and |G| = |p k · p l | = 0 (C.4) for some constant λ. Evaluating the second Landau condition on this solution ( * ) of the first gives v ij ∂ ∂v ij (U + F) * = v ij ∂U ∂v ij + v ij k<l ∂ 2 U ∂v ij ∂v kl V kl * = (λ + n − 2)λ n−2 V ij ∂U| v→V ∂V ij (C.5) using again the homogeneity of U. The second Landau condition is thus solved for all i, j when λ = 2 − n, and indeed this is the leading singularity since the v kl are generically nonzero. Returning to (C.3), on the solution ( * ) we have (U + F) * = (2 − n) n−2 |G| = 0, (C. 6) so to solve the first Landau condition we do indeed need the Gram determinant to vanish. Generally this requires analytic continuation to non-physical momentum configurations, since the only physical configurations (in Euclidean signature) for which the Gram determinant vanishes are collinear ones, and on physical grounds there are no collinear singularities. There is no contradiction here since the Landau equations are necessary, but not sufficient, conditions for a singularity.

D Bernstein-Sato operators
In this appendix, we construct a Cayley-Menger analogue of the classic identity det(∂)(det X) a = a(a + 1) . . . (a + n − 1)(detX) a−1 , (D.1) where X = (x ij ) is an n × n matrix of independent variables and ∂ = (∂/∂x ij ) is the corresponding matrix of partial derivatives. For proofs and variants of this identity, traditionally attributed to Cayley, see, e.g., [45,46]. From a modern perspective, (D.1) is an example of a Bernstein-Sato operator, a differential operator whose action lowers the power a to which some polynomial of interest is raised, generating in the process an auxiliary polynomial in a known as the b-function [47]. Thus we have where for (D.1), B f = det(∂), f = det X and b f (a) = a(a + 1) . . . (a + n − 1). In the following, we construct analogous operators for the Cayley-Menger determinant and other polynomials arising in our parametric representations (2.45) and (2.14). Such relations are potentially a source of new weight-shifting operators, see e.g., [48,49]. Our starting point is the observation that The operator B |m| thus corresponds to evaluating the Kirchhoff polynomial U = |g| and replacing all v ij → ∂ s ij to generate a polynomial differential operator in the ∂ s ij . We have verified (D.4) by direct calculation for matrices up to and including n = 5. Moreover, the leading behaviour at order a n−1 follows by noting that such terms can only arise from all n − 1 partial derivatives in B |m| hitting a power of |m| rather than a derivative of |m|.
Using (2.32) in the form ∂ s ij |m| a = av ij |m| a along with (2.31), then gives B |m| |m| a = a n−1 |m| a |g| + O(a n−2 ) = (−1) n 2 n−1 a n−1 |m| a−1 + O(a n−2 ) (D. 5) in agreement with (D.4). 12 Similarly, we find B |g| = (|m|) The b-function here is the same as that in (D.4), and the leading a n−1 behaviour can be understood via the analogous argument to that in (D.5). We note the result (D.7) is equivalent to Theorem 2.15 of [45], since |m| = |m (n+1,n+1) | − |m (n+1,n+1) + J| where J is the n × n all-1s matrix, and |m (n+1,n+1) | is the Cayley-Menger minor formed by deleting the final row and column consisting of 1s and 0s. In addition, we find with the same b-function (D.11), but this operator does not appear to act simply (for n > 3) on (∂ s ij |m|) a , in contrast to (D.10). Finding a Bernstein-Sato operator for (∂ s ij |m|) a would be useful since by (2.44) this corresponds to the Cayley-Menger minors featuring in (2.45).
In principle, given a Bernstein-Sato relation such as (D.4), one might hope to apply it inside the parametric representation (2.45) and integrate by parts to obtain an operator acting solely on the Schwinger exponential. Since the exponential is diagonal in the representation (2.45), the result could then be translated to a differential operator in the external momenta. This would then yield a new weight-shifting operator.
In practice, however, we must account for all the other powers of Cayley-Menger minors present in (2.45), as well as the arbitrary function. Either we must find a modified Bernstein-Sato operator that acts appropriately on the entire non-exponential prefactor in (2.45), which seems hard to do, or else we must find some means of removing and then restoring these other factors. The Cayley-Menger minors, for example, can be removed and then restored via a conjugation Ω B |m| Ω −1 where Ω = n i<j |m (i,j) | −α ij −1 . After multiplying out, however, this conjugated operator is not in the Weyl algebra (i.e., is non-polynomial in the s ij and their derivatives) and so does not trivially translate into an operator in the external momenta. On the other hand, if we include additional powers of the |m (i,j) | on the left, so as to recover an operator in the Weyl algebra, besides lowering α in (2.45) we also lower some of the α ij . The operator then does not lower the spacetime dimension d. Thus we have not succeeded in finding new weight-shifting operators via this route, though with some variation the method might yet be successful.