Off-shell CHY amplitudes and Feynman graphs

A polynomial form is established for the off-shell CHY scattering equations proposed by Lam and Yao. Re-expressing this in terms of independent Mandelstam invariants provides a new expression for the polynomial scattering equations, immediately valid off shell, which makes it evident that they yield the off-shell amplitudes given by massless 𝜙3 Feynman graphs. A CHY expression for individual Feynman graphs, valid even off shell, is established through a recurrence relation.


Introduction
Cachazo, He and Yuan (CHY) [1][2][3] have shown that the scattering equations, originally introduced by Fairlie and Roberts [4][5][6], describe the kinematics of massless particles in an arbitrary space-time dimension, D, by proposing remarkable formulae for tree amplitudes, which have been proved for φ 3 theory and for gauge theory [7]. In a subsequent paper [8], we showed how to re-express these in terms of a polynomial form for the scattering equations. Here we show how to write this polynomial form so that it is valid when the external particles are off shell, by expressing it in terms of the independent Mandelstam variables associated with a particular cyclic ordering of the external particles, rather than the set of Mandelstam variables for all possible channels and orderings used in [8]. A particular use for these new off-shell polynomial scattering equations is in establishing recurrence relations and we apply them to establish CHY formulae for individual off-shell Feynman diagrams. These formulae for individual diagrams may help illuminate the underlying structure of the CHY formalism.
Consider N massless particles, labelled by a ∈ A = {1, 2, . . . , N }, with momenta, k a , a ∈ A, which are on-shell, k 2 a = 0, and satisfy momentum conservation, a∈A k a = 0. Introduce a variable z a ∈ C for each a ∈ A. Then the scattering equations are the N equations f a (z, k) = 0, a ∈ A, where (1.1) This system of N equations is Möbius invariant, and consequently the functions f a satisfy three identities, a∈A f a (z, k) = 0; a∈A z a f a (z, k) = 0; a∈A z 2 a f a (z, k) = 0, (1.2) JHEP04(2020)141 so that only N −3 are linearly independent. They are equivalent [8] to the N −3 polynomial scattering equationsĥ m (z, k) = 0, wherê the sum is over all N !/m!(N − m)! subsets S ⊂ A with m elements, and We shall refer to theĥ m (z, k), 2 ≤ m ≤ N − 2, as the scattering polynomials. In terms of these, the CHY formula for the sum of planar tree amplitudes in φ 3 is where dω = dz i dz j dz k /(z i − z j )(z i − z k )(z j − z k ) is the Möbius invariant measure corresponding to fixing z i , z j , z k , and the integral encircles the zeros ofĥ m (z, k), 2 ≤ m ≤ N − 2.
Taking z 1 → ∞, z 2 = 1, z N = 0, we have (1.6) whereh m (z, k) = lim and the integration contour now encircles the zeros ofh m (z, k), 1 ≤ m ≤ N −3, and amounts to a sum over those zeros, of which there are (N − 3)!. Using the polynomial form of the equation is less singular and more stable for numerical calculations in particular. Note that the polynomialsĥ m are invariant under simultaneous permutations of the momenta, k a , and the variables, z a , whereas the integral depends on the specific choice of the ordering of the momenta, k 1 , k 2 , . . . , k N , and to get the complete tree amplitude one has to sum over the possible orderings. The definition (1.3) ofĥ m involves the 2 N − 2N − 2 Mandelstam invariants but, given momentum conservation and the massless condition, just 1 2 N (N −3) of these can be taken to be independent, provided that N −1 ≤ D. We can re-expressĥ m in terms of an independent set of invariants but not while maintaining its manifest permutation symmetry. We can pick an independent set of momenta as follows. Given the specified order of the momenta, k 1 , k 2 , . . . , k N , consider consecutive sets, where I − 1 is identified with N when I = 1, [I, J] o is the complement of {I − 1, I, J, J + 1} in A, and Π n V is the symmetric function, where V ⊂ A and n ≤ |V |. Thus, Π 0 V = 1 and Π |V | V = i∈V z i . We use (1.9) to defineh m as the off-shell polynomial form of the scattering equations, using (1.3) only on shell. Note thath m =ĥ m in general only if k 2 a = 0, a ∈ A. Both h m , 2 ≤ m ≤ N − 2, and the value of the integral expression (1.5) for A N only involve the consecutive invariants, k 2 [I,J] , 1 ≤ I < J < N . This implies that, if we replaceĥ m withh m in (1.5), the integral gives the sum of planar tree diagrams for massless φ 3 theory, even when the external momenta are off shell.
To show this, note that if the momenta, k a , 1 ≤ a ≤ N, satisfy momentum conservation, but are not necessarily massless, take another set of momenta, a , 1 ≤ a ≤ N, satisfying momentum conservation, and such that 2 [I,J] = k 2 [I,J] , 1 ≤ I < J < N , and 2 a = 0, 1 ≤ a ≤ N . Thenh m as defined by (1.9) is the same for {k a } and for { a }. For { a }, but not {k a }, h m as defined by (1.9) is the same asĥ m as defined by (1.3). Then, forĥ m with momenta { a }, (1.5) gives a sum of planar tree diagrams, each of which is a product of propagators 1/ 2 U , where U is a consecutive subset of A. Hence, this equals the corresponding product of propagators 1/k 2 U . It follows that (1.11) gives the sum of planar diagrams with the off-shell momenta.
In section 2, we also show that the off-shell scattering equations derived in this way, h m = 0, withh m as in (1.9), are the polynomial form of the off-shell scattering equations proposed by Lam and Yao [9].
In section 3, as an application of these off-shell scattering equations we establish CHY integrals associated for individual Feynman tree diagrams using an off-shell recurrence relation. Associating as usual the momentum k a and complex variable z a with the a-th external leg of an individual Feynman diagram, ∆, we define the corresponding integrand by first associating a cross-ratio with each internal line or propagator of the diagram, in the way originally introduced by Koba and Nielsen [10]. Thus, 12) JHEP04(2020)141 Figure 1. Internal line of a Feynman diagram. and taking u ∆ to be the product of such cross-ratios for the N − 3 propagators of the diagram. The corresponding integral is then  [11][12][13][14][15].]

Off-shell scattering equations
To rewriteĥ m , as defined by (1.3), in terms of independent invariants we need to write each k 2 S in terms of k 2 [I,J] , 1 ≤ I ≤ J < N . First note that, for any three vectors K 1 , K 2 , K 3 , From this it follows by induction that, for vectors K i , 1 ≤ i ≤ 2n − 1,

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Now, we can write a given S ⊂ A * as the union of consecutive subsets S r , r = 1, 3, . . . , 2n−1, of A * , and denote the indices between S 2r−1 and S 2r+1 by S 2r , 1 ≤ r ≤ n − 1, which is nonempty. Taking K r = k Sr , 3), if 1 ≤ I < J < N , and the consecutive subset [I, J] ⊂ A * occurs as S i,j , with i, j both odd, z S contains a factor z I z J and z S contains a factor z I−1 z J+1 , while if i, j are both even, z S contains a factor z I−1 z J+1 , and z S contains a factor z I z J ; if i is odd and j is even z S contains a factor z I z J+1 and z S contains a factor z I−1 z J , while if i is even and j is odd it is the other way round. If i and j have different parity, the term has a negative sign.
provided that k 2 a = 0 for each a ∈ A. Relaxing this condition, we take the vanishing of (2.5) as the definition of the off-shell polynomial scattering equations because, inserted in (1.11), they give the correct off-shell tree diagrams, as argued in the Introduction. To fix the Möbius invariance, let z 1 → ∞, z 2 = 1, z N = 0, and now write

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For the simplest case, N = 4, writing z 3 = x, and s ij = k 2 {i,j} , we only have h 1 , whereas the 'on-shell' polynomialh 1 , defined by (1.7), is (2.12) is equivalent to (2.5) as a definition of the off-shell polynomial scattering equations. Note that, if |S| = 1 or 0, σ S , as defined by (2.13), equals 0. To see how the original form of the scattering equations (1.1) should be modified in order to work off shell, we reverse the line of argument that was used in the derivation of the polynomial form in [8]. Consider (2.14) Thus Thus the off-shell polynomial scattering equations are equivalent to the identical vanishing of the right hand side of (2.15). Since this only has single poles and vanishes for large z, this is equivalent to the vanishing of its residue at each z = z a , that is to This is the form of the off-shell scattering equations proposed by Lam and Yao [9]. It follows from our discussion that they do indeed yield the correct off-shell amplitudes. [For other discussion of off-shell amplitudes see [16] and [17].] The Möbius invariance of the system of off-shell scattering equations follows from the Möbius invariance of the condition that (2.15) vanish identically. It can also be verified directly on (2.16). Reflecting this invariance, f a , a ∈ A, satisfy the same relations (1.2) that the f a , a ∈ A, do.

An example
As an example of the recurrence relation we will prove in subsection 3.3, we consider the N = 6 Mercedes diagram, M , shown in figure 2(a), where we take (z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ) = (∞, 1, x, y, z, 0). Then the integral (1.13) takes the form and the integral is round their common zeros. In order to remove the pole of the integrand at z = 0, we replace h 6 2 , h 6 3 , by h 6 (z 5 ) = h 6 (z) , h 6 (z 6 ) = h 6 3 , where h 6 (z) = h 6 3 − zh 6 2 + z 2 h 6 1 is the z 1 → ∞ limit of (2.14). This introduces a Jacobian factor of −z, to give . (z) = s 56 xy and this is cancelled by factors in the numerator so that we may write This example illustrates the techniques we shall use to establish a general recurrence relation. (For a further example, with N = 8, see appendix B.) But before doing this we need to analyze the structure of the integrand in the next subsection.

Form of the integrand
The sum of off-shell tree diagrams is given by (1.11), which can be rewritten We associate the individual Feynman diagram ∆ with the integral, A ∆ N , obtained by replacing Φ N by 1/u ∆ , both of which are Möbius invariant, where u ∆ is the product of the cross-ratios (1.12). [In the case N = 4, the first product in the integrands of (3.7) and (3.9) should be replaced by (z 1 − z 3 ) −1 (z 2 − z 4 ) −1 .] The product 1/u ∆ has only simple poles in the variables z a , and these only occur at locations of the form z a = z b where |a − b| = 2. To see this is true, suppose ∆ has i 1 vertices with one internal leg (type 1), i 2 vertices with two internal legs (type 2), and i 3 vertices with three internal legs (type 3), so that total number of vertices, V = i 1 + i 2 + i 3 = N − 2. As each internal leg or propagator connects two vertices, the number of propagators, P = 1 2 (i 1 + 2i 2 + 3i 3 ) = N − 3. Thus As in figure 1, each propagator corresponds to a consecutive subset S = [I, J] ⊂ A, where 1 ≤ I < J < N (and to its complement, but choose the representative subset so that N / ∈ S. The graph has at least two type 1 vertices and we label the graph so that the legs 1, N JHEP04(2020)141 meet at one of these vertices. Then each propagator corresponds to a consecutive subset of [2, N − 1], including [2, N − 1] itself, and we denote these subsets by S i , 1 ≤ i ≤ N − 3. They are such that, if S i ∩ S j = ∅, either S i ⊂ S j , or S j ⊂ S i . Thus, the propagator subsets, S i , are partially ordered: given a propagator S i 0 , all the propagators S j ⊃ S i 0 can be ordered into an ascending sequence, At The propagator [I, J] is associated with a factor in 1/u ∆ given by the cross-ratio, , the product of cross-ratios for S i 0 , S i 1 , and S i 2 only has a simple pole at z I = z J . Since this does not occur as a pole in the cross-ratio associated with any other propagator, the integrand itself has at most a simple pole at z I = z J .
The factor (z I −z I+2 ) only occurs in the denominator of the cross-ratio associated with [I, I + 2]. If this is a propagator, either [I, I + 1] or [I + 1, I + 2] must be in the sequence. The cross-ratios of each of these contains a factor of (z I − z I+2 ) in the numerator, so the product of the cross-ratios for all propagators does not contain a pole at z I = z I+2 . Similar considerations exclude poles at z 2 = z N and z 1 = z N −1 . It follows that the product of cross-ratios contains only simple poles and these are at z a = z b , for some values of a, b satisfying |a − b| = 1 or |a − b| > 2. The latter poles are cancelled in the integrand of (1.13) leaving just i 1 simple poles at positions z I = z I+1 corresponding to the type 1 vertices.

Recurrence relation
We now establish a recurrence relation for A ∆ N , which we use to establish that it equals the corresponding Feynman diagram. Consider an N -point Feynman diagram, ∆, with possibly off-shell momenta, k a , and associated variables z a , a ∈ A, labeled so that the legs associated with z 3 and z 4 meet in a vertex. Fix Möbius invariance by setting z 1 → ∞, z 2 = 1, and z N = 0, and write x = z 3 , y = z 4 . Further consider the (N − 1)-point diagram, ∆ , JHEP04(2020)141 obtained by removing the z 3 , z 4 external legs of ∆, so that the propagator at which they join becomes an external leg and the external legs of ∆ are labeled z 1 , z 2 , z 4 , z 5 , . . . , z N , and the associated momenta are k 1 , k 2 , k 3 + k 4 , k 5 , . . . , k N . [Note that (k 3 + k 4 ) 2 = 0 in general, even if k 2 a = 0 for 1 ≤ a ≤ N , so that an off-shell recurrence relation is needed.] After fixing Möbius invariance, the amplitude associated with ∆ is given by 13) and h N m = lim z 1 →∞h N m+1 /z 1 , the superscript N indicating that these are the scattering polynomials associated to the N -point amplitude, with momenta, k 1 , . . . , k N , and variables, z 1 , . . . , z N rather than the (N −1)-point one, which we shall denote by h N −1 m , with momenta, k 1 , k 2 , k 3 +k 4 , k 5 , . . . , k N , and variables, z 1 , z 2 , z 4 , z 5 , . . . , z N . The amplitude associated with ∆ is given by The extra cross-ratio factor in F ∆ N relative to F ∆ N −1 corresponds to the propagator factor and other factors from the product in (3.13)

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To establish the recurrence relation, as in subsection 3.1, it is useful to make use of the functionh (z) introduced in (2.14), or rather its asymptotic form as z 1 → ∞, in the denominator of (3.12) introduces a Jacobian factor of (−1) .
for some a, b with 5 ≤ a < b ≤ N , but in such a case the contribution to the integrand from such a point is cancelled by the product of such factors z a − z b in the integrand. From (3.16), we see that F ∆ N has a pole at x = y and, from subsection 3.2, we know that its other possible singularities are simple poles at z a−1 = z a , where 6 ≤ a ≤ N . However we shall show that such poles are cancelled by a zero in F ∆ N , and so they do not contribute to the integral. Assuming this to be the case, and using (3.16), . (3.20) Now, the only pole of the integrand, other than where h N 1 = h N (z 5 ) = h N (z 6 ) = . . . = h N (z N ) = 0, is at x = y and so, using the global residue theorem, we can replace h N (z 5 ) with x − y in the specification of the integration contour, We can now replace h N −1 and an appropriate Jacobian factor, To complete the argument, we need to show that there are no contributions to the A ∆ N from poles that F ∆ N may have at z a−1 = z a , for some 6 ≤ a ≤ N . To consider the possible contribution to (3.19) from such a pole in F ∆ N , note, as in (3.16), that (3.26) and h N (za) is one of the zero denominators in A ∆ N , which demonstrates that this contribution to the integral vanishes.
From (3.24), we have the recurrence relation,

Comments
In non-abelian gauge theory, the expressions we have given would be dressed by a group factor. Similar extra factors would also occur if the scalar field transformed in the adjoint representation of some global internal symmetry. In the scalar case, to the extent that the complete tree amplitude is a linear combination of 'double-partial amplitudes' with coefficients given by group factors, and that the 'double-partial amplitudes' are themselves sums of subsets of the individual ordinary φ 3 planar graphs, then the off-shell version is given by replacing those with individual off-shell Feynman graphs. A key feature of our analysis in this paper is to provide the off-shell formula for any individual Feynman tree diagram in ordinary φ 3 theory. To arrive at a consistent picture for individual graphs in the gauge theory would require suitable Möbius invariant numerators with dependence on the polarization and momenta. fully, one needs to specify the value of a, designating which pair k a−1 + k a is joined. In (3.4), a = 4, and the associated momenta and variables are k 1 , k 2 , k 3 + k 4 , k 5 , k 6 with z 1 , z 2 , z 4 , z 5 , z 6 so