Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension

A proof is given of the formula, recently proposed by Cachazo, He and Yuan (CHY) for gluon tree amplitudes in pure Yang-Mills theory in arbitrary dimension. The approach is to first establish the corresponding result for massless ϕ3 theory using the BCFW recurrence relation and then to extend this to the gauge theory case. Additionally, it is shown that the scattering equations introduced by CHY can be generalized to massive particles, enabling the description of tree amplitudes for massive ϕ3 theory.


Introduction
Recently Cachazo, He and Yuan (CHY) have proposed a compact and relatively simple formula for the sum of all N -point tree diagrams for pure Yang-Mills theory in an arbitrary dimension of space-time and a similar formula for the sum of all N -point tree diagrams for gravity [1]. These formulae express the N -point amplitudes as sums over the solutions for z a to the equationŝ f a (z, k) = 0, a ∈ A, wheref a (z, k) = b∈A b =a and where k a are the momenta of the particles labeled by a ∈ A, with N = |A|, which CHY call the scattering equations. It follows from momentum conservation, a∈A k a = 0, and the condition that the particles are massless, k 2 a = 0, a ∈ A, that this system of equations is invariant under Möbius transformations, z a → ζ a = αz a + β γz a + δ , a ∈ A, (1.2) in that, if z = (z a ) is a solution to (1.1), ζ = (ζ a ) also provides a solution:f a (ζ, k) = 0, a ∈ A. Because of this symmetry under the three-dimensional Möbius group, only N − 3 of the equations (1.1) are independent, and we can restrict to their solutions by means of the delta functions a∈A ′ δ f a (z, k) , (1.4) so that the integrand of is Möbius invariant provided that the function Ψ N (z; k; ǫ) is itself Möbius invariant. In (1.5), we have chosen a cyclic ordering on A, with a + 1 indicating the label next after a in the ordering, say (1, 2, . . . , N ) with N + 1 ≡ 1, Ψ N (z; k; ǫ) may depend on the momenta k a and polarizations, ǫ a , if any, of the particles, and dω denotes the invariant measure on the Möbius group, . (1.6) In [1], CHY propose, and present evidence for, an explicit form for Ψ N (z; k; ǫ) for the sum of N -gluon tree diagrams in pure gauge theory, in an arbitrary dimension of space-time, for a given order of the external gluons (to be multiplied by an appropriate color factor). In order to give a precise interpretation of (1.5) that includes a sum over all, possibly complex, solutions of the equations (1.1), CHY rewrite the expression as a contour integral where the contour O encloses all the solutions of (1.1) and ′ is defined as in (1.3).
Here, following the approach we used in [2], to establish the complete equivalence of gluon tree amplitudes in open twistor string theory [3] to the corresponding amplitudes in N = 4 super Yang-Mills theory, we prove these CHY formulae using the BCFW relations [4], by first showing that taking Ψ N (z; k; ǫ) constant in (1.5) gives the tree amplitudes for massless φ 3 theory. [In [5], CHY have also noted that taking Ψ N (z; k; ǫ) constant in (1.5) can provide the amplitudes for a scalar theory, but they consider a theory with U(N ) × U(Ñ ) symmetry.] The CHY equationsf a (z, k) = 0, a ∈ A, can be modified to describe massive particles (although we only need the massless case as a precursor to our proof of the CHY proposal for gauge theory tree amplitudes). Having chosen a cyclic ordering on A, we consideř , a ∈ A, (1.8) where, in the signature we are using, k 2 a = m 2 , a ∈ A. This system of equations is again Möbius invariant, and now (1.7) produces the tree amplitudes for a φ 3 theory for a scalar JHEP05(2014)010 particle of mass m. However, unlike the original, these modified CHY equations depend on the order of particles which previously only entered through what one might regard as the measure factor in the integrand of (1. 5) or (1.7).
After reviewing the general properties of the CHY scattering equations in section 2, we give a calculation in this formalism of the elementary cases of the 4− and 5-point amplitudes in massless φ 3 in section 3, noting that if we have chosen z 1 → ∞, the formulae would apply even when k 1 is off-shell. In section 4, we give a direct inductive proof of the formula for φ 3 theory, using a form of the result that works when one of the momenta is off shell. This approach would be very cumbersome for gauge theory, so we give a second proof in section 5 using the BCFW relations. This analysis is extended in section 6 to give a proof of the formula of CHY [1] for the tree amplitudes of pure gauge theory. Results on Pfaffians, necessary for the discussion of section 6 are described in appendix A. The extension of the results on φ 3 theory to the massive case is discussed in appendix B.

Properties of the CHY scattering equations
We review some properties of the CHY scattering equations (1.1).
Möbius invariance. This system of equations is Möbius invariant provided that momentum is conserved, i.e. a∈A k a = 0, and all the particles are massless, i.e. k 2 a = 0, a ∈ A. To see this note that provided that k 2 a = 0 for all a ∈ A. Thus, then the system of N equations (1.1) determines the N variables z a only up to Möbius transformation, implying only N − 3 of them are independent and there must be three relations between them. The equations determine the N − 3 independent cross-ratios of the z a ; we are free to specify any three of the z a as we wish and the others are then determined by (1.1).
For general values of z, the functionsf a (z, k) satisfy the relations: independently of the momentum conservation and zero-mass conditions on the k a ;

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using first the zero-mass conditions, k 2 a = 0, and then momentum conservation; and From (2.4) and (2.5), we have that Relation to tree amplitude integrands in string theory. The N -point tree amplitude in the original bosonic open string theory is given, in Koba-Nielsen variables, by an integral where the leading trajectory, α(s) = α 0 + 1 2 α ′ s, with intercept α 0 and slope parameter α ′ , and the particles satisfy the mass condition k 2 a = m 2 , where m 2 = −2α 0 /α ′ = −2a 0 . [Scherk obtained φ 3 theory by taking α ′ → 0, at fixed a 0 , in (2.7) in his original paper on the zero-slope (infinite-tension) limit [6]. In this connection see the recent papers by Mason and Skinner [8] and by Berkovits [9].] Crucially, V (z, k), or equivalently, is Möbius invariant, that is, if ζ a is given by thenf a (z, k) is given by (1.8) and the Möbius invariance of U (z, k) implies so that the set of equationsf a (z, k) = 0, a ∈ A, is Möbius invariant.f a (z, k) =f a (z, k) when a 0 = 0.

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The Möbius invariance of U (z, k) immediately implies the relations (2.3)-(2.5), because, for an infinitesimal Möbius transformation δz a = ǫ 1 + ǫ 2 z a + ǫ 3 z 2 a , δU = a∈A δz a ∂U (z, k) ∂z a = 0, implying a∈A δz afa = 0, Factorization. Consider the situation in which, as the k a , a ∈ A, vary in some specific way maintaining the zero-mass conditions k 2 a = 0, two or more of the z a to tend to the same value, z S , say. Specifically, suppose z a = z S + ǫx a + O(ǫ 2 ), as ǫ → 0, for a ∈ S, where S ⊂ A, and z a → z S for a / ∈ S. Then implying that ifĝ a = 0 for a ∈ S, a = s (e.g. as the limit of conditionsf a = 0 for a ∈ S, a = s, as ǫ → 0), where s is some particular element of S, then it follows that g s (x) = 0 as well. Further, multiplying (2.14) by x a − x r , for a choice of r ∈ S, and summing over a ∈ S, as in (2.4), gives Then, ifĝ a (x, k) = 0, a ∈ S, a = r, s, we have that and, if additionally g r (x, k) = 0, then k 2 S = 0. Imposing the equationsf a (z, k) = 0, for all a ∈ A, and taking the limit ǫ → 0, So, in this limit, the CHY equations (1.1) have in effect factored into two sets: the first set of equations are the CHY equations for the momenta (k a , a ∈ S; −k S ), with associated variables (x a , a ∈ S; ∞); and the second set are the equations for the momenta (k S ; k a , a / ∈ S), with associated variables (z S ; z a , a / ∈ S).

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Specialization. In what follows, in general it will be convenient to take (r, s, t) in (1.6) to be the same as (i, j, k) in (1.3) and further to take i = 1, j = 2 and k = N . Then (1.7) becomes (2.20) and making the particular choice z 1 = ∞, z 2 = 1, z N = 0, this becomes where Ψ o N (z; k; ǫ) denotes the specialization of Ψ N (z; k; ǫ) to these values and To be clear about the position of singularities, it is convenient to rewrite this by setting

Massless φ theory
In this section we discuss (1.7) in the simplest case, in which Ψ N (z; k; ǫ) is constant, showing how, for N = 4, 5, (2.24) produces the amplitudes of massless φ 3 theory in a form in which the mass-shell condition k 2 1 = 0 can be relaxed. In the next section, we will give an inductive proof of this partly off-shell formula for general N . Specifically, taking Ψ N (z; k; ǫ) = (−2) 3−N in (1.7) and (2.24): , f a is given by (2.23), the contour O encircles the simultaneous solutions of f a (z, k) = 0, 3 ≤ a ≤ N − 1 (but excludes z 3 = 1 and z N −1 = 0), and we need to sum A φ N over the different cyclic orderings of the external legs. For N = 4, writing z = z 3 , this gives Summing over the 6 different cyclic orderings (by permuting 2,3,4), and multiplying by g 2 /4, where g is the coupling constant, gives the total amplitude A φ tot 4 = g 2 /s + g 2 /t + g 2 /u (because each term occurs four times), the appropriate result for φ 3 theory, L = 1 2 ∂ µ φ∂ µ φ + (g/3!)φ 3 . For the N -point function, A φ tot N , we need to multiply (3.2) by (g/2) N −2 as we sum over the possible different orderings keeping 1 fixed, because, as we shall prove, A φ N gives the sum of the tree diagrams in φ 3 theory that can be drawn in planar form for a given ordering of the external momenta [as 1/s in (3.3) can be for the ordering (1, 2, 3, 4) but not the ordering (1, 3, 2, 4)]. Each tree diagram will contribute to 2 N −2 orderings, treating orderings related by cyclic rotations as equivalent. (If we treat orderings related by anticyclic or cyclic rotations as equivalent, we sum over half as many orderings and each diagram occurs 2 N −3 times, so, in that case, we multiply by g N −2 /2 N −3 .) This is the same counting as in Scherk's original discussion of the zero-slope limit of what became string theory [6]. It would be good to be able to derive (3.2) directly from (1.7) by taking α ′ → 0. (See also [8] and [9].) [We have chosen the sign of Ψ N so that A φ N will equal a sum of tree diagrams with propagators 1/k 2 , where k is the momentum. With this normalization, A φ tot N equals i(−1) N times the amplitude defined using the usual field theory convention.] For N = 5, writing z = z 3 , w = z 4 , we have This is most conveniently evaluated using the global residue theorem. We write for the residue of the integrand at α = β = 0, where O αβ is an appropriately oriented contour about α = β = 0; α, β are chosen from the factors in the denominator of the integrand; and we have displaced the factors 1 − z and w by ǫ and δ, respectively, in order to avoid singular configurations of these factors. Then The term R(w, 1 − z), for which the displacements by ǫ and δ are unnecessary, is easily evaluated: where we use the notation

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Then Noting that the integrand of (3.4) is symmetric under the interchanges Putting together (3.9), (3.12) and (3.13), we see that A φ 5 gives the sum of planar tree φ 3 diagrams for momentum ordering (k 1 , k 2 , k 3 , k 4 , k 5 ), This formula does not explicitly involve k 1 and hold when k 1 is off-shell, i.e. when k 2 1 = 0, and the same is true of the form (3.3) for A φ 4 .

Inductive proof for (partly) off-shell massless φ 3 theory
We shall prove by induction that (3.2) provides a formula for the sum of planar tree diagrams that is valid when k 1 is off-shell. We shall do this by establishing the recurrence relation which holds if A φ N denotes the sum of planar tree diagrams with the first momentum offshell. This relation follows from the observation that, for N > 3, the external line carrying the off-shell momentum k 1 enters the tree diagram either at a vertex in which one of the other two lines meeting there is internal or both are. In the former case, because the diagrams are planar with the given ordering of momenta, the other external line must either carry momentum k 2 or momentum k N , and the internal line is a propagator carrying offshell momentum k 1 +k 2 or k 1 +k N joining it to an (N −1)-point tree amplitude in which all the other momenta are the remaining k a in order, and so on shell, providing the first and last terms in (4.1). In the latter case, the two off-shell internal lines have momenta π m andπ m+1 .

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To establish (4.1) holds for A φ N = A φ N defined by (3.2), we use the global residue theorem as we did in case N = 5, again displacing the factors of 1 − z 3 and z N −1 by ǫ and δ, respectively, in order to avoid singular configurations Denoting the right hand side of (4.
Applying the global residue theorem again, Repeating this process, we obtain After substituting this form for , we shall show that the various terms correspond to the terms in the recurrence relation (4.1).
for general values of the external momenta satisfying the zero-mass condition, we must have that z 3 also vanishes, for otherwise we would have k 2 S = 0, as in (2.18) with s = N , where S is the set of a for which z a vanishes as δ → 0 (including N ), and we have stipulated that the k a are in general position. In fact the residue comes from the region where all z a = O(δ), 3 ≤ a ≤ N − 1. We shall first calculate the contribution from this region and then show it is necessary for all z a , 3 ≤ a ≤ N − 1, to vanish to get a non-zero contribution.

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Then at g 4 = . . . = g N −1 = 0. In (4.9), in the limit δ → 0, a factor of δ (N −4) 2 in the numerator, coming from the products in (4.3), is cancelled by an equal factor in the denominator, coming from leaving a finite answer. As in (2.17), So, when g 4 = . . . = g N −1 = 0, and, provided that there are no other contributions to this residue from other regions, by induction. If we consider contributions to R(z δ N −1 , f 4 , . . . , f N −2 , f N −1 ) from regions in which not all the z a , 3 ≤ a ≤ N − 1 vanish as δ → 0. As we have argued, z 3 → 0 as well as z N −1 as δ → 0. If the set S = {a : z a → 0} contains only M of the z a , 3 ≤ a ≤ N , where M < N − 2, the product (4.11) in (4.3) will contribute a factor of δ (M −2) 2 to the denominator, but the product (4.10) will contribute a factor of δ L where L > (M − 2) 2 , which implies that the contribution vanishes as δ → 0. This is because the products in (4.10) would contribute an exactly balancing contribution of δ (M −2) 2 if the a ∈ S were consecutive but the second product in (4.10) contributes a factor of δ 2 in the limit for every a, b ∈ S with 3 ≤ a < b ≤ N − 1 and b − a > 1, and this number is at a minimum for a JHEP05(2014)010 given M when the a are consecutive. However, as 3, N − 1 ∈ S and M < N − 2, the a ∈ S cannot be consecutive and so the contribution vanishes in the limit, establishing (4.14).
Because of the symmetry under z a ↔ 1 − z N +2−a , k a ↔ k N +2−a , 2 ≤ a ≤ N, we have the corresponding result . This contribution comes from the region z a = O(δ), m + 1 ≤ a ≤ N − 1; 1 − z a = O(ǫ), 3 ≤ a ≤ m, for reasons similar to those that determined the appropriate regions for the residues already considered: if S 0 denotes the set of a for which z a → 0 (including a = N ) and S 1 denotes the set of a for which z a → 1 (including a = 2), then each of S 0 , S 1 much contain one of m or m + 1 or we should have k 2 S 0 = 0 or k 2 S 1 = 0, contradicting the assumption that the k a are in general position. If S 0 does not consist of consecutive elements we should find again that the contribution will vanish in the limit and similarly for S 1 . Thus the only way to get a non-zero contribution is to have S 0 = {a : m + 1 ≤ a ≤ N } and S 1 = {a : 2 ≤ a ≤ m}, which is the region we are considering.

BCFW proof for massless φ 3
The proof given in section 4 that, with Ψ N constant, we obtain the tree amplitudes of massless φ 3 , would be difficult to extend to the gauge theory case, because the presence of four-point gluon vertices proliferates the number of terms involved in the induction. So, as a preliminary to discussing this case in section 6, we now give a proof using the BCFW relations. To this end, choose a (possibly complex) momentum vector ℓ such that ℓ 2 = ℓ · k 2 = ℓ · k N = 0, and consider where A φ N denotes the sum of planar tree diagrams. Then A φ N (ζ) is meromorphic in ζ with poles arising because of the propagators in the tree diagrams it comprises. The residues of these poles are given by the product of amplitudes A φ m , where m < N and so A φ N (ζ) can be expressed as a sum over these residues, and this provides a recursion relation, which is the BCFW relation [4,7].
If it were not the case that A φ N (ζ) → 0 as ζ → ∞, a constant would have to be included, in essence a boundary term from infinity. This would indeed arise if we were considering the sum of all N -point tree functions regardless of the order of the external momenta, but, the ordering of momenta in A φ N and the fact that the displaced momenta k 2 + ζℓ and k N − ζℓ are not adjacent means that at least one propagator denominator in each tree diagram included in A φ N depends linearly on ζ and so A φ N (ζ) → 0 as ζ → ∞. Integrating around a small contour O ζ enclosing ξ = ζ and then moving the contour out to infinity, where the poles of A φ N (ζ) occur at the points ζ = ζ i . Define π m ,π m , s m ,s m as in (4.2), and π ζ m andπ ζ m by replacing k 2 by k ζ 2 in π m and k N by k ζ N inπ m , respectively, where

5) and
This relation is sufficient to determine A φ N for N > 3 from A φ 3 = 1. We now seek to show A φ N = A φ N , as defined by (3.2) satisfies (5.7), with Res ζ L m A φ N and Res ζ R m A φ N given by (5.5) and (5.6), respectively. Defining and the contour encloses the solutions of f a (z, ζ) = 0, 3 ≤ a ≤ N − 1, (5.10) for given ζ, but excludes the poles at z N −1 = 0 and z 3 = 1. The poles of A φ N (ζ) occur when the contour is pinched between the zeros of the f a and either z N −1 = 0 or z 3 = 1. First consider a solution of (5.10) which is such that, as ζ → ζ 0 , z N −1 → 0. It may be that as ζ → ζ 0 some other z a → 0. Suppose again that S denotes the set of a, with 3 ≤ a ≤ N , for which z a → 0. For essentially the same reasons used in section 4 to determine which contributions to the recurrence relation considered there were nonzero, having z a → 0 for a ∈ S will not produce a pole unless the a in S are consecutive. Thus we must have S = {a : m ≤ a ≤ N } for some m with 3 ≤ m ≤ N − 1. Then, from (2.18), k 2 S = (π ζ 0 m ) 2 = 0, so that ζ 0 = ζ R m , as defined by (5.4), and the solution to (5.10) has z a → 0, m ≤ a ≤ N − 1, as ζ → ζ R m . Similarly, there are solutions to (5.10) that have the property that z a → 1, 3 ≤ a ≤ m, as ζ → ζ L m , as defined by (5.4).

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We now investigate the (potential) pole at ζ = ζ R m , which comes from the region of integration in (5.8) given by As in (2.15) and (4.7), for m ≤ a ≤ N − 1, As in (2.17) and (4.12), Integrating with respect to ζ and z m shows that Res ζ R m A φ N provides the appropriate contribution to the BCFW relation,

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since the pole at z m = 0 is exterior to the contour. The residue at ζ = ζ L m follows similarly, using the symmetry under m ↔ N + 2 − m, z a ↔ 1 − z a , L ↔ R, establishing the full BCFW relation and so proving the formula for A φ N by induction.
6 Proof for pure gauge theory The CHY proposal [1,5] for a choice of Ψ N (z; k; ǫ) that gives the tree amplitudes for pure gauge theory in any space-time dimension employs the Pfaffian of the antisymmetric matrix [Properties of Pfaffians are reviewed in appendix A, together with further some results needed here for our discussion.] The Pfaffian of Ψ N (z; k; ǫ) itself is zero when the conditions (1.1) hold, because then it has two null vectors arising from common null vectors of A and C: the vectors (1, 1, . . . , 1) and (z 1 , z 2 , . . . , z N ). So, CHY consider the Pfaffian of the ( N (z; k; ǫ) obtained by omitting the a-th and b-th rows and columns of Ψ N (z; k; ǫ) and demonstrate that is independent of a, b for 1 ≤ a < b ≤ N . Then they give evidence for the proposition that the tree amplitudes, A YM N , for gauge theory are given by taking (z a − z a+1 ) (6.5) in (1.7), where the normalization factor has been chosen for the convenience of the present discussion. The required Möbius invariance of Ψ N (z; k; ǫ) is straightforward to verify. As in (3.2), where again ε N = (−1)

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Now, lim and independent of a, b for 1 < a < b ≤ N , Ψ o (ab) being defined relative to Ψ o in the same way that Ψ (a,b) N is relative to Ψ N . Thus, in (6.6), (6.7), We now give a proof that A YM N , defined by (6.6), does indeed give the N -gluon Yang-Mills tree amplitude by establishing that it satisfies a BCFW recurrence relation, as in (5.5)-(5.7). To this end, we need to consider A YM N (ζ), obtained from A YM N by deforming the momenta k 2 → k ζ 2 = k 2 + ζℓ, k N → k ζ N = k N − ζℓ, as in (5.3), where, again, ℓ 2 = ℓ · k 2 = ℓ · k N = 0, and also by deforming the basis of polarization vectors ǫ ζ1 2 , ǫ ζ2 2 , ǫ j 2 , 3 ≤ j ≤ d − 2, for k ζ 2 , as in [7], so that where we have introducedl such thatl 2 =l · k 2 =l · k N = 0,l · ℓ = 2, so that ǫ ζi 2 · ǫ ζj 2 = δ ij , ǫ ζj 2 · k ζ 2 = 0, 1 ≤ i, j ≤ N − 2, with ǫ ζj 2 ≡ ǫ j 2 , 3 ≤ j ≤ N − 2, constant; and similarly for the basis of polarization vectors ǫ ζ1 The difference between A YM N (ζ) here and A φ (ζ) in section 5 is the factor of Ψ o N (z; k; ǫ), as defined by (6.13), in the integrand of (6.6), and this does not affect the nature of the singularities in ζ, which are thus still poles at ζ = ζ L m , ζ = ζ R m , 3 ≤ m ≤ N − 1, as defined in (5.4). [The singularities of Ψ o N (z; k; ǫ) occur when two or more of the z a coincide. When the z a are adjacent the calculation in appendix A shows that factors of z a − z a+1 cancel those in the denominator, so that Ψ o N (z; k; ǫ) is regular there. If they are not adjacent, these are cancelled by the factors of (z a − z b ) 2 , a > b + 1, as in section 4.] To demonstrate that A YM N satisfies the BCFW recurrence relation we need to show that Ψ o N factorizes appropriately at these values of ζ. so that Θ ab = −Θ ba . Then (4.2), and F ab = α a · β b , with , (6.24) where E (a) indicates that the a-th row and column have been removed from E, and similarly for G (a) , and F [a,b] indicates that the a-th row and b-th column have been removed from F .
So, using the Factorization Lemma of appendix A, as z m → 0, , (6.27) where α r = α · ǫ r , β r = β · ǫ r and α r [a] , β r [a] denote α r , β r , respectively, omitting the a-th entry. Then In principle, in (6.26), ǫ r should run over the d elements of a basis for the whole space, but we can restrict to a basis for the polarization states forπ m , since the metric tensor is where k is chosen so that k · ǫ r = k 2 = 0, k ·π m = 0, and the contributions from the last two terms vanish because This, together with the corresponding result for the pole at ζ = ζ R m , which holds by symmetry, establishes the BCFW recurrence relation.
where S M denotes the group of permutations of 1, 2, . . . , M . We can write Pf A as a sum over the pairings ̺ = {(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i N , j N )} of the integers 1, 2, . . . , 2N − 1, 2N . Denote the set of such pairings by P N . We can choose a unique description of the labeling of the pairing by specifying that i a < j a and i a < i b if 1 ≤ a < b ≤ N . Given ̺ ∈ P N , define ρ ̺ ∈ S 2N by ρ ̺ (2a−1) = i a , ρ ̺ (2a) = j a , 1 ≤ a ≤ N , and set sign̺ = signρ ̺ . Then If A (ij) denotes the matrix obtained from A by deleting the i-th and j-th rows and i-th and j-th columns. Then A (ij) is an antisymmetric matrix with dim A (ij) = dim A − 2. Then we have the expansion In order to establish the factorization result in section 6, we need a result on the factorization of Pfaffians in suitable limits, which we prove here. First we establish a preliminary result, which may serve to elucidate the subsequent argument.

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Preliminary Lemma. Suppose E, G are antisymmetric matrices of dimensions of m × m and n × n, respectively; a, b are vectors of dimensions m and n, respectively; and Then, if m + n is odd, Pf Φ = 0; if m + n, m and n are even, Pf Φ = Pf E × Pf G; if m + n is even, and m and n are odd, Proof. Suppose m + n even, m + n = 2N .
where ̺ = {(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i N , j N )}. Let C ̺ denote the number of a for which i a ≤ m and j a > m. m and n are either both odd or both even and C ̺ has to be odd if m, n are odd, and even if m, n are even. For each ̺ ∈ P N with a given value of C ̺ > 1, we associate to ̺ ′ ∈ P N , which is obtained by replacing the pairs (i c , j c ), (i d , j d ), where c, d are the largest values of a such that i a ≤ m and j a > m, by the pairs (i c , j d ), (i d , j c ). Then sign̺ = −sign̺ ′ , but the value of the product is the same for ̺ ′ as for ̺. Thus the contributions to the sum over ̺ ∈ P N from those ̺ with C ̺ > 1 cancel in pairs. Since C ̺ has the same parity as m and n, all the nonzero contributions come from C ̺ = 0 if m, n are even and from C ̺ = 1 if m, n are odd. If m, n are even, each pairing ̺ ∈ P N making a nonzero contribution to Pf Φ is the union of parings ̺ 1 ∈ P 1 2 m , of 1, 2, . . . , m, and ̺ 2 ∈ P 1 2 n , of m + 1, 2, . . . , m + n. So, the sum over such ̺ ∈ P N is the product of sums over ̺ 1 ∈ P 1 2 m and over ̺ 2 ∈ P 1 2 n , with sign̺ = sign̺ 1 sign̺ 2 . The product (A.7) factors into two products corresponding to the pairings ̺ 1 and ̺ 2 , respectively. Thus Pf Φ = Pf E × Pf G.

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the second set are the equations for the momenta (k S ; k a , a / ∈ S), with associated variables (z S ; z a , a / ∈ S). The results (B.2) to (B.8) make it straightforward to adapt the analysis of sections 3, 4 and 5 to the massive case. If we replacef a (z, k) in (3.1) byf a (z, k)to obtain the effect in general is to replace k a · k b by k a · k b − a 0 whenever b = a ± 1, so that, for example, (3.3) becomes (B.10) Further the effect is to replace s ab = (k a + k b ) 2 by s ab − m 2 = (k a + k b ) 2 − m 2 and s abc = (k a + k b + k c ) 2 by s abc − m 2 = (k a + k b + k c ) 2 − m 2 in (3.9) to (3.14) [Note that a, b and a, b, c are consecutive where they occur in s ab and s abc .], giving the correct result for A φ,m 5 . Because of (B.5) and (B.6), the effect is to replace s m with s m − m 2 ands m with s m − m 2 throughout sections 4 and 5, so that those sections provide inductive proofs of that A φ,m N provides the correct N -point tree amplitudes for massive φ 3 theory.
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