Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism

In this paper we extend our techniques, developed in a previous paper (Du, etc, JHEP 05(2016)086) for direct evaluation of arbitrary $n$-point tree-level MHV amplitudes in 4d Yang-Mills and gravity theory using the Cachazo-He-Yuan (CHY) formalism, to the 4d Einstein-Yang-Mills (EYM) theory. Any single-trace color-ordered $n$-point tree-level MHV amplitude in EYM theory, obtained by a direct evaluation of the CHY formula, is of an elegant factorized form of a Parke-Taylor factor and a Hodges determinant, much simpler and more compact than the existing formulas in the literature. We prove that our new expression is equivalent to the conjectured Selivanov-Bern-De Freitas-Wong (SBDW) formula, with the help of a new theorem showing that the SBDW generating function has a graph theory interpretation. Together with Ref. (Du, etc, JHEP 05(2016)086), we provide strong analytic evidence for hidden simplicity in quantum field theory.

compact formula for single-trace EYM amplitudes in the MHV configuration with two negative helicity gluons. Bern, De Freitas and Wong [54] extended KLT relation to EYM theory. As a result, they were able to conjecture a compact formula for the single-trace MHV amplitudes with one negative-helicity gluon and one negative-helicity graviton. This formula is very similar to one of Selivanov such that we will refer both formulas together as Selivanov-Bern-De Freitas-Wong (SBDW) formula in this paper. The authors of [54] further argued that the tree-level single-trace MHV amplitudes with two negative-helicity gravitons should vanish. The SBDW formula associates the amplitudes to the Taylor expansion coefficients of a multivariate generating function. As a result, the calculation involves an ordeal of high-order derivatives while the final expression is neither explicit nor compact. In [46,47], the CHY formula for EYM amplitudes was proposed. Our motivation is thus to study whether the CHY formalism can give a simpler and more compact expression for EYM amplitudes.
In this paper, we focus on the tree-level single-trace EYM amplitudes in MHV configurations. Based on the CHY formula for EYM theory, we have derived a new compact expression for the MHV amplitudes with at least one negative helicity gluon: where r is the number of gluons (labeled by g's) and the number of gravitons (labeled by h's) is thus s = n − r. The above equation is of a factorized form of a Parke-Taylor factor, whose denominator only depends on gluons, and a minor of Hodges matrix, det(φ h + ), whose indices range within the positivehelicity gravitons only. One of the two negative-helicity particles, say, g j , is a gluon, while the other one, say, i, can either be a gluon or graviton. We then carry on to prove analytically that the (h − h − ) MHV amplitudes, in which gluons all carry positive helicities, have to vanish. In the SBDW formula, the same amplitude as (1.1) is derived from a generating function. We prove a new theorem, showing that such a generating function leads to a weighted sum of spanning forests. It has been shown in [8] that the Hodges minor in eq. (1.1) has exactly the same graph theory interpretation. Thus we are able to show the equivalence between our new formula and the SBDW prescription for arbitrary n.
The structure of this paper is the following. In Sec. 2, we review the CHY formula in Yang-Mills theory, gravity and EYM theory. In Sec. 3, we derive the new formula for (g − g − ) and (h − g − ) single-trace MHV amplitudes. We also prove that (h − h − ) amplitudes vanish identically. In Sec. 4, we provide a graphic representation of the SBDW formula, which is used to prove its equivalence to our new formula. The conclusion of this paper is given in Sec. 5. Finally, we prove that the special solution makes the reduced pfaffian vanish at non-MHV configurations in Appendix A.

Scattering Equations and CHY Formalism
The scattering equations for n massless particles are a set of n equations on the complex variables z a : where s ab ≡ 2k a · k b are the Mandelstam variables and z ab ≡ z a − z b . These equations are Möbius covariant such that we can use this freedom to fix the value of three z's, which implies that among the n equations only n − 3 of them are linearly independent. The number of solutions to (2.1) is (n − 3)!. This fact was first demonstrated by a semi-analytic inductive method in [12], and later proved by more elegant algebraic methods in [20,22]. In four dimensions, there are always two special solutions, written in spinor variables: where the arbitrary projective spinors η, θ and ξ encode the Möbius freedom in the solutions. This spinorial form was first written down by Weinzierl [18], while it has appeared earlier in other context or forms in [16,[55][56][57]. For these two solutions, we have: which will be used frequently when studying MHV amplitudes. In the following, We will use ω a for generic solutions to eq. (2.1), while σ a andσ a are used only for the special solutions (2.2). CHY formalism states that generic tree-level n-point massless amplitudes are supported only by the solutions to the scattering equations. Namely, it can be calculated by: , (2.4) where I n is the CHY integrand, defining various theories, and the matrix Φ is: This matrix has rank deficiency three so that to have a nonzero determinant we need to delete three rows (i, j, k) and three columns (p, q, r). With the resultant submatrix Φ ijk pqr , the det is defined as: where perm(ijk) is the signature of the permutation that moves (1, 2, . . . , n) into (i, j, k, . . . , n), with the (. . .) keeping the original ascending order. It has been shown that det (Φ) is independent of the choice of (i, j, k) and (p, q, r) [13]. In this work, our main subject is single-trace tree-level EYM amplitudes, whose CHY integrand has a close relation with those of pure Yang-Mills and pure gravity. Next, we are going to give a brief review on the CHY formalism of these theories.

Yang-Mills and pure gravity
The integrands for color-ordered Yang-Mills and pure gravity amplitudes are: color-ordered Yang-Mills, where {k} is the set of external momenta and { } (both { } and { }) is the set of polarizations for gluons (gravitons). The 2n × 2n antisymmetric matrix Ψ is given by: where the blocks are: The upper half of Ψ, (A, −C T ), has two null vectors such that we need to delete two rows and columns in the first n rows and columns to obtain a nonzero pfaffian. Thus Pf is defined as: where 1 i < j n. It is also independent of the choice of (i, j). Since both Pf (Ψ) and det (Φ) are invariant under permutations, the total Yang-Mills amplitudes can be obtained by where T a i is the Lie algebra generator of the gauge group.

Single-trace Einstein-Yang-Mills
The EYM amplitude, M s,r , is characterized by the number of external gravitons s and gluons r, with s + r = n. For convenience, we define the set of gravitons and gluons to be h and g, while the set of external particles be p = h ∪ g = {1, 2, . . . , n}. By convention, we use: . . , g r } .
We will also use the sets of + and − helicity gravitons h ± and gluons g ± , as well as p ± = h ± ∪ g ± . The orders of these sets are denoted as In [46], the authors proposed an integrand for single-trace EYM amplitudes, which reads:  Tr (T a s+1 . . . T an ) ω g 1 g 2 ω g 2 g 3 . . . ω grg 1 + non-cyclic perm.
where Φ and Ψ are given by eq. (2.5) and (2.8) respectively, the same as the Yang-Mills case. The 2s × 2s matrix Ψ h is given by: where A h , B h and C h are s×s dimensional diagonal submatrices of A, B, and C, whose indices range within the graviton set h. In this formula, we assume that the gluons have the polarization µ a and gravitons have µν a = µ a ν a . We note that if there is only one gluon, namely, s = n − 1, the amplitudes are identically zero due to Pf (Ψ h ) = 0, independent of helicity configurations and solutions. In this case, Ψ h can be obtained from Ψ by deleting the n-th and 2n-th row and column. However, as discussed below eq. (2.9), there are two null vectors in the upper half of Ψ such that after deleting the n-th row and column, there is still one, which makes the pfaffian vanish. Physically, the vanishing of this amplitude is easy to understand from the conservation of color quantum numbers.
Also in [46], the authors have shown that eq. (2.13) can give the correct soft limit, and checked numerically that the amplitudes agree with the known results. In the next section, we are going to derive analytically the single-trace MHV amplitudes. We will show that they agree with those given by Selivanov, Bern, De Freitas and Wong (SBFW) [52][53][54] in Sec. 4.

Single-Trace MHV Amplitudes for Einstein-Yang-Mills
At MHV, the EYM amplitudes can be put into three categories: 1. two negative helicity gluons, hereafter (g − g − ).

two negative helicity gravitons, hereafter (h
For the first two cases, SBFW [52][53][54] have provided a compact formula to calculate the amplitudes, while in Sec. 3.1 and 3.2, we provide a more explicit expression in terms of the Hodges determinant [7], using the CHY integrand (2.13). For the last case, [54] argued that the amplitude vanishes by imposing the required factorization properties. In Sec. 3.3, we prove analytically that this is the case.
Since the factor Pf (Ψ) is shared by both the single-trace EYM integrand (2.13) and the Yang-Mills one (2.7), EYM amplitudes must be supported by the same set of solutions at most, if not less. It has been conjectured that only the special solution σ (σ) shown in eq. (2.2) supports the Yang-Mills MHV (anti-MHV, hereafter MHV) amplitudes [16,48,58]. In [1], the present authors proved this point analytically, namely, using the CHY prescription, the solutions in eq. (2.2) do reproduce the correct Parke-Taylor formula [2] for Yang-Mills and Hodges formula [7] for pure gravity at MHV. In particular, if the two negative helicities are located at position i and j, we have where Ψ and Φ are evaluated on σ (with k and dependency suppressed). Next, F and P ξ contain only the Möbius degrees of freedom: which will not appear in the final expression for amplitudes. Finally, s(n) = (n 2 − 3n + 8)/2 provides an unobservable overall sign to the amplitudes. On the other hand, the vanishing of Pf (Ψ) has been proved for the solution {σ a } in (2.2) while it remains a numerical fact for the others. More generally, there is an intriguing Eulerian number pattern on how Pf (Ψ) is supported by the solutions. Such an observation has been indicated in some earlier work, for example, [59,60], but a full understanding is still elusive.
As an immediate application of the discussion above, since the EYM amplitude is proportional to Pf (Ψ), at MHV we only need to calculate Pf (Ψ h ) at the special solution σ, although in general, Pf (Ψ h ) is nonzero at other solutions.
In this case, suppose the two negative helicity gluons are g i and g j while all the gravitons have positive helicities. We choose the polarization to be: namely, all the reference vectors are the same. Then we always have · = 0 such that B h is identically zero. The 2s × 2s matrix Ψ h now has the form: Then Pf (Ψ h ) = 0 if and only if C h is of full rank. Indeed, if C h has rank deficiency, we can always make one row of it zero by elementary transformations, and thus Ψ h has one row of zeros such that det(Ψ h ) = 0. Next, independent of the solutions, we always have: namely, we can pretend that A h = 0. The reason is that if C h is of full rank, there always exists an elementary transformation that makes A h vanish, and then we can use the formula for the pfaffian of an off-diagonal block matrix. After plugging in σ, we get: When calculating the determinant, we can pull out F h a ξ / h a q from each row and h b ξ h b q from each column. This calculation leads to where φ h is the s × s diagonal submatrix of the Hodges matrix φ [7] with all gluon rows and columns deleted: The n × n Hodges matrix φ is given by: where the spinors |ξ and |η represent gauge freedom and the diagonal elements φ aa do not depend on it.
We can even choose them to be an external momentum spinor |i , as long as the i-th row and column have been deleted for some reason. The matrix φ has rank n − 3 so that its nonzero minors are at most n − 3 dimensional. Combining eq. (3.7) with eq. (3.1), we get the color-ordered (g − g − ) MHV amplitude as: 1 For h = ∅, we define φ ∅ = 1. In Sec. 4, we will show that det(φ h ) can arise from the SBFW prescription.
In this case, the particle h i and g j have negative helicities while all the other particles have positive helicities. The set of positive helicity gravitions is thus h + = h\{h i }. We fix the gauge freedom in as: such that we still have B h = 0 and eq. (3.5) holds. In addition, the i-th column of C h is zero except for the diagonal element C ii under our gauge choice. Therefore, det(C h ) evaluates to where in the last equality, we have plugged in the special solution σ, such that: This calculation leads to such that the color-ordered amplitude reads: Comparing eq. (3.10) with (3.15), we find that for at least one negative helicity gluon g j : where φ h + is a submatrix of the Hodges matrix φ whose indices belong to the set of positive helicity gravitons. For h + = ∅, we define φ ∅ = 1. After expanding the determinants, we can check the boundary cases with only two gluons: vanishes since now φ h + is n − 2 dimensional while φ only has rank n − 3, which agrees with the analysis in [61].
, matches perfectly with those existing results [61,62] for three-and four-point cases.

(h − h − ) amplitudes
In this case, the two negative helicity particles are h i and h j such that h − = {h i , h j }. We are going to prove that Pf [Ψ h (σ)] = 0 such that the amplitude vanishes identically. Actually, here we prove a stronger statement than this: if |h − | 2, we always have Pf [Ψ h (σ)] = 0. First, we choose the polarizations as where p and q are two arbitrary reference vectors that do not coincide with any graviton momentum. By plugging in σ and extracting common factors in each row and column, we can reach at where Ψ h is composed of the following blocks in the same way as eq. (2.8): • A-part: (3.20) • C-part: (3.21) We immediately observe that in C the rows that belong to h − are identical. Using this feature, we can perform the following elementary transformations, after choosing one reference particle i ∈ h − : 1. For all j ∈ h − and j = i, subtract the (s + j)-th row and column of Ψ h by the (s + i)-th row and column. This operation makes the j-th row of C and the j-th column of − C T zero.
2. Subtract the first s rows and columns (except for the i-th) of Ψ h by a multiple of the i-th row (denoted by the subscript i×) and column (denoted by the subscript ×i): This operation makes the i-th row of C and the i-th column of − C T zero, except for C ii . Less obviously, it also makes A zero, except for the i-th row and column: where the Schouten identity has been used in the numerator.
3. Subtract the first s rows and columns of Ψ h by a multiple of the (s + i)-th row and column: This operation makes the entire A zero.
4. Finally, for j ∈ h + , subtract the (s + j)-th row and column by a multiple of the (s + i)-th: Now the (s + i)-th row and column of Ψ h are zero except for C ii = [iq]/ ip .
We can then delete this row and column after pulling C ii out of the pfaffian, and we call the left-over 2s − 2 dimensional submatrix Ψ h : where in Ψ h the upper left diagonal block is identically zero, with dimension s + s − − 2. The lower left off-diagonal blocks has dimension s + × (s + s − − 2), in which the columns are more than the rows when s − 2 and s 3. Therefore, we can always find an elementary transformation to make at least one column zero in this block, such that Pf ( Ψ h ) = det( Ψ h ) = 0. Finally, for the simplest case s = s − = 2, the last two lines and columns of Ψ h are proportional since B = 0, which makes Pf ( Ψ h ) vanish.
Therefore, we have proved that Pf [Ψ h (σ)] = 0 for all possible s with at least two negative helicity gravitons. As a result, the (h − h − ) amplitude of EYM vanishes. Using the CHY formalism, we are now able to give an direct proof to the statement in [54]: the gluon all-plus single-trace MHV amplitudes of EYM are identically zero.
Before we proceed, we note that this technique can also be used to prove that Pf [Ψ(σ)] = 0 for all non-MHV amplitudes. The proof will be given in Appendix A.

Summary of Results
Here we first give a summary on what we have done in this section. The CHY formalism states that the color-ordered EYM amplitude with s gravitons and r gluons can be obtained from: At MHV, Pf (Ψ) is only supported by the special solution σ. By calculating Pf [Ψ h (σ)], we find that: • the (g − g − ) and (h − g − ) MHV amplitudes can be written as: • the (h − h − ) MHV amplitude vanishes, due to Pf [Ψ h (σ)] = 0, which analytically justifies the very well motivated conjecture given in [54].
In Sec. 4, we we will prove the equivalence between eq. (3.24) and the SBDW formula.

The Connection to the SBDW Formula
In [52,53], Selivanov proposed an exponential formula for tree-level single trace (g − g − ) MHV amplitudes. Bern, De Freitas and Wong generalized KLT relation and then proposed that the (g − h − ) MHV amplitudes also have a similar exponential expression [54]. In this paper, we refer this exponential formula as Selivanov-Bern-De Freitas-Wong (SBDW) formula, which states that: 2 where S is given by: where h + is the complement of h + in p = {1, 2, . . . , n}. The matrix element ψ ab is related to φ ab through: Our S(h + ; p) is of a more general form comparing with the original one in [54] since ours allows an arbitrary choice of the reference spinors. Using ψ ab , We can define another matrix: One then immediately sees that all the diagonal minors of W n are equal to those of φ, independent of the choice of |ξ and |η . In particular, we have: 2 Here we neglect an overall factor which depends on coupling constants.
Then to prove the equivalence of eq. (4.1) and eq. (3.24), we only need to show that: We will show that the both sides have the same graph theory interpretation and thus equal. First, the following identity holds for S: where K n is a weighted complete graph 3 with the vertex set {v 1 , . . . , v n } and weight ψ ab assigned to the edge v a v b . The summation on the right hand side of (4.6) is over F Ir (K n ), the set of spanning forests of K n rooted on the vertices with labels I r = {i 1 , i 2 , . . . , i r }. Some examples of spanning forests are presented in Fig. 1. Such a forest F has the same vertex set as K n while its edge set is denoted as E(F ). The forest F has exactly r trees, each of which contains one and only one vertex of I r . Then G(a Ir ; p), as defined in eq. (4.2), is a multivariate generating function of spanning forests in the sense that at fixed n and ψ ab = 1, its Taylor expansion coefficient of a i r+1 a i r+2 · · · a in equals the number of spanning forests of K n with r trees rooted in I r . In Sec. 4.1, we study a 7-point example to help readers understand the idea of the general inductive proof, which will then be given in Sec. 4.2.
On the other hand, it has been demonstrated in [8] that the evaluation of det[(W n ) h + ] has the same graph theory interpretation as S. The matrix W n actually represents the weighted complete graph K n , while ψ ab is the weight assigned to the edge v a v b and (W n ) aa is the total weight associated to the vertex v a . We have a beautiful matrix-forest theorem 4 , as given in [8]: (4.7) Thus if we choose h + = I r , the statement (4.5) is the direct consequence of our new theorem (4.6) and the matrix-forest theorem (4.7).

A Seven-point Example of The Matrix-forest Theorem
In this section, we explicitly calculate the factor S(h + ; p) of the amplitude: and demonstrate the graphic correspondence of the expansion (4.6). To simplify the presentation, we relabel the particles as: The root set is {4, 5, 6, 7}. However, by choosing the reference spinor in eq. (4.3) to be: we can substantially reduce the number of the graphs involved since now ψ 6a = ψ 7a = 0 for all a ∈ {1, 2, 3, 4, 5} .
In other words, the vertex 6 and 7 are always disjoint to the others. We have thus reduced the problem to the spanning forests of K 5 (instead of K 7 ) with vertex set 5  by brute force, enumerate the spanning forests, and demonstrate that eq. (4.6) indeed holds in this case. Instead, we use a strategy that helps to better understand the inductive proof in the next subsection: we divide the graphs into lower order ones, check the theorem and then add them up. First, we pick up the root 5 and expand the right hand side of eq. (4.6) with respect to those ψ's whose subscripts contain 5: ψ 5l ψ 5k C lk + ψ 51 ψ 52 ψ 53 D . (4.14) We can verify that B 3 can also be generated by: At the moment, we have verified that Theorem (4.6) holds for all 4-vertex spanning forests. Finally, one can verify by brute force that the expansion (4.10) indeed equals eq. (4.9) such that Theorem (4.6) also holds for our 5-vertex forests. Astute readers may immediate see that if the final step holds, we have just completed an inductive-style proof from 4-vertex graphs to 5-vertex ones. The general inductive proof from level n − 1 to level n uses exactly the same construction.
To close this subsection, we note that in this example we have encountered an evaluation of |A A A| + |B B B| + |C C C| + |D D D| = 16 + 24 + 9 + 1 = 50 forests, which keep proliferating for larger n. Thus without Theorem (4.6), the SBDW-style calculation would be very involved and it would be very difficult to tell that such a massive summation over 50 terms simply equals a much simpler 3 × 3 determinant according to (4.7). This example also demonstrates that in this case the CHY-style direct evaluation of EYM is much more powerful than the SBDW formula.

General Proof
In this subsection, we present the general inductive proof of eq. (4.6). The method is parallel to the one used in [8] to prove eq. (4.7). To start the induction, we first show that the n = 2 case holds. The only nontrivial scenario at n = 2 is I = {1} and I = {2} (or vice versa), such that: . (4.18) Next, we assume that eq. (4.6) holds for n−1. At n vertices, we can construct a forest with roots I r by first constructing a forest of n − 1 vertices with roots I r−1+t = {i 1 , . . . , i r−1 , p 1 , . . . , p t }, and then connecting the vertex i r to P t = {p 1 , . . . , p t } ⊂ I r : -18 -Consequently, we have: where the second line is obtained according to our induction assumption. The set I r−1+t is the complement of I r−1+t in the set p = p\{i r }. In the following, a bar with a prime always means the complement of a set in p . In particular, we have I r = I r−1 . Our next job is to show that eq. (4.20) is nothing but the expansion of S(I r ; p) with respect to ψ irp k . First, we observe that ψ irp k only appear in the out-most level of G(a Ir , ψ): By setting all ψ irp k to zero, we obtain the zero-th order of S(I r ): which agrees with the first term in eq. (4.20). At a generic order t with P t = {p 1 , . . . , p t }, we have: In the exponent part, we can leave only those ψ irp k in P t be nonzero such that eq. (4.21) further transforms into: n 1 ∈Ir a n 1 l∈Ir ψ ln 1 = n 1 ∈I r−1+t a n 1 Then the t-th order term corresponds to acting the underlined derivatives of eq. (4.23) onto the underlined part of eq. (4.24), and then setting all a p k = 0:  All the other ways of distributing derivatives result in lower order terms in the expansion. We also note that (4.25) is the only t-th order term we can have with P t specified. These two observations indicate that given the set P t , the highest order in the expansion of ψ irp k ∈ P t is t. Since P t cannot be larger than I r , the expansion of S(I r ; p) must terminate at the order n − r. Therefore, we have:  which is exactly eq. (4.20). The inductive proof of eq. (4.6) is thus complete, and the desired equality: follows immediatly by choosing I r = h + in eq. (4.6).

Conclusion and Discussion
In this paper, we proposed a new compact formula of the tree-level single-trace MHV amplitudes of Einstein-Yang-Mills theory, which results from a direct evaluation using the CHY formalism. The amplitudes with (g − g − ) and (h − g − ) configurations are expressed by multiplying a Parke-Taylor factor with a Hodges minor. We proved analytically that the amplitudes with (h − h − ) configuration have to vanish. We also established a graph theoretical interpretation of the SBDW formula for MHV amplitudes of Einstein-Yang-Mills, and further proved that our new formula, eq. (3.24), is equivalent to the SBDW formula.
There are some problems that still deserve further investigation: • In the work [14], CHY formula for amplitudes beyond single trace was discussed. Is there similar compact formula for more general amplitudes in EYM?
Next, we perform the same set of elementary transformations as in Sec. 3.3 and we can also reach eq. (3.22). In this case, the resultant matrix Ψ has a zero block with dimension n − 4 + n − in the upper left corner, and its lower left corner have dimension n + × (n − 4 + n − ). This block is square when n − = 2, which is exactly the MHV configuration, and the result has been given in eq. (3.1). For non-MHV (n − 3), we have more columns than rows in Ψ such that we can always find an elementary transformation to make at least one column of the lower left corner zero, which leads to Pf [Ψ(σ)] ∝ Pf ( Ψ ) = 0.