A note on multi-trace EYM amplitudes in four dimensions

To show the two formulas (2.1) and (2.7) are equivalent to each other, we introduce the following three transformations of spanning forests.

Transformation-1:

When the $k_{c}\cdot k_{d}$ in (2.2) is expressed by spinor products, the expression corresponding to the structure on the LHS of Fig. 3 is given by

Note that the minus of the factor for the type-3 edge $e(v_{j},u_{i})$ is absorbed due to the antisymmetry of the spinor product $\left[u_{i},v_{j}\right]=-\left[v_{j},u_{i}\right]$. After dividing out the common factors in the numerator and denominator, the factors involving the reference spinors $\zeta$ and $\chi$ are collected as

where the Schouten identity $\left\langle a,b\right\rangle\left\langle c,d\right\rangle+\left\langle a,c\right\rangle\left\langle d,b\right\rangle+\left\langle a,d\right\rangle\left\langle b,c\right\rangle=0$ has been applied. When the RHS of eq. (2.9) are inserted, the expression (2.8) splits into two terms corresponding to the graphs on the RHS of Fig. 3, where the first contains the type-3 and -4 edges introduced in eq. (2.7) and the second still contains the type-1 and -2 edges in eq. (2.1) but the choices of gauge on both sides of the type-1 edge are exchanged.

Transformation-2:

For given $a$, $b$ on the LHS of Fig. 4, all nodes in the boxed region are considered as roots. Other nodes are connected to the roots via tree structures. The summation over all possible such spanning forests are implied by the box. In the first graph on the RHS, we sum over all possible spanning forests with keeping the chain structure $v_{j}\to v_{j-1}\to...\to v_{1}$ and then connect the node $v_{j}$ to $u_{i}$ via a type-1 edge. The contributions where $b$ (the root of the chain $v_{j}\to v_{j-1}\to...\to v_{1}$) is (i). a node outside the boxed regions, (ii). a node belonging to $\{u_{1},...,u_{i}\}$ as well as (iii). a node in the trace $\boldsymbol{1}$ are subtracted as the last three graphs on the RHS of Fig. 4. Then only the cases where $b$ is a node in trace $\boldsymbol{2}$ survive, thus the equality of Fig. 4 holds. An explicit example for this transformation is given by Fig. 5: According to the definition, the first graph on the RHS of Fig. 4 where two gravitons $l_{1}$ and $l_{2}$ belong to the set $L$ (the set of gravitons that are not in set $\{u_{1},u_{2}\dots u_{i},v_{1},v_{2},\dots v_{j}\}$) gives the equation of Fig. 5, which can be verified by moving the last four graphs on the RHS to the LHS.

Transformation-3:

If the set $\{v_{1},...,v_{j}\}$ is empty, the first term on the RHS of transformation-2 contains a factor

where momentum conservation has been applied. Transformation-2 then turns into the relation shown by Fig. 6.

To prove the formula (2.7), we note that the RHS of (2.1) has the general pattern of the LHS of Fig. 4 or Fig. 6. One can therefore transform these factors according to transformation-2 and -3, whose terms on the RHS can be classified into the following three categories.

• •

  Category-1  The first two graphs on the RHS of Fig. 4 and the first graph on the RHS of Fig. 6 have a common structure: all possible spanning forests while a chain structure ($v_{j}\to v_{j-1}\to...\to v_{1}$, $v_{j}\to v_{j-1}\to...\to b$ in the first two terms of Fig. 4 and the single node $b$ in Fig. 6) is preserved are summed over and the starting node of this chain is connected to $u_{i}$ via a type-1 edge. All these graphs belong to category-1.

• •

  Category-2  In the third graph on the RHS of Fig. 4 and the second graph on the RHS of Fig. 6, the chain $v_{j}\to v_{j-1}\to...\to v_{1}$ is planted at a node in the set of gravitons $\{u_{1},...,u_{i}\}$ and $v_{j}$ is connected to $u_{i}$ via a type-1 edge. Thus there is a loop structure with gravitons on it for a graph in this category.

• •

  Category-3  In the last graph on the RHS of Fig. 4 and the last graph on the RHS of Fig. 6, all graphs where the chain $v_{j}\to v_{j-1}\to...\to v_{1}$ pointing towards a gluon in trace $\boldsymbol{1}$ are summed over. The starting node $v_{j}$ is also connected to $u_{i}$, via a type-1 edge. Thus there is a bridge between two gluons belonging to the trace $\boldsymbol{1}$ in such a graph.

Now let us track back to the origins of these graphs in distinct categories and see the cancellations among graphs:

• •

  The typical graph of category-1 shown by Fig. 7 (a) where the nodes on the bridge between $a$ and $b$ are $u_{1},...,u_{i},v_{j},...,v_{1}$ in turn, can only be obtained from the expansions of Fig. 7 (b) and (c). Since Fig. 7 (a) is the first term in the expansion of Fig. 7 (b) but the second term in the expansion of Fig. 7 (c) when transformation-2 is applied, one such figure must appear twice with opposite signs and then cancel in pairs.

• •

  Graphs of category-2 appear in pairs as shown by Fig. 8 (b) and (e), which respectively come from the decomposition of spanning forests (a) and (d). These two graphs are related to one another by exchanging the roles between the chain structures $v_{j}\to v_{j-1}\to...\to v_{1}\to u_{l}$ and $u_{i}\to u_{i-1}\to...\to u_{l+1}\to u_{l}$. The graph (e) further splits into the two graphs (c) and (f), when transformation-1 is applied. The graph (c) cancels with (b), while the graph (f) has to vanish due to the antisymmetry of spinor products. Thus graphs of category-2 have to cancel out.

• •

  Graphs of category-3 also come in pairs as shown by Fig. 9 (b) and (f), which are obtained from the spanning forests (a) and (e). When applying transformation-1 to the graph (f), one gets the two graphs (c) and (g). The graph (c) cancels with (b), while the graph (g) is further expanded into graphs (d) and (h). This is because the summation over $a\in\boldsymbol{1},b\in\boldsymbol{1}$ can split as

  $\displaystyle\sum\limits_{\begin{subarray}{c}a\in\boldsymbol{1}\\ b\in\boldsymbol{1}\end{subarray}}\to\sum\limits_{\begin{subarray}{c}a\prec b\\ a,b\in\boldsymbol{1}\end{subarray}}+\sum\limits_{\begin{subarray}{c}b\prec a\\ a,b\in\boldsymbol{1}\end{subarray}}.$

  (2.11)

  When $a$, $b$ are renamed by $b$, $a$, respectively, the second summation on the RHS turns into a summation over $a\prec b~{}(a,b\in\boldsymbol{1})$ but the structure between $a$ and $b$ is reflected, as shown by the graph (d) in Fig. 9 (the direction of the arrow line from $u_{i}$ to $v_{j}$ is adjusted by absorbing a minus into the coefficient $\left\langle b,a\right\rangle=-\left\langle a,b\right\rangle$).

There is a boundary case of category-3 when both sets $\{u_{1},...,u_{i}\}$ and $\{v_{1},...,v_{j}\}$ are empty, as shown by the graph (b) in Fig. 10, which is obtained after applying transformation-3 to spanning forest (a). When we apply transformation-1 to this graph (b), the LHS of Fig. 3 is same with the second term on the RHS (upto a minus), thus the LHS is half of the first term on the RHS, as shown by the first equality of Fig. 10. The summation over all $a,b\in\boldsymbol{1}$ is further written as twice of the summation over $a,b$ ($a,b\in\boldsymbol{1},a\prec b$).

To sum up, only the graphs of structures Fig. 9 (d), (h) and Fig. 10 (d) survive after cancellations (An explicit one-graviton example is given in appendix A.). These graphs are those characterizing eq. (2.7), hence proof of the equivalence between eq. (2.1) and eq. (2.7) is completed.

In the Cachazo-He-Yuan (CHY) framework, an $n$-point scattering amplitude $M_{n}$ can be given by the following (integrated) formula

In the above expression:

• •

  One have summed over solutions $\{\sigma_{a}\}$ ($a=1,...,n$) of the following scattering equations for scattering variables $\{z_{a}\}$

  $\displaystyle\sum\limits^{n}_{\begin{subarray}{c}b=1\\ b\neq a\end{subarray}}\frac{s_{ab}}{z_{ab}}=0\,,\,\,a\in\{1,2,\dots,n\}\,,$

  (3.2)

  where $s_{ab}=2k_{a}\cdot k_{b}$ and $z_{ab}=z_{a}-z_{b}$. In four dimensions, the scattering equations have a special solution (which is called MHV solution)

  $\displaystyle{\sigma^{\text{MHV}}_{a}}=\frac{\left\langle a,\eta\right\rangle\left\langle\theta,\zeta\right\rangle}{\left\langle a,\zeta\right\rangle\left\langle\theta,\eta\right\rangle}\,,$

  (3.3)

  where the spinors $\eta$, $\theta$ and $\zeta$ represent the M$\ddot{\text{o}}$bius freedom of scattering equations.

• •

  The reduced determinant $\text{det}\,^{\prime}[\Phi]$ is defined by

  $\displaystyle\text{det}\,^{\prime}[\Phi]\equiv perm(ijk)perm(pqr)\frac{\text{det}[\Phi^{ijk}_{pqr}]}{\sigma_{ij}\sigma_{jk}\sigma_{ki}\sigma_{pq}\sigma_{qr}\sigma_{rp}},$

  (3.4)

  where $perm(ijk)$ is the signature of the permutation that moves the standard ordering $(1,2,...,n)$ to $(i,j,k,...)$ and keeps $(...)$ ascending. The $n\times n$ matrix $\Phi$ is given by

  $\displaystyle\Phi_{ab}=\Bigg{\{}\begin{array}[]{cc}\frac{s_{ab}}{\sigma^{2}_{ab}}&,a\neq b\\ -\sum\limits_{c\neq a}\frac{s_{ac}}{\sigma^{2}_{ac}}&,a=b\\ \end{array}\,,$

  (3.7)

  and $\Phi^{ijk}_{pqr}$ is the submatrix obtained by removing the $(i,j,k)$-th row and the $(p,q,r)$-th column from $\Phi$.

• •

  The CHY integrand ${\cal I}_{n}$ relies on theories. In EYM theory, the CHY integrand $\mathcal{I}_{\text{EYM}}$ for color-odered $m$-trace amplitudes have the following form

  $\displaystyle\mathcal{I}_{\text{EYM}}(\boldsymbol{1}|\dots|\boldsymbol{m}\|\mathsf{H})=\underbrace{\big{(}\mathcal{C}_{\boldsymbol{1}}\dots\mathcal{C}_{\boldsymbol{m}}\,{\sum\limits}^{\prime}_{\{a,b\}}P_{\{a,b\}}\big{)}}_{\mathcal{I}_{L}}\underbrace{\text{Pf}\,^{\prime}[\Psi]}_{\mathcal{I}_{R}}\,,$

  (3.8)

  where $\mathcal{C}_{\boldsymbol{i}}\equiv\frac{1}{z_{a_{1}a_{2}}z_{a_{2}a_{3}}\dots z_{a_{s}a_{1}}}$ is the Parke-Taylor factor for the gluon trace $\boldsymbol{i}=\{a_{1},a_{2},...,a_{s}\}$ and the reduced Pfaffian $\text{Pf}\,^{\prime}[\Psi]$ is defined as

  $\displaystyle\text{Pf}\,^{\prime}[\Psi]\equiv\frac{perm(ij)}{z_{ij}}\text{Pf}[\Psi_{ij}^{ij}],\,(1\leq i<j\leq n).$

  (3.9)

  The $\Psi$ is an $2n\times 2n$ antisymmetric matrix with the following structure

  $$\Psi=\begin{pmatrix}A&-C^{T}\\ C&B\\ \end{pmatrix},$$

  where the three blocks $A$, $B$ and $C$ are $n\times n$ matrices defined as follows

  $\displaystyle A_{ab}=\Bigg{\{}\begin{array}[]{cc}\frac{k_{a}\cdot k_{b}}{z_{ab}}&,a\neq b\\ 0&,a=b\\ \end{array}\,,~{}~{}B_{ab}=\Bigg{\{}\begin{array}[]{cc}\frac{\epsilon_{a}\cdot\epsilon_{b}}{z_{ab}}&,a\neq b\\ 0&,a=b\\ \end{array}\,,~{}~{}C_{ab}=\Bigg{\{}\begin{array}[]{cc}\frac{\epsilon_{a}\cdot k_{b}}{z_{ab}}&,a\neq b\\ -\sum\limits_{c\neq a}\frac{\epsilon_{a}\cdot k_{c}}{z_{ac}}&,a=b\\ \end{array}\,.$

  (3.16)

  The $\epsilon_{a}$ denotes half polarization of the graviton $a$. In the integrand (3.8),

  $\displaystyle{\sum\limits}^{\prime}_{\{a,b\}}P_{\{a,b\}}=\sum\limits_{\begin{subarray}{c}a_{1}\prec b_{1}\in\boldsymbol{1}\\ \dots\\ a_{m-1}\prec b_{m-1}\in\boldsymbol{m-1}\end{subarray}}sgn(\{a,b\})z_{a_{1}b_{1}}\dots z_{a_{m-1}b_{m-1}}\text{Pf}\,[\Psi]_{\mathsf{H},a_{1},b_{1},\dots,a_{m-1},b_{m-1};\mathsf{H}}\,,$

  (3.17)

  where $\{a,b\}$ represents $\{a_{1},b_{1},\dots,a_{m-1},b_{m-1}\}$.

It is worth pointing out that Pfaffians can be defined recursively

A simple but helpful property is the following: if a $2t\times 2t$ antisymmetric matrix has the form

where both $A$ and $C$ are $t\times t$ blocks, $O$ is an all-zero block. The Pfaffian of (3.19) can be reduced as

Now let us evaluate the double-trace amplitude with $(g^{-},g^{-})$-configuration by CHY formula straightforwardly. When the EYM integrand (3.8) for double-trace amplitude is substituted into the integrated expression (3.1), the amplitude becomes⁴⁴4The sign in (3.17) is absorbed if the $a$, $b$ in the Pfaffian are arranged in canonical order.

For the sector supported by the MHV solution (3.3), we have

where we omit the superscript of $\sigma^{\text{MHV}}_{a}$ for convenience. In the above expression, we have used the property introduced in Du:2016wkt

where the $i$, $j$ are the two negative-helicity particles, and

The submatrix $[\Psi]_{\mathsf{H},a,b;\mathsf{H}}$ of $\Psi$ consists of three blocks $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$. The indices of square matrices $A^{\prime}$ and $B^{\prime}$ correspondingly take values as (i). elements of $\mathsf{H}\cup\{a,b\}$ in order, and (ii). elements in $\mathsf{H}$. The block $C^{\prime}$ is the submatrix of $C$ with row and column indices taking values in $\mathsf{H}$ and $\mathsf{H}\cup\{a,b\}$ respectively.

In the case that the negative-helicity particles $i$, $j$ are gluons, the Lorentz contraction of momenta and polarizations in $\text{Pf}\,[\Psi]_{\mathsf{H},a,b;\mathsf{H}}$ are expressed by

where we choose the reference spinor for all positive-helicity particles as $\xi$, while that for the negative-helicity particles as $q$. When $(\left\langle h_{1},\zeta\right\rangle\dots\left\langle h_{t},\zeta\right\rangle)^{2}\left\langle a,\zeta\right\rangle\left\langle b,\zeta\right\rangle F^{2t+2}$ and $(\left\langle h_{1},\zeta\right\rangle\dots\left\langle h_{t},\zeta\right\rangle)^{2}\left\langle a,\zeta\right\rangle\left\langle b,\zeta\right\rangle$ are extracted out from rows and columns in $\text{Pf}\,[\Psi]_{\mathsf{H},a,b;\mathsf{H}}$, respectively, the amplitude turns into

where

In the coming subsection, we prove that the summation in eq. (3.26) can be expanded into the expression inside the square brackets of eq. (2.7). Then the result from the MHV sector (i.e. the sector supported by the MHV solution) of CHY formula precisely reduces into the expansion formula (2.7) and is therefore equivalent to the spanning forest form (2.1).

In this subsection, we expand the $\text{Pf}\,[\widetilde{\Psi}]_{\mathsf{H},a,b;\mathsf{H}}$ and show that this expansion results in the expression in the square brackets of eq. (2.7). We first provide the examples with one and two gravitons, then the general proof.