Note on the Hopf-Algebra-Based Formula of Yang–Mills-Scalar Amplitudes

Abstract

In this note, we study the Hopf-algebra-based (HAB) formula of Yang–Mills-Scalar (YMS) amplitudes, which expands a YMS amplitude with massive scalars as a combination of propagator matrices that mix massless scalars corresponding to gluons with the original massive scalars. We propose a recursive formula that conveniently expresses the HAB formula. In this formula, gluons are converted into massless scalars. Thus it expresses a YMS amplitude with massive scalars by amplitudes with fewer gluons, massive scalars and massless scalars. We verify this formula by using the soft behavior of amplitudes. We further show the equivalence between the massless limit of the HAB formula and an earlier proposed recursive expansion formula through explicit calculations on amplitudes with one and two gluons.

1. Introduction

The study of scattering amplitudes in Yang–Mills-scalar (YMS) theory plays a crucial role in understanding the perturbative relation between Yang–Mills (YM) theory and general relativity (GR). The reason is demonstrated as follows. Once YMS amplitudes are expanded in terms of bi-adjoint scalar (BS) [1,2] amplitudes, one can immediately write down an expression of Yang–Mills amplitude, also written in terms of BS ones [3]. Coefficients in the formula have been used to generate all Bern–Carrasco–Johansson (BCJ) [4,5] numerators [6], kinematic numerators satisfying Jacobi identities with the corresponding color factors, which are the key objects in the double-copy relation between YM and GR.

A concrete realization of the expansion of massless-scalar YMS amplitudes into BS ones is the recursive expansion formula [3]; more on this recursive formula can be found in [7,8,9,10,11], which is based on the manifesting of gauge invariance. According to this approach, any tree-level single-trace YMS amplitude (with massless scalars) is expressed into a combination of YMS amplitudes with fewer gluons. This formula, when applied iteratively, finally results in a BS expansion of YMS amplitudes. It has been shown that the recursive expansion relation is effective for the construction of tree-level Bern–Carrasco–Johansson (BCJ) [4,5] numerators. They are also generalized to one-loop integrands by the forward limit strategy [12,13,14,15,16,17]. An interesting perspective is that the recursive expansion formula can be determined by the universal soft behaviors; see [18,19,20,21]. More related work can be found in [22,23,24].

The Hopf-algebra-based (HAB) formula [25] is an alternative approach to the YMS amplitude (but with general massive scalars). This formula originated from the study of heavy-mass effective field theory [26,27]. With the help of the HAB formula, YMS amplitudes with massive scalars (in this note, all massive scalars are supposed to have the same mass), are expressed in terms of scalar propagator matrices combining massless scalars corresponding to gluons and the original massive scalars. The expansion coefficients are generated by a systematic rule based on the framework of Hopf algebra. With the help of the double-copy statement, the HAB approach has been successfully applied in the study of massive particles scattering against gravitons [25,27,28,29,30,31] and has been used to calculate classical gravitational processes [32,33,34,35,36,37].

In this note, we study the HAB formula of YMS amplitudes. We arrange the HAB formula for YMS amplitudes with massive scalars into a convenient recursive relation, through which a YMS amplitude is written as a combination of amplitudes (these ‘amplitudes’ are intermediate objects; we call them amplitudes in the sense that (i) the massless limit gives the usual YMS amplitudes with massless scalars, and (ii) they are gauge-invariant), mixing massive scalars, massless scalars and fewer gluons. The HAB formula can be regarded as an iterative result of this recursive formula. We verify this formula by the soft limit approach. We further prove that the HAB formula in the massless limit is equivalent to the other approach [3] by explicit calculations on amplitudes with one and two gluons. This work provides a convenient expression of the BCJ numerators for amplitudes with gluons coupling to massive scalars, which have been successfully applied in the study of their double-copy process with gravitons coupling to massive scalars. In addition, we partly fill the gap between the HAB formula [25] and the recursive expansion formula [3]. We hope this work may provide more hints for understanding the double-copy relation between gauge field and gravity.

The structure of this paper is organized as follows. In Section 2, we briefly review the recursive expansion formula proposed in [3], the HAB formula [25] as well as the construction of amplitudes based on universal soft behaviors [18,19,20,21]. The soft behavior of amplitudes is also introduced specifically for YMS amplitudes with massive scalars. In Section 3, we construct a recursive version of the HAB formula. A verification of this recursive formula is provided in Section 4, based on the soft behavior approach. In Section 5, we instead derive a recursive expansion of YMS amplitudes involving only massless scalars, starting from the known formula [3]. Through explicit calculation, we demonstrate that this expansion matches the massless limit of the HAB formula for one- and two-gluon cases. Finally, we summarize this work in Section 6. Helpful relations are included in Appendix A and Appendix B.

Convention of Notations

$k_i^{\mu }$: momentum of gluon $g_i$, ${ϵ}_i^{\mu }$: polarization vector of gluon $g_i$,

$p_i^{\mu }$: momentum of scalar i, $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_a)$: a permutation of a elements,

$F_i^{\mu \nu }\equiv k_i^{\mu }{ϵ}_i^{\nu }-{ϵ}_i^{\mu }{k}_i^{\nu }$: field strength tensor of gluon $g_i$,

$F_{\mathit{\alpha }}^{\mu \nu }\equiv F_{{\alpha }_1}^{\mu {\mu }_1}{F}_{{\alpha }_2}^{{\mu }_1{\mu }_2}\dots F_{{\alpha }_a}^{{\mu }_{a-1}\nu }$: contractions of strength tensors for ordered gluons

$\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_a)$,

$A⧢B$: shuffle permutations of A, B with keeping the relative ordering in each set,

$X_{\mathit{\alpha }}^L$ or $X_{\mathit{\alpha }}^R$: the total momentum of scalars on the left-hand side (or right hand side), of gluon ${\alpha }_1$ (or ${\alpha }_a$) in a given permutation where the relative order of gluons is $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_a)$,

For a YMS amplitude involves scalars in the ordering $1,\dots ,n$, we introduce $P^{\mu }$ to denote $p_1^{\mu }+p_2^{\mu }+\dots +p_{n-1}^{\mu }$.

If we have a permutation in $\{1,\dots ,n\}⧢{\mathit{\alpha }}_1⧢{\mathit{\alpha }}_2⧢\dots ⧢{\mathit{\alpha }}_{l-1}$ where ${\mathit{\alpha }}_1,\dots ,{\mathit{\alpha }}_{l-1}$ are ordered subsets of gluons that have been transformed into scalars in the previous steps, we define $P_l^{\mu }$ as $P_l^{\mu }\equiv P^{\mu }+k_{{\alpha }_1}^{\mu }+\dots +k_{{\alpha }_l}^{\mu }$.

${\mathbf{P}}_{{\mathit{\rho }}^g}^r$: ordered partitions that splits the gluon sequence ${\mathit{\rho }}^g$ into r ordered subsets, preserving the relative ordering in ${\mathit{\rho }}^g$

The notation ${\overline{g}}_i$ for a gluon $g_i$ is introduced in some places in order to emphasize that $g_i$ in the amplitude is regarded as a scalar.

2. A Review of Recursive Expansion, HAB Formula and Soft Behaviors

This section provides a review of two distinct approaches to the YMS amplitudes, the recursive expansion [3] formula and the Hopf-algebra-based approach [25,27,28,29,30,31]. Furthermore, we review the soft behavior construction of amplitudes, which plays a role in this paper.

2.1. The Recursive Expansion Formula of YMS

As proposed in [3], massless EYM amplitudes satisfy the recursive expansion formula, which also holds for YMS amplitudes (with $U\left(N\right)$ Yang–Mills fields coupled to $U\left(N\right)\times U\left(N\right)$ bi-adjoint scalar field, as defined in [1,2,38]). Concretely, for a tree-level doubly color-ordered massless YMS amplitude [3] involving n scalars ($1,2,\dots ,n$) and m gluons denoted by $\left\{g_i\right\}$, and a given right permutation $\mathit{\sigma }$ of all scalars and gluons, we have

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,\dots ,n\parallel \left\{g_i\right\} | \mathit{\sigma })= \left({ϵ}_f·X_f^L\right)A_{\mathrm{YMS}}(1,\{2,\dots ,n-1\}⧢g_f,n\parallel \left\{g_i\right\}\setminus g_f | \mathit{\sigma })\hfill \\ \hfill & +\sum _{\mathit{\alpha }}\left({ϵ}_f·F_{{\mathit{\alpha }}^{\mathrm{T}}}·X_{\mathit{\alpha }}^L\right)A_{\mathrm{YMS}}(1,\{2,\dots ,n-1\}⧢\{\mathit{\alpha },g_f\},n\parallel \left\{g_i\right\}\setminus (g_f\cup \mathit{\alpha }) | \mathit{\sigma }),\hfill \end{array}$$

(1)

in which, $g_f$ is an arbitrarily chosen gluon (fiducial gluon), and all non-empty ordered subsets $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_a)$ of gluons $\left\{g_i\right\}\setminus g_f$ on the second line have been summed over. The shuffle symbol ⧢ denotes the sum over all possible permutations of scalars $\{2,\dots ,n-1\}$ and gluons $\{\mathit{\alpha },g_f\}$ preserving the relative ordering in each ordered set. For example, $\{2,3\}⧢\{g_1,g_f\}$ implies that we sum over the following permutations

$$\begin{array}{cc}\hfill \{2,3\}⧢\{g_1,g_f\}: & \{g_1,g_f,2,3\},\{g_1,2,g_f,3\},\{g_1,2,3,g_f\},\hfill \\ \hfill & \{2,g_1,g_f,3\},\{2,g_1,3,g_f\},\{2,3,g_1,g_f\}.\hfill \end{array}$$

The ${\left(X_f^L\right)}^{\mu }$ is the sum of the momenta of scalars on the left-hand side of the fiducial gluon $g_f$ in a given permutation of scalars and gluon $g_f$. Similarly, ${\left(X_{\mathit{\alpha }}^L\right)}^{\mu }$ is the total momentum of scalars on the left-hand side of gluon ${\alpha }_1$ (the leftmost element in $\mathit{\alpha }$) in a given ordering of scalars and gluons $\{\mathit{\alpha },g_f\}$. In (1), we use ${ϵ}_i^{\mu }$ to denote the polarization vector for gluon $g_i$. The tensor $F_{{\mathit{\alpha }}^{\mathrm{T}}}^{\mu \nu }$ denotes the contraction of strength tensors $F^{\mu \nu }=k^{\mu }{ϵ}^{\nu }-{ϵ}^{\mu }{k}^{\nu }$ of gluons in $\mathit{\alpha }$ and takes the form

$$F_{{\mathit{\alpha }}^{\mathrm{T}}}^{\mu \nu }={(F_{{\alpha }_a}·\dots ·F_{{\alpha }_1})}^{\mu \nu },$$

(2)

where the ordering ${\mathit{\alpha }}^{\mathrm{T}}=({\alpha }_a,\dots ,{\alpha }_1)$ is the inverse of $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_a)$. In the recursive expression (1), amplitude must satisfy the gauge invariance identity

$$A_{\mathrm{YMS}}(1,\dots ,n\parallel \left\{g_i\right\}{ | \mathit{\sigma })|}_{{ϵ}_j\to k_j}=0$$

(3)

associating with any external gluon $g_j$. This can be understood as follows. In each term on the RHS of (1), $g_j$ can act as (i) a gluon (i.e., $g_j\in \left\{g_i\right\}\setminus (g_f\cup \alpha )$), (ii) a scalar in $\mathit{\alpha }$ or (iii) the fiducial scalar $g_f$. In case (i), the gauge invariance condition is satisfied according to inductive assumption. In case (ii), the gauge invariance is manifested through the antisymmetry of $F_j^{\mu \nu }$. In case (iii), the gauge invariance is far from obvious and has been proved in [3,10]. This identity guarantees the cancellation of the possible longitudinal parts of polarization vectors.

Applying (1) iteratively, a doubly color-ordered YMS amplitude can be expanded into pure color-ordered BS ones

$$A_{\mathrm{YMS}}(1,\dots ,n\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\rho }}C\left(\mathit{\rho }\right)A_{\mathrm{BS}}(1,\mathit{\rho },n | \mathit{\sigma }),$$

(4)

with 1 and n fixed as the two ends. Here we have summed over all possible permutations of the scalars and gluons preserving the scalar ordering $\{2,\dots ,n-1\}$. Each coefficient $C\left(\mathit{\rho }\right)$ arises from the corresponding class of expansion routes, which leads to the specific BS amplitude $A_{\mathrm{BS}}(1,\mathit{\rho },n | \mathit{\sigma })$. For example, YMS amplitude with two scalars and two gluons $A_{\mathrm{YMS}}(1,2\parallel \{g_1,g_2\} | \mathit{\sigma })$ can be expanded with the recursive expansion Formula (1) applied repeatedly

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}(1,2\parallel \{g_1,g_2\} | \mathit{\sigma })& =({ϵ}_1·p_1)A_{\mathrm{YMS}}(1,g_1,2\parallel g_2 | \mathit{\sigma })+({ϵ}_1·F_2·p_1)A_{\mathrm{BS}}(1,g_2,g_1,2 | \mathit{\sigma }),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}(1,g_1,2\parallel g_2 | \mathit{\sigma })& =\left[{ϵ}_2·(p_1+k_1)\right]A_{\mathrm{BS}}(1,g_1,g_2,2 | \mathit{\sigma })+({ϵ}_2·p_1)A_{\mathrm{BS}}(1,g_2,g_1,2 | \mathit{\sigma }),\hfill \end{array}$$

(6)

where we denote the momentum of scalar i by $p_i$ and that of gluon j by $k_j$. The second line shows a further expansion of YMS amplitude $A_{\mathrm{YMS}}(1,g_1,2\parallel g_2 | \mathit{\sigma })$, which is on the RHS of the first expansion. Two expansions are combined to derive the expansion to pure BS amplitudes. According to the specific BS amplitude $A_{\mathrm{BS}}(1,\mathit{\rho },2 | \mathit{\sigma })$, we can rearrange the expansion coefficients from the corresponding expansion routes into $C\left(\mathit{\rho }\right)$. The coefficient $C(g_1,g_2)$ arises from the two-step expansion route $(1,2)\to (1,g_1,2)\to (1,g_1,g_2,2)$, while $C(g_2,g_1)$ collects the contributions from $(1,2)\to (1,g_2,g_1,2)$ and the two-step expansion route $(1,2)\to (1,g_1,2)\to (1,g_2,g_1,2)$. Concretely, we have

$$\begin{array}{cc}\hfill C(g_1,g_2)=& ({ϵ}_1·p_1)\left[{ϵ}_2·(p_1+k_1)\right],\hfill \\ \hfill C(g_2,g_1)=& ({ϵ}_1·F_2·p_1)+({ϵ}_1·p_1)({ϵ}_2·p_1),\hfill \end{array}$$

(7)

and the explicit expression of the expansion of YMS amplitudes to pure BS amplitudes as

$$A_{\mathrm{YMS}}(1,2\parallel \{g_1,g_2\} | \mathit{\sigma })=C(g_1,g_2)A_{\mathrm{BS}}(1,g_1,g_2,2 | \mathit{\sigma })+C(g_2,g_1)A_{\mathrm{BS}}(1,g_2,g_1,2 | \mathit{\sigma }).$$

(8)

As shown by [3], this formula also holds for amplitude with a pair of massive scalars.

2.2. Hopf-Algebra-Based Formula

The Hopf-algebra framework [25,27,28,29,30,31] provides a systematic way to express the scattering amplitudes involving gluons and massive scalars or fermions. In this framework, the color ordering (in [25], mentioned as flavor orderings. To agree with the convention of many works on massless YMS amplitudes, we adopt ‘color ordering’ instead. We hope this will not cause any misunderstanding) is kept in the theory-independent Hopf-algebra structure, while the kinematic data of amplitudes is collected in the theory-dependent linear map. In this paper, we focus on the YMS amplitudes with massive scalars (where all scalar particles have the same mass) and massless gluons.

As proposed in [25], singly colored YMS amplitudes can be represented as

$$A_{\mathrm{YMS}}(\mathit{\varsigma },n)=\sum _{\Gamma \in R_{\mathit{\varsigma }}}{\displaystyle \frac{〈\widehat{\mathcal{N}}(\Gamma )〉}{d_{\Gamma }}}.$$

(9)

Here $\mathit{\varsigma }$ denotes the ordering of gluons and scalars with scalar n fixed due to the cyclic symmetry. We sum over all possible tri-graphs $\Gamma$ following the ordering $\mathit{\varsigma }$. Each tri-graph $\Gamma$ corresponds to a unique propagator in the denominator, and implies the nested commutator structure in the numerator. The following examples illustrate the tri-graphs along with their corresponding numerators and denominators.

(10)

(11)

Here, the Mandelstam variables $s_{i\dots g_j\dots }\equiv {(p_i+\dots +k_j+\dots )}^2$ (where $p_i,k_j,\dots$ denote the momenta of particles) are used to simplify expressions. Each single-particle generator $K_i$ is defined by $T_{\left(i\right)}^{\left(i\right)}$ for gluon $g_i$ and $T^{\left(i\right)}{t}^{a_i}$ for scalar i, where T refers to the kinematic part and $t^{a_i}$ is the generator of the color group for scalars; see [25,27,28]. The commutator of the single-particle generators is defined as $[K_i,K_j]=K_i★K_j-K_j★K_i$, while for the details of the fusion product ★, one can refer to [25,27,28]. Since each numerator is defined as the properly nested commutator determined by the corresponding tri-graph, these numerators satisfy the Jacobi identity naturally inherited from commutators and thus qualify as BCJ numerators [25].

The above singly color-ordered YMS amplitude (9) are connected to doubly color-ordered amplitudes in (1) and (4) via the color decomposition

$$A_{\mathrm{YMS}}(\mathit{\varsigma },n)=\sum _{\mathit{\eta }\in S_{n-1}}{A}_{\mathrm{YMS}}(\mathit{\eta },n\parallel \left\{g_i\right\} | \mathit{\varsigma },n)tr\left(t^{\mathit{\eta }}{t}^n\right).$$

(12)

On the LHS, $\mathit{\varsigma }$ is a permutation of all elements in $\{1,\dots ,n-1\}\cup \left\{g_i\right\}$. On the RHS, we use $tr\left(t^{\mathit{\eta }}{t}^n\right)$ as shorthand for $tr(t^{{\eta }_1}\dots t^{{\eta }_{n-1}}{t}^n)$, where each $\mathit{\eta }=\{{\eta }_1,\dots ,{\eta }_{n-1}\}$ is a permutation of scalars $\{1,\dots ,n-1\}$.

Considering that the propagator structure (10) and (11) of tri-graphs can be captured by the propagator matrix, an equivalent representation of YMS amplitudes was derived [25] from the expression (9)

$$A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\rho }}\mathbf{m}(1,\mathit{\rho },n-1,n | \mathit{\sigma }) {\mathcal{N}}_{\mathbb{I}}\left(1,\mathit{\rho },n-1,n\right),$$

(13)

where $\mathit{\rho }=({\rho }_1,\dots ,{\rho }_{n+m-3})$ denotes the permutation of $(n-3)$ scalars $\{2,\dots ,n-2\}$ and m gluons doubly color-ordered sum over all $\mathit{\rho }$ preserving the scalar ordering ${\mathit{\rho }}^s=\{2,\dots ,n-2\}$. The $\mathbf{m}(1,\mathit{\rho },n-1,n | \mathit{\sigma })$ denotes the propagator matrix [25] involving the original massive scalars and massless scalars, which are converted from gluons. For example, the propagator matrix $\mathbf{m}(1,2,g_1,g_2,3 | 1,2,g_1,g_2,3)$, involving massive scalars $1,2,3$ and massless scalars corresponding to $g_1,g_2$, is given as

$$\begin{array}{cc}\hfill & \mathbf{m}(1,2,g_1,g_2,3 \parallel 1,2,g_1,g_2,3)={\displaystyle \frac{1}{(s_{12}-m_s^2) s_{g_1{g}_2}}}+{\displaystyle \frac{1}{(s_{12}-m_s^2) (s_{12g_1}-m_s^2)}}\hfill \\ \hfill & +{\displaystyle \frac{1}{(s_{2g_1}-m_s^2) (s_{12g_1}-m_s^2)}}+{\displaystyle \frac{1}{(s_{2g_1}-m_s^2) (s_{2g_1{g}_2}-m_s^2)}}+{\displaystyle \frac{1}{s_{g_1{g}_2} (s_{2g_1{g}_2}-m_s^2)}} .\hfill \end{array}$$

(14)

The kinematic coefficient ${\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },n-1,n)$ in (13) is defined by applying the linear map ${〈·〉}_{\mathrm{m}}$ to the fusion product $\langle {\mathrm{K}}_1★{\mathrm{K}}_{{\mathit{\rho }}_1}★\dots ★{\mathrm{K}}_{{\mathit{\rho }}_{n+m-3}}★{\mathrm{K}}_{n-1}{\rangle }_{\mathrm{m}}$ , with respect to the ordering $\mathit{\rho }$. For details of the fusion product, see [25,27,28]. The explicit expression of the kinematic coefficient ${\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },n-1,n)$ for amplitudes with n massive scalars and m gluons is

$$\begin{array}{cc}\hfill {\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },n-1,n)=& {(-1)}^m\sum _{r=1}^m{(-1)}^r\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g}^r}\prod _{l=1}^r{\displaystyle \frac{2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}},\hfill \\ \hfill =& \sum _{r=1}^m\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g}^r}\prod _{l=1}^r-{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^l|}2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}}.\hfill \end{array}$$

(15)

On the RHS of the above expression, ${\mathit{\rho }}^g$ denotes the relative ordering of gluons in the original ordering of gluons and scalars $\mathit{\rho }$. For example, the overall ordering $(1,\mathit{\rho },3,4)=(1,g_1,2,g_2,3,4)$ implies gluon ordering ${\mathit{\rho }}^g=(g_1,g_2)$ and preserves the scalar ordering ${\mathit{\rho }}^s=(1,2,3)$ (including every scalar except n) in the original amplitude (13). We first sum, for fixed r, over all ordered partitions ${\mathbf{P}}_{{\mathit{\rho }}^g}^r$ that divide the gluon sequence ${\mathit{\rho }}^g$ into r subsets, each preserving the relative ordering in ${\mathit{\rho }}^g$, and then sum over the all possible r. For example, all ordered partitions of ${\mathit{\rho }}^g=(g_1,g_2,g_3)$ are given as

$$\begin{array}{cc}\hfill & {\mathbf{P}}_{{\mathit{\rho }}^g}^1:\left\{(g_1,g_2,g_3)\right\},\hfill \\ \hfill & {\mathbf{P}}_{{\mathit{\rho }}^g}^2:\mathrm{Perm}\{(g_1,g_2),\left(g_3\right)\},\mathrm{Perm}\{(g_1,g_3),\left(g_2\right)\},\mathrm{Perm}\{(g_2,g_3),\left(g_1\right)\},\hfill \\ \hfill & {\mathbf{P}}_{{\mathit{\rho }}^g}^3:\mathrm{Perm}\{\left(g_1\right),\left(g_2\right),\left(g_3\right)\}.\hfill \end{array}$$

(16)

Here Perm denotes all possible permutations of the subsets. Notice that the ordering within each subset inherits from the overall ordering ${\mathit{\rho }}^g=(g_1,g_2,g_3)$. The denominators in (15) are in the form of massive propagators with the scalar mass $m_s$. For a specific l, $P_l$ is defined as the sum of momenta carried by all scalar particles (excluding n) and all gluons contained in the sets ${\mathit{\alpha }}^1,\dots ,{\mathit{\alpha }}^{l-1}$. In the numerator, the tensor $F_{{\mathit{\alpha }}^l}^{\mu \nu }$ denotes the consecutive contraction of strength tensors ${(F_{{\alpha }_1^l}·\dots ·F_{{\alpha }_a^l})}^{\mu \nu }$ preserving the gluon ordering ${\mathit{\alpha }}^l=({\alpha }_1^l,\dots ,{\alpha }_a^l)$. The $X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}$ collects the momenta of the scalars in ${\mathit{\rho }}^s$ and gluons in ${\mathit{\alpha }}^1,\dots ,{\mathit{\alpha }}^{l-1}$ that lie to the left of gluon ${\alpha }_1^l$ with respect to $\mathit{\rho }$, while $X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}$ does the same for those particles on the right of gluon ${\alpha }_a^l$. For example, for the ordering $\mathit{\rho }=(1,g_1,2,g_2,3)$, the following shows the terms corresponding to different partitions $\left\{{\mathit{\alpha }}^l\right\}$:

$$\begin{array}{cc}\hfill \left\{(g_1,g_2)\right\}& :{\displaystyle \frac{2p_1·F_1·F_2·p_3}{p_{123}^2-m_s^2}},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \{\left(g_1\right),\left(g_2\right)\}& :{\displaystyle \frac{2p_1·F_1·p_{23}}{p_{123}^2-m_s^2}}{\displaystyle \frac{2(p_{12}+k_1)·F_2·p_3}{{(p_{123}+k_1)}^2-m_s^2}}.\hfill \end{array}$$

(18)

From the first line of (15) to the second line, we just re-expressed the sign ${(-1)}^{m+r}$ as ${\prod }_{l=1}^r(-1){(-1)}^{|{\mathit{\alpha }}_l|}$, recalling that the sum of $|{\mathit{\alpha }}_l|$ (the number of elements in ${\mathit{\alpha }}_l$) for a given partition with r subsets is just the total number of gluons, i.e., m.

The Formula (13) is suitable for YMS amplitudes with at least three scalars, since three scalars 1, $n-1$, and n are selected to be fixed. For YMS amplitudes with two scalars, the expansion of amplitudes [25] takes the form of

$$A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\rho }}\mathbf{m}(1,\mathit{\rho },g_1,2 | \mathit{\sigma }){\mathcal{N}}_{\mathbb{I}}\left(1,\mathit{\rho },g_1,2\right).$$

(19)

Here, the gluon $g_1$ is selected as the fiducial gluon, with its position held fixed. Concretely, the kinematic coefficient [25] is given by

$$\begin{array}{cc}\hfill {\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2)=& {(-1)}^m\sum _{r=1}^m{(-1)}^r\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g\setminus g_1}^r}-{\displaystyle \frac{2p_1·F_1·F_{{\mathit{\alpha }}^1}·p_1}{P_1^2-m_s^2}}\prod _{l=2}^m{\displaystyle \frac{2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}},\hfill \\ \hfill =& \sum _{r=1}^m\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g\setminus g_1}^r}{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^1|+1}2p_1·F_1·F_{{\mathit{\alpha }}^1}·p_1}{P_1^2-m_s^2}}\prod _{l=2}^m-{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^l|}2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}},\hfill \end{array}$$

(20)

where we sum over all ordered partitions of gluons ${\mathit{\rho }}^g\setminus g_1$ since $g_1$ is already considered as a scalar. $P_1\equiv k_1+p_1$ is defined as the momentum sum of the scalar 1 and the gluon $g_1$. For $l>2$, $P_l$ is extended to include the momenta of all gluons in the sets ${\mathit{\alpha }}_1$ through ${\mathit{\alpha }}_{l-1}$.

The decomposition Formulas (13) and (19) with the corresponding kinematic coefficients (15) and (20) are equivalent to (9). This can be verified by expanding the nested commutators in (9) and then working out the explicit expression of the fusion products therein, as shown by the appendix of [25]. After extracting the color factors associated with the scalars, the kinematic products precisely match with (15) (and (20) for the case with only gluons). Another understanding is the following. Since the numerators have been expressed via nested commutators (thus Jacobi identities), one can apply Jacobi identities to reduce them in terms of numerators corresponding to half-ladder diagrams and extract the color factors for scalars. The propagators associated with each numerator with respect to the half-ladder diagram are collected as the propagator matrix $\mathbf{m}$. This resulting expression is written with two fixed scalars; a further application of the BCJ relation will reduce it into amplitudes with three fixed elements.

In the current paper, we derive a recursive expansion formula of YMS amplitudes, starting from the (Hopf-algebra-based) HAB Formulas (13) and (19) with coefficients (15) and (20), respectively.

2.3. The Soft Behavior Approach to YMS Amplitudes

Scattering amplitudes satisfy the universal soft behavior [39,40], which states that an n-point amplitude is factorized into an $n-1$ point amplitude and a soft factor when the momentum of a massless particle tends to zero. In this subsection, we review the soft scalar and soft gluon behaviors of BS and YMS amplitudes, which will be useful in the reconstruction of amplitudes in upcoming Section 4.

The leading soft behavior of BS amplitudes with $p_i\to \tau p_i$ ($\tau \to 0$) is given by [18]

$$A_{\mathrm{BS}}^{{\left(0\right)}_i}(1,\dots ,n | \mathit{\sigma })=S_i^{\left(0\right)}{A}_{\mathrm{BS}}(1,\dots ,i-1,\cancel{i},i+1,\dots ,n | \mathit{\sigma }),$$

(21)

where $\tau$ has been absorbed into the leading soft factor for later convenience. The soft factor (in many works about soft behaviors, such as [41,42], the $\tau$ dependence is separated from the soft factor, and with this definition the soft factor does not have divergence caused by ${\displaystyle \frac{1}{\tau }}$; in this work, the dependence of $\tau$ is absorbed into the soft factor for convenience) for scalar $s_i$ takes the form

$$S_i^{\left(0\right)}={\displaystyle \frac{1}{\tau }}\left({\displaystyle \frac{{\delta }_{(i-1)i}}{s_{(i-1)i}}}+{\displaystyle \frac{{\delta }_{i(i+1)}}{s_{i(i+1)}}}\right).$$

(22)

Here the operator ${\delta }_{ij}$ is defined as

$${\delta }_{ij}=\left\{\begin{array}{cc}1\hfill & i\prec j\hfill \\ -1\hfill & j\prec i\hfill \\ 0\hfill & i,j \mathrm{not} \mathrm{adjacent}\hfill \end{array}\right.,$$

(23)

which is determined by the relative order of scalars i and j in the given ordering $\mathit{\sigma }$. The notation $i\prec j$ means that i is on the left-hand side of j in $\mathit{\sigma }$. Similarly, the leading soft behavior of YMS amplitudes with a soft scalar i are represented as

$$A_{\mathrm{YMS}}^{{\left(0\right)}_i}(1,\dots ,n\parallel \left\{g_j\right\} | \mathit{\sigma })=S_i^{\left(0\right)}{A}_{\mathrm{YMS}}(1,\dots ,i-1,\cancel{i},i+1,\dots ,n\parallel \left\{g_j\right\} | \mathit{\sigma }),$$

(24)

where the leading scalar soft factor is the same as that of BS amplitudes (22) due to the universality of the scalar soft factor.

Analogous to the soft behavior of scalars, the YMS amplitudes are factorized under the soft limit of gluons. Specifically, when gluon $g_i$ is the soft one, i.e., $k_i\to \tau k_i$, the leading and subleading behaviors of YMS amplitudes are presented by

$$A_{\mathrm{YMS}}(1,\dots ,n\parallel \left\{g_j\right\} | \mathit{\sigma })=\left[S_{g_i}^{\left(0\right)}+S_{g_i}^{\left(1\right)}\right]A_{\mathrm{YMS}}(1,\dots ,n\parallel \left\{g_j\right\}\setminus g_i | \mathit{\sigma }\setminus g_i)+\mathcal{O}\left({\tau }^1\right),$$

(25)

The factors $S_{g_i}^{\left(0\right)}$ and $S_{g_i}^{\left(1\right)}$ are leading and subleading gluon soft factors [41,42]

$$S_{g_i}^{\left(0\right)}={\displaystyle \frac{1}{\tau }}\sum _{a\ne g_i}{\displaystyle \frac{{\delta }_{a{g}_i}({ϵ}_i·p_a)}{s_{a{g}_i}}}, S_{g_i}^{\left(1\right)}=\sum _{a\ne g_i}{\displaystyle \frac{{\delta }_{a{g}_i}({ϵ}_i·J_a·{\mathsf{K}}_{s_i})}{s_{a{g}_i}}},$$

(26)

where the operator ${\delta }_{a{g}_i}$ is defined similarly as ${\delta }_{ij}$ in (23). We sum over all external particles a except for the soft gluon $g_i$ and introduce the angular momentum $J_a^{\mu \nu }$ for each particle a. It is appropriate to regard the subleading soft gluon factor as a differential operator acting on the kinematic variables of amplitudes excluding the gluon $g_i$. Explicitly, the soft factor [18] acts in the form of

$$\begin{array}{cc}\hfill \left(S_{g_i}^{\left(1\right)}{k}_a\right)·V=& -{\displaystyle \frac{{\delta }_{a{g}_i}}{s_{a{g}_i}}}(k_a·F_i·V),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \left(S_{g_i}^{\left(1\right)}{ϵ}_a\right)·V=& -{\displaystyle \frac{{\delta }_{a{g}_i}}{s_{a{g}_i}}}({ϵ}_a·F_i·V),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill V_1·\left(S_{g_i}^{\left(1\right)}{F}_j\right)·V_2=& -{\displaystyle \frac{{\delta }_{g_j{g}_i}}{s_{g_j{g}_i}}}{V}_1·(F_j·F_i-F_i·F_j)·V_2.\hfill \end{array}$$

(29)

Here, $V^{\mu }$ represents an arbitrary Lorentz vector, and the field strength tensor $F^{\mu \nu }$ has been previously defined.

In [18], a bottom–up construction of YMS amplitudes was proposed by considering the soft behavior of amplitudes. By studying the YMS amplitudes in the soft limit, one can build an expansion formula of amplitudes from amplitudes with a chosen gluon or scalar removed. The explicit expressions of YMS amplitudes were verified from the soft behaviors of other particles and the permutation symmetry among gluons. Specifically, the expansion formula of YMS amplitudes (1) was constructed with the soft bootstrap [18]. Additionally, following the bottom–up construction, another form of the expansion is given by [22]

$$A_{\mathrm{YMS}}(1,\dots ,n\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\alpha }}{\displaystyle \frac{p_r·F_{{\mathit{\alpha }}^{\mathrm{T}}}·X_{\mathit{\alpha }}^L}{p_r·K}}{A}_{\mathrm{YMS}}(1,\{2,\dots ,n-1\}⧢\mathit{\alpha },n\parallel \left\{g_i\right\}\setminus \mathit{\alpha } | \mathit{\sigma }),$$

(30)

where we sum over all ordered subsets $\mathit{\alpha }=({\alpha }_1,\dots ,{\alpha }_a)$ of gluons $\left\{g_i\right\}$. The arbitrary reference momentum $p_r$ is introduced here and reflects the gauge invariance of the amplitude [22]. The K collects the momenta of all gluons $\left\{g_i\right\}$. As analyzed in [22], the $X_{\mathit{\alpha }}^L$ collects the momenta of scalars at the LHS of gluon ${\alpha }_1$. We emphasize that the full amplitude is independent of the reference momentum $p_r$ since the second term in (A5) always results in a gauge invariance identity. Nevertheless, the explicit expression of expansion coefficients relies on the choice of $p_r$. Compared to the expansion (1) with the specific selection of the fiducial gluon $g_f$, the expansion Formula (30) manifestly respects both permutation symmetry among gluons and gauge invariance. However, the coefficients of the expansion Formula (30) contain spurious poles [22]. As pointed out in [22], the spurious pole involved in coefficients in (30) in fact only cancels when the permutations are summed over, due to gauge invariance identities (3). This is because $p_r·K$ in the denominator is introduced after taking the average all choices of the fiducial gluon, see [22].

3. A Recursive Expansion Relation from the HAB Formula

In this section, we show that the HAB Formulas (13) and (19) with coefficients (15) and (20) induce recursive expansion relations for YMS amplitudes mixing gluons, massive scalars, and massless scalars. By this recursion, some gluons in the original amplitude are converted into massless scalars, accompanied by proper kinematic coefficients that absorb the polarization vectors. We present a simple example and then sketch the general verification for Formula (13) with at least three scalars. The boundary case (19) with only two massive scalars is further studied.

3.1. Example

We now demonstrate the recursive expansion relation by YMS amplitude involving three massive scalars and two gluons $A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })$. According to the HAB Formula (13), the amplitude is expressed as a sum over the two gluon orderings $(g_1,g_2)$ and $(g_2,g_1)$ as

$$A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })=\mathbf{m}(1,g_1,g_2,2,3 | \mathit{\sigma }) {\mathcal{N}}_{\mathbb{I}}\left(1,g_1,g_2,2,3\right)+(g_1\leftrightarrow g_2).$$

(31)

The explicit expressions of the kinematic coefficients follow from (15) when the ordered partitions of gluons $g_1$ and $g_2$, ${\mathbf{P}}_{{\mathit{\rho }}^g}^1:\{g_1,g_2\}$ and ${\mathbf{P}}_{{\mathit{\rho }}^g}^2:\{\left(g_1\right),\left(g_2\right)\},\{\left(g_2\right),\left(g_1\right)\}$ are taken into account:

$$\begin{array}{cc}\hfill & {\mathcal{N}}_{\mathbb{I}}\left(1,g_1,g_2,2,3\right)\hfill \\ \hfill =& -{\displaystyle \frac{2p_1·F_{12}·p_2}{p_{12}^2-m_s^2}}+{\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2(p_1+k_1)·F_2·p_2}{{(p_{12}+k_1)}^2-m_s^2}}+{\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2p_1·F_1·(p_2+k_2)}{{(p_{12}+k_2)}^2-m_s^2}}.\hfill \end{array}$$

(32)

Exchanging $g_1$ and $g_2$, we get the other coefficient

$$\begin{array}{cc}\hfill & {\mathcal{N}}_{\mathbb{I}}\left(1,g_2,g_1,2,3\right)\hfill \\ \hfill =& -{\displaystyle \frac{2p_1·F_{21}·p_2}{p_{12}^2-m_s^2}}+{\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2p_1·F_2·(p_2+k_1)}{{(p_{12}+k_1)}^2-m_s^2}}+{\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2(p_1+k_2)·F_1·p_2}{{(p_{12}+k_2)}^2-m_s^2}}.\hfill \end{array}$$

(33)

Note that these terms fall into two classes: (i) a single-factor term, e.g., ${\displaystyle \frac{2p_1·F_{12}·p_2}{p_{12}^2-m_s^2}}$, in which the strength tensors of the two gluons are contracted together, and (ii) two-factor terms, e.g., ${\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2(p_1+k_1)·F_2·p_2}{{(p_{12}+k_1)}^2-m_s^2}}$, in which the strength tensors of the two gluons appear in different factors.

A key observation is that whenever a gluon occupies a single factor, it plays the role of a scalar from the perspective of the remaining gluon. As an illustration, consider the term ${\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2(p_1+k_1)·F_2·p_2}{{(p_{12}+k_1)}^2-m_s^2}}$; the first factor is occupied by gluon $g_1$, whereas the second factor contains the combined momentum $p_1+k_1$. Hence, from the viewpoint of $g_2$, gluon $g_1$ behaves as a scalar. Consequently, the two-factor terms can be related to YMS amplitudes in which the prior gluon has already been converted into a massless scalar. For example, the second term of the kinematic coefficient (32) can be identified as a reduced kinematic coefficient with a factor

$${\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{\displaystyle \frac{2(p_1+k_1)·F_2·p_2}{{(p_{12}+k_1)}^2-m_s^2}}={\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{\mathcal{N}}_{\mathbb{I}}\left(1,{\overline{g}}_1,g_2,2,3\right),$$

(34)

where we used ${\overline{g}}_1$ to emphasize the role of $g_1$ as a scalar. Associating this kinematic coefficient as well as the one obtained by exchanging $g_1$, $g_2$ (i.e., the second term of (33)) with the corresponding propagator matrix, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}\left[\mathbf{m}(1,g_1,g_2,2,3 | \mathit{\sigma }){\displaystyle \frac{2(p_1+k_1)·F_2·p_2}{{(p_{12}+k_1)}^2-m_s^2}}+\mathbf{m}(1,g_2,g_1,2,3 | \mathit{\sigma }){\displaystyle \frac{2p_1·F_2·(p_2+k_1)}{{(p_{12}+k_1)}^2-m_s^2}}\right]\hfill \\ \hfill =& {\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}\left[\mathbf{m}(1,g_1,g_2,2,3 | \mathit{\sigma }){\mathcal{N}}_{\mathbb{I}}\left(1,{\overline{g}}_1,g_2,2,3\right)+\mathbf{m}(1,g_2,g_1,2,3 | \mathit{\sigma }){\mathcal{N}}_{\mathbb{I}}\left(1,g_2,{\overline{g}}_1,2,3\right)\right]\hfill \\ \hfill \equiv & {\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3\parallel \left\{g_2\right\} | \mathit{\sigma }).\hfill \end{array}$$

(35)

Here, the expression inside the square brackets is just the YMS amplitude $A_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3\parallel \left\{g_2\right\} | \mathit{\sigma })$ with gluon $g_2$, massless scalar ${\overline{g}}_1$, and massive scalars 1, 2, 3.

Following a similar discussion, other terms in (32) and (33) also result in YMS amplitudes with fewer gluons and more massless gluons. Altogether, the original YMS amplitude turns into

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\}|\mathit{\sigma })=-{\displaystyle \frac{2p_1·F_{12}·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{BS}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3|\mathit{\sigma })-{\displaystyle \frac{2p_1·F_{21}·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{BS}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3|\mathit{\sigma })\hfill \\ \hfill & +{\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3\parallel \left\{g_2\right\}|\mathit{\sigma })+{\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_2,2,3\parallel \left\{g_1\right\}|\mathit{\sigma }).\hfill \end{array}$$

(36)

The BS amplitudes $A_{\mathrm{BS}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })$ and $A_{\mathrm{BS}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })$ are defined by the propagator matrices $\mathbf{m}(1,g_1,g_2,2,3 | \mathit{\sigma })$ and $\mathbf{m}(1,g_2,g_1,2,3 | \mathit{\sigma })$, with massive scalars 1, 2, 3 and the massless scalars ${\overline{g}}_1,{\overline{g}}_2$ coming from gluons $g_1$, $g_2$. The amplitudes $A_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3\parallel \left\{g_2\right\} | \mathit{\sigma })$ and $A_{\mathrm{YMS}}(1,{\overline{g}}_2,2,3\parallel \left\{g_1\right\} | \mathit{\sigma })$ stand for the YMS amplitudes where $g_1$ and $g_2$ are already converted into scalars ${\overline{g}}_1$ and ${\overline{g}}_2$, respectively. It is not hard to see that the relation (36) can be arranged into the following compact form.

$$A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })=\sum _{\mathit{\alpha }}-{\displaystyle \frac{{(-1)}^{|\mathit{\alpha }|}2p_1·F_{\mathit{\alpha }}·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,\overline{\mathit{\alpha }},2,3\parallel \left\{g_i\right\}\setminus \mathit{\alpha } | \mathit{\sigma }).$$

(37)

Note that YMS amplitudes with no gluons degenerate to BS amplitudes.

The above example makes it explicit how gluons are converted into massless scalars and how the full YMS amplitude expands into amplitudes with fewer gluons and more massless scalars. In the next subsection, we extend (37) to a general formula with at least three massive scalars and an arbitrary number of gluons.

3.2. General Formula for YMS Amplitudes with at Least Three Massive Scalars

Inspired by the example, we now provide a general recursive expansion formula for the YMS amplitudes with at least three scalars from the HAB Formula (13) with coefficients (15).

We begin with the HAB Formula (13) of YMS amplitudes with n massive scalars and m massless gluons:

$$A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\rho }}\mathbf{m}(1,\mathit{\rho },n-1,n | \mathit{\sigma }) {\mathcal{N}}_{\mathbb{I}}\left(1,\mathit{\rho },n-1,n\right),$$

(38)

with the explicit expression (15) of the kinematic coefficient

$$\begin{array}{cc}\hfill {\mathcal{N}}_{\mathbb{I}}\left(1,\mathit{\rho },n-1,n\right) & =\sum _{r=1}^m\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g}^r}\prod _{l=1}^r-{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^l|}2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}}.\hfill \end{array}$$

(39)

As pointed out in the previous subsection, gluons in the first ordered subset ${\mathit{\alpha }}^1=\mathit{\beta }$ are treated as scalars for the remaining gluons ${\mathit{\alpha }}^2,\dots ,{\mathit{\alpha }}^r$. For all terms that share the same first subset $\mathit{\beta }$, one can collect the remaining terms together as follows:

$$\begin{array}{cc}\hfill & {\left[ \sum _{r=1}^m\left.\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g}^r} \prod _{l=1}^r-{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^l|}2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}} \right]\right|}_{{\mathit{\alpha }}^1=\mathit{\beta }}\hfill \\ \hfill =& -{\displaystyle \frac{{(-1)}^{|\mathit{\beta }|}2X_{\mathit{\beta }}^{L\left(\mathit{\rho }\right)}·F_{\mathit{\beta }}·X_{\mathit{\beta }}^{R\left(\mathit{\rho }\right)}}{P^2-m_s^2}}\sum _{t=1}^{m-\left|\mathit{\beta }\right|}\sum _{{\mathit{\gamma }}^{\mathit{\ell }}\in {\mathbf{P}}_{{\mathit{\rho }}^g\setminus \mathit{\beta }}^t}\prod _{\mathit{\ell }=1}^t-{\displaystyle \frac{{(-1)}^{|{\mathit{\gamma }}^{\mathit{\ell }}|}2X_{{\mathit{\gamma }}^{\mathit{\ell }}}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\gamma }}^{\mathit{\ell }}}·X_{{\mathit{\gamma }}^{\mathit{\ell }}}^{R\left(\mathit{\rho }\right)}}{P_{\mathit{\ell }}^2-m_s^2}}\hfill \end{array}.$$

(40)

Here, the factors associated with $\mathit{\beta }$ in distinct terms are identical and have been extracted from the sum. The P denotes the summation of the momenta of all scalars except for n, whereas the $P_l$ further includes the momenta of gluons in $\mathit{\beta }$ and ${\mathit{\gamma }}^1,\dots ,{\mathit{\gamma }}^{\mathit{\ell }-1}$. Those same momenta of gluons in $\mathit{\beta }$ also enter in the $X_{{\mathit{\gamma }}^{\mathit{\ell }}}^{L\left(\mathit{\rho }\right)}$ and $X_{{\mathit{\gamma }}^{\mathit{\ell }}}^{R\left(\mathit{\rho }\right)}$. For the remaining gluons ${\mathit{\gamma }}^{\mathit{\ell }}$, gluons in $\mathit{\beta }$ are treated as scalars. Consequently, the summations on the second line yield a reduced kinematic coefficient

$${\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho },n-1,n)|}_{\overline{\mathit{\beta }}}=\sum _{t=1}^{m-\left|\mathit{\beta }\right|}\sum _{{\mathit{\gamma }}^{\mathit{\ell }}\in {\mathbf{P}}_{{\mathit{\rho }}^g\setminus \mathit{\beta }}^t}\prod _{\mathit{\ell }=1}^t-{\displaystyle \frac{{(-1)}^{|{\mathit{\gamma }}^{\mathit{\ell }}|}2X_{{\mathit{\gamma }}^{\mathit{\ell }}}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\gamma }}^{\mathit{\ell }}}·X_{{\mathit{\gamma }}^{\mathit{\ell }}}^{R\left(\mathit{\rho }\right)}}{P_{\mathit{\ell }}^2-m_s^2}}.$$

(41)

The bar over the subscript $\mathit{\beta }$ denotes that gluons in $\mathit{\beta }$ are effectively massless scalars in the remaining factors. We define the trivial boundary case, in which the first subset includes all gluons, by unity:

$${\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho },n-1,n)|}_{\overline{{\mathit{\rho }}^g}}=1.$$

(42)

This corresponds to the special term of the kinematic coefficient (39), which involves all gluons:

$$-{\displaystyle \frac{{(-1)}^{|{\mathit{\rho }}^g|}2X_{{\mathit{\rho }}^g}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\rho }}^g}·X_{{\mathit{\rho }}^g}^{R\left(\mathit{\rho }\right)}}{P^2-m_s^2}}.$$

(43)

Then, the kinematic coefficient is written as a sum over all compatible ordered subsets of gluons:

$${\mathcal{N}}_{\mathbb{I}}\left(1,\mathit{\rho },n-1,n\right)=\sum _{\mathit{\beta }\left({\mathit{\rho }}^g\right)}-{\displaystyle \frac{{(-1)}^{|\mathit{\beta }({\mathit{\rho }}^g)|}2X_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}^{L\left(\mathit{\rho }\right)}·F_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}·X_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}^{R\left(\mathit{\rho }\right)}}{P^2-m_s^2}}{\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho },n-1,n)|}_{\overline{\mathit{\beta }}\left({\mathit{\rho }}^g\right)},$$

(44)

where each term comprises a reduced kinematic coefficient multiplied by a compatible factor. Here $\mathit{\beta }\left({\mathit{\rho }}^g\right)$ denotes the gluon subset compatible with the overall gluon ordering ${\mathit{\rho }}^g$. For example, for the overall ordering $\mathit{\rho }=(2,g_1,g_2)$ and gluon ordering ${\mathit{\rho }}^g=(g_1,g_2)$, we have $\mathit{\beta }\left({\mathit{\rho }}^g\right)$: $\left(g_1\right),\left(g_2\right),(g_1,g_2)$.

As a result, the original expansion of the amplitude (38) can now be viewed as

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })\hfill \\ \hfill =& \sum _{\mathit{\rho }}\mathbf{m}(1,\mathit{\rho },n-1,n | \mathit{\sigma })\left[ \sum _{\mathit{\beta }\left({\mathit{\rho }}^g\right)}-{\displaystyle \frac{{(-1)}^{|\mathit{\beta }({\mathit{\rho }}^g)|}2X_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}^{L\left(\mathit{\rho }\right)}·F_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}·X_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}^{R\left(\mathit{\rho }\right)}}{P^2-m_s^2}}{\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho },n-1,n)|}_{\overline{\mathit{\beta }}\left({\mathit{\rho }}^g\right)} \right].\hfill \end{array}$$

(45)

We now rewrite the sum over $\mathit{\rho }$ by first shuffling together the scalars $\{2,\dots ,n-2\}$ and a given ordering ${\mathit{\rho }}^g=\mathit{\theta }$ involving all gluons, and subsequently summing over all possible $\mathit{\theta }$. Consequently, (45) becomes

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })\hfill \\ \hfill =& \sum _{\mathit{\theta }}\sum _{\mathit{\rho }\left(\mathit{\theta }\right)}\mathbf{m}(1,\mathit{\rho }\left(\mathit{\theta }\right),n-1,n | \mathit{\sigma })\sum _{\mathit{\beta }\left(\mathit{\theta }\right)}-{\displaystyle \frac{{(-1)}^{|\mathit{\beta }(\mathit{\theta })|}2X_{\mathit{\beta }\left(\mathit{\theta }\right)}^{L\left(\mathit{\rho }\right(\mathit{\theta }\left)\right)}·F_{\mathit{\beta }\left(\mathit{\theta }\right)}·X_{\mathit{\beta }\left(\mathit{\theta }\right)}^{R\left(\mathit{\rho }\right(\mathit{\theta }\left)\right)}}{P^2-m_s^2}}{\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho }\left(\mathit{\theta }\right),n-1,n)|}_{\overline{\mathit{\beta }}\left(\mathit{\theta }\right)}.\hfill \end{array}$$

(46)

Here, we summed over all $\mathit{\rho }\left(\mathit{\theta }\right)\in \{2,\dots ,n-2\}⧢\mathit{\theta }$ for a given $\mathit{\theta }$, and then summed over all possible $\mathit{\theta }$ (in other words, all non-empty ordered subsets of gluons).

An immediate observation is that one can change the ordering of the summations in (46) as follows. First, we pick an ordered gluon subset $\mathit{\beta }$ and then collect all compatible gluon orderings $\mathit{\theta }\left(\mathit{\beta }\right)$ together with the attendant shuffle of scalars $\{2,\dots ,n-2\}$ and gluons $\mathit{\theta }\left(\mathit{\beta }\right)$. As an illustration, for a subset $\mathit{\beta }=(g_1,g_2)$ of gluons $\{g_1,g_2,g_3\}$, the compatible gluon orderings include $(g_3,g_1,g_2)$, $(g_1,g_2,g_3)$ and $(g_1,g_3,g_2)$. That is to say, the complete collection of $\mathit{\theta }\left(\mathit{\beta }\right)$ is precisely the shuffle of ordered gluons $\mathit{\beta }$ and the unordered remaining gluons $\left\{g_i\right\}\setminus \mathit{\beta }$. Exploiting this freedom, we reorganize the series of summations (46) as

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })\hfill \\ \hfill =& \sum _{\mathit{\beta }}\sum _{\mathit{\theta }\left(\mathit{\beta }\right)}\sum _{\mathit{\rho }\left(\mathit{\theta }\right(\mathit{\beta }\left)\right)}-{\displaystyle \frac{{(-1)}^{|\mathit{\beta }|}2X_{\mathit{\beta }}^{L\left(\mathit{\rho }\right(\mathit{\theta }\left(\mathit{\beta }\right)\left)\right)}·F_{\mathit{\beta }}·X_{\mathit{\beta }}^{R\left(\mathit{\rho }\right(\mathit{\theta }\left(\mathit{\beta }\right)\left)\right)}}{P^2-m_s^2}} \hfill \\ \hfill & \times \mathbf{m}(1,\mathit{\rho }\left(\mathit{\theta }\left(\mathit{\beta }\right)\right),n-1,n | \mathit{\sigma }) {\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho }\left(\mathit{\theta }\left(\mathit{\beta }\right)\right),n-1,n)|}_{\overline{\mathit{\beta }}},\hfill \end{array}$$

(47)

where, the three summations were taken over all $\mathit{\beta }$, all $\mathit{\theta }\left(\mathit{\beta }\right)\in \mathit{\beta }⧢\mathrm{perms}(\left\{g_i\right\}\setminus \mathit{\beta })$ ($\mathrm{perms}A$ for a set A denotes all permutations of elements in A) and all $\mathit{\rho }\left(\mathit{\theta }\right(\mathit{\beta }\left)\right)\in \{2,\dots ,n-2\}⧢\mathit{\theta }\left(\mathit{\beta }\right)$.

Noting that for a given $\mathit{\beta }$ in the first summation, the second and the third summations can be achieved by summing over all possible $\mathit{\rho }\in \{2,\dots ,n-2\}⧢\mathit{\beta }⧢\mathrm{perms}(\left\{g_i\right\}\setminus \mathit{\beta })$. According to the properties of shuffling, this summation further split into two steps: (i) summing over all $\mathit{\varrho }\left(\mathit{\beta }\right)\in \{2,\dots ,n-2\}⧢\mathit{\beta }$, and then (ii) summing over $\mathit{\rho }\left(\mathit{\varrho }\left(\mathit{\beta }\right)\right)\in \mathit{\varrho }\left(\mathit{\beta }\right)⧢\mathrm{perms}(\left\{g_i\right\}\setminus \mathit{\beta })$ for a given $\mathit{\varrho }\left(\mathit{\beta }\right)$ in (i). We therefore re-express (47) as

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })\hfill \\ \hfill =& \sum _{\mathit{\beta }}\sum _{\mathit{\varrho }\left(\mathit{\beta }\right)}-{\displaystyle \frac{{(-1)}^{|\mathit{\beta }|}2X_{\mathit{\beta }}^{L\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}·F_{\mathit{\beta }}·X_{\mathit{\beta }}^{R\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}}{P^2-m_s^2}}\hfill \\ \hfill & \times \sum _{\mathit{\rho }\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}\mathbf{m}(1,\mathit{\rho }\left(\mathit{\varrho }\left(\mathit{\beta }\right)\right),n-1,n | \mathit{\sigma }) {\mathcal{N}}_{\mathbb{I}}{(1,\mathit{\rho }\left(\mathit{\varrho }\left(\mathit{\beta }\right)\right),n-1,n)|}_{\overline{\mathit{\beta }}}.\hfill \end{array}$$

(48)

The last summation is nothing but the reduced YMS amplitude $A_{\mathrm{YMS}}(1,\mathit{\varrho }\left(\mathit{\beta }\right),n-1,n\parallel \left\{g_i\right\}\setminus \mathit{\beta } | \mathit{\sigma })$, in which the gluons in $\mathit{\beta }$ are treated as additional massless scalars, while elements of $\left\{g_i\right\}\setminus \mathit{\beta }$ act as gluons. Hence, we finally arrive at the following recursive expression of YMS amplitude $A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })$:

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })\hfill \\ \hfill =& \sum _{\mathit{\beta }}\sum _{\mathit{\varrho }\left(\mathit{\beta }\right)\in \mathit{\beta }\{2,\dots ,n-2\}}-{\displaystyle \frac{{(-1)}^{|\mathit{\beta }|}2X_{\mathit{\beta }}^{L\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}·F_{\mathit{\beta }}·X_{\mathit{\beta }}^{R\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}}{P^2-m_s^2}}{A}_{\mathrm{YMS}}(1,\mathit{\varrho }\left(\mathit{\beta }\right),n-1,n\parallel \left\{g_i\right\}\setminus \mathit{\beta } | \mathit{\sigma }).\hfill \end{array}$$

(49)

It is worth pointing out that the sign ${(-1)}^{|\mathit{\beta }|}$ depending on the number of gluons in $\mathit{\beta }$ will be absorbed when reflecting the ordering of contraction of strength tensors, i.e., ${(-1)}^{|\mathit{\beta }|}2X_{\mathit{\beta }}^{L\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}·F_{\mathit{\beta }}·X_{\mathit{\beta }}^{R\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}=2X_{\mathit{\beta }}^{R\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}·F_{{\mathit{\beta }}^T}·X_{\mathit{\beta }}^{L\left(\mathit{\varrho }\right(\mathit{\beta }\left)\right)}$, where ${\mathit{\beta }}^T$ denotes the reversed ordering of $\mathit{\beta }$. The resulted kinematic numerators with this reverse ordering were used in [25]. We take the five-point amplitude $A_{\mathrm{YMS}}(1,2\parallel \{g_1,g_2,g_3\} | \mathit{\sigma })$ with two scalars 1, 2 and three gluons $g_1$, $g_2$ and $g_3$ as an example to demonstrate this difference. When we expand $A_{\mathrm{YMS}}(1,2\parallel \{g_1,g_2,g_3\} | \mathit{\sigma })$ into BS amplitudes using the recursive expansion formulas, $A_{\mathrm{YMS}}(1,2\parallel \{g_1,g_2,g_3\} | \mathit{\sigma })$ is finally expressed in terms of $\mathbf{m}(1,\mathit{\rho },g_1,2 | \mathit{\sigma })$ with coefficients ${\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2)$ ($\mathit{\rho }$ can be $(g_2,g_3)$ or $(g_3,g_2)$). To relate this with those presented in the appendix of [25], we flip $\mathit{\rho },g_1$ in $\mathbf{m}(1,\mathit{\rho },g_1,2 | \mathit{\sigma })$ outside of 1, 2, then obtain ${(-1)}^{|\mathit{\rho }|+1}\mathbf{m}(g_1,{\mathit{\rho }}^T,1,2 | \mathit{\sigma })$, where the sign ${(-1)}^{|\mathit{\rho }|+1}$ is a consequence of the reflection property of $\mathbf{m}$. Absorbing this extra sign ${(-1)}^{|\mathit{\rho }|+1}$ into the coefficient ${\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2)$, we get the ${\mathcal{N}}_{\mathbb{I}}(g_1,{\mathit{\rho }}^T,1,2)$ whose expressions agree with those presented in the appendix of [25]. This sign is involved in ${\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2)$, and should not be confused with signs when we use nested commutators to express other numerators.

Comments on the result (49): It is straightforward to see that the Formula (49) also holds when some of the scalars in the color-ordered YMS amplitude $A_{\mathrm{YMS}}(1,\dots ,n-1,n\parallel \left\{g_i\right\} | \mathit{\sigma })$ are replaced by massless scalars. One therefore reconstructs the HAB formula by applying (49) iteratively.

The above discussion only concentrated on YMS amplitudes with at least three scalars (with the fixed ones 1, $n-1$, and n). In the following subsection, we study the boundary case in which the YMS amplitude only involves two scalars.

3.3. Boundary Case: YMS Amplitudes with Two Massive Scalars

Analogous to the case of YMS amplitudes with at least three scalars, amplitudes with two scalars and m gluons are also represented as a sum of products of the kinematic coefficients and the propagator matrices. In this case, we rewrite the explicit HAB Formula (19) in the following:

$$A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\rho }}\mathbf{m}(1,\mathit{\rho },g_1,2 | \mathit{\sigma }){\mathcal{N}}_{\mathbb{I}}\left(1,\mathit{\rho },g_1,2\right),$$

(50)

where the gluon $g_1$ acts as one of the fixed elements on the RHS. The kinematic coefficient (20) takes the form

$$\begin{array}{cc}\hfill {\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2)=& {(-1)}^m\sum _{r=1}^m{(-1)}^r\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g\setminus g_1}^r}-{\displaystyle \frac{2p_1·F_1·F_{{\mathit{\alpha }}^1}·p_1}{P_1^2-m_s^2}}\prod _{l=2}^m{\displaystyle \frac{2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}},\hfill \\ \hfill =& \sum _{r=1}^m\sum _{{\mathit{\alpha }}^l\in {\mathbf{P}}_{{\mathit{\rho }}^g\setminus g_1}^r}{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^1|+1}2p_1·F_1·F_{{\mathit{\alpha }}^1}·p_1}{P_1^2-m_s^2}}\prod _{l=2}^m-{\displaystyle \frac{{(-1)}^{|{\mathit{\alpha }}^l|}2X_{{\mathit{\alpha }}^l}^{L\left(\mathit{\rho }\right)}·F_{{\mathit{\alpha }}^l}·X_{{\mathit{\alpha }}^l}^{R\left(\mathit{\rho }\right)}}{P_l^2-m_s^2}}.\hfill \end{array}$$

(51)

recalling that the massive scalars 1 and 2 have equal mass. It is worth pointing out that such an expression can be generalized to amplitudes with a fermion line.

Following a discussion parallel with the previous subsection, we can perform a resummation of terms over the first subset ${\mathit{\alpha }}^1=\mathit{\beta }$. This is permissible due to the observation that the fiducial gluon $g_1$ and the gluons in $\mathit{\beta }$ are effectively treated as scalars with respect to the remaining factors. The resulting factorization of the kinematic coefficient takes the form

$${\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2)=\sum _{\mathit{\beta }\left({\mathit{\rho }}^g\right)}{\displaystyle \frac{{(-1)}^{|\mathit{\beta }({\mathit{\rho }}^g)|+1}2p_1·F_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}·F_1·p_1}{P^2-m_s^2}}{\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2){|}_{\overline{\mathit{\beta }}\left({\mathit{\rho }}^g\right)}.$$

(52)

Here, the reduced kinematic coefficients are constructed from the corresponding factors. The YMS amplitudes with two scalars can then be expressed as a nested double summation as follows:

$$A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\}|\mathit{\sigma }) = \sum _{\mathit{\rho }}\mathbf{m}(1,\mathit{\rho },g_1,2|\mathit{\sigma })\sum _{\mathit{\beta }\left({\mathit{\rho }}^g\right)}{\displaystyle \frac{{(-1)}^{|\mathit{\beta }({\mathit{\rho }}^g)|+1} 2p_1·F_{\mathit{\beta }\left({\mathit{\rho }}^g\right)}·F_1·p_1}{P^2-m_s^2}}{\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho },g_1,2){|}_{\overline{\mathit{\beta }}\left({\mathit{\rho }}^g\right)}.$$

(53)

Exchanging the order of the two summations yields

$$A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\beta }}{\displaystyle \frac{{(-1)}^{|\mathit{\beta }|+1} 2p_1·F_{\mathit{\beta }}·F_1·p_1}{P^2-m_s^2}}\sum _{\mathit{\rho }\left(\mathit{\beta }\right)}\mathbf{m}(1,\mathit{\rho }\left(\mathit{\beta }\right),g_1,2 | \mathit{\sigma }){\mathcal{N}}_{\mathbb{I}}(1,\mathit{\rho }\left(\mathit{\beta }\right),g_1,2){|}_{\overline{\mathit{\beta }}}.$$

(54)

The second summation is the reduced YMS amplitude $A_{\mathrm{YMS}}(1,\mathit{\beta },g_1,2\parallel \left\{g_i\right\}\setminus (g_1\cup \mathit{\beta }) | \mathit{\sigma })$ with additional massless scalars converted from $g_1$ and gluons in $\mathit{\beta }$. As a result, we express the amplitude $A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })$ with two massive scalars 1, 2 by amplitudes with fewer gluons and additional massless scalars, through the following recursive expansion relation:

$$A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\beta }}{\displaystyle \frac{{(-1)}^{|\mathit{\beta }|+1}2p_1·F_{\mathit{\beta }}·F_1·p_1}{P^2-m_s^2}}{A}_{\mathrm{YMS}}(1,\mathit{\beta },g_1,2\parallel \left\{g_i\right\}\setminus (g_1\cup \mathit{\beta }) | \mathit{\sigma }).$$

(55)

This formula has a form analogous to the general Formula (49) derived previously, but has distinct expansion coefficients. As clarified in the previous subsection, the sign ${(-1)}^{|\mathit{\beta }|+1}$ will be absorbed when we reverse the contraction of the field strength tensors in the coefficients.

In the next section, we reconstruct the expansion Formulas (49) and (55) by analyzing the soft behavior of the YMS amplitudes.

4. Understanding the Recursive Expansion Formula by Soft Behavior

In this section, we understand the expansion Formula (49) proposed in the previous section by soft behavior of YMS amplitudes, through a similar discussion with [18]. For brevity, we demonstrate this approach by a simple example, the YMS amplitude $A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })$ with three massive scalars and two gluons. General understanding just follows from a similar discussion.

We begin with the soft behavior of the YMS amplitude $A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })$, where gluon $g_2$ is the soft one, i.e., $k_2\equiv \tau {\widehat{k}}_2$ ($\tau \to 0$), and focus on the subleading order behavior. (One may try to begin with the leading order. However, there exists a subleading contribution since the coefficient may contain $F_{g_2}^{\mu \nu }$. As demonstrated in [18], the subleading behavior reveals the full information of the expansion coefficients.)

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}^{{\left(1\right)}_{g_2}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })& =S_{g_2}^{\left(1\right)}{A}_{\mathrm{YMS}}(1,2,3\parallel \left\{g_1\right\} | \mathit{\sigma }\setminus g_2)\hfill \\ \hfill & =S_{g_2}^{\left(1\right)}{\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2)\hfill \\ \hfill & =B_{\mathrm{A}}+B_{\mathrm{N}}+B_{\mathrm{D}}.\hfill \end{array}$$

(56)

The expression can be decomposed into three distinct contributions from the action of the soft factor $S_{g_2}^{\left(1\right)}$ on the amplitude, on the numerator, and on the denominator, denoted by $B_{\mathrm{A}}$, $B_{\mathrm{N}}$, and $B_{\mathrm{D}}$, respectively. In the following, we evaluate these terms individually.

First, the term $B_{\mathrm{A}}$ arises from the action of the soft factor of gluon $g_2$ on the reduced amplitude

$$B_{\mathrm{A}}={\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}\left[S_{g_2}^{\left(1\right)}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2)\right]={\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(1\right)}_{g_2}}(1,{\overline{g}}_1,2,3\parallel g_2 | \mathit{\sigma }),$$

(57)

Next, we consider the numerator contribution $B_{\mathrm{N}}$. This term arises from the action of the soft factor on $p_1·F_1·p_2$. Using the properties (27) and (29) of the soft factor, we have

$$\begin{array}{cc}\hfill \left(S_{g_2}^{\left(1\right)}{p}_1\right)·F_1·p_2& =-{\displaystyle \frac{{\delta }_{1g_2}}{s_{1g_2}}}{p}_1·F_2·F_1·p_2,\hfill \\ \hfill p_1·\left(S_{g_2}^{\left(1\right)}{F}_1\right)·p_2& =-{\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}{p}_1·(F_1·F_2-F_2·F_1)·p_2,\hfill \\ \hfill p_1·F_1·\left(S_{g_2}^{\left(1\right)}{p}_2\right)& ={\displaystyle \frac{{\delta }_{2g_2}}{s_{2g_2}}}{p}_1·F_1·F_2·p_2.\hfill \end{array}$$

(58)

Combining these terms gives the total action of the soft factor

$$S_{g_2}^{\left(1\right)}(p_1·F_1·p_2)=\left(-{\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}+{\displaystyle \frac{{\delta }_{2g_2}}{s_{2g_2}}}\right)(p_1·F_1·F_2·p_2)+\left(-{\displaystyle \frac{{\delta }_{1g_2}}{s_{1g_2}}}+{\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}\right)(p_1·F_2·F_1·p_2).$$

(59)

Utilizing the antisymmetry of ${\delta }_{ij}$, we reorganize the terms as

$${\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}-{\displaystyle \frac{{\delta }_{2g_2}}{s_{2g_2}}}={\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}+{\displaystyle \frac{{\delta }_{g_{2}2}}{s_{g_{2}2}}}=\tau {\left.S_{g_2}^{\left(0\right)}\right|}_{g_1{g}_{2}2},$$

(60)

where the subscript denotes the relative order of particles as $(g_1,g_2,2)$. This soft factor, once combined with the YMS amplitude $A_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2)$, yields the soft behavior of the amplitude

$$A_{\mathrm{YMS}}^{{\left(0\right)}_{{\overline{g}}_2}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma }\setminus g_2)={\left.S_{g_2}^{\left(0\right)}\right|}_{g_1{g}_{2}2}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2).$$

(61)

Here, the $g_2$ is treated as a soft scalar. This procedure works the same for ${\displaystyle \frac{{\delta }_{1g_2}}{s_{1g_2}}}-{\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}=\tau {\left.S_{g_2}^{\left(0\right)}\right|}_{1g_2{g}_1}$. Therefore, by identifying these terms as the soft behavior of reduced YMS amplitudes, the total contribution $B_{\mathrm{N}}$ from the numerator becomes

$$B_{\mathrm{N}}=-\tau \left[{\displaystyle \frac{2p_1·F_1·F_2·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })+{\displaystyle \frac{2p_1·F_2·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })\right].$$

(62)

Finally, we calculate the denominator contribution $B_{\mathrm{D}}$.

$$B_{\mathrm{D}}=-{\displaystyle \frac{2p_1·F_1·p_2}{{(p_{12}^2-m_s^2)}^2}}\left[S_{g_2}^{\left(1\right)}(p_{12}^2-m_s^2)\right]A_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2)$$

(63)

Due to the antisymmetry of the field strength, the action of the soft factor on the momentum squared is identically zero. Consequently, the non-vanishing contribution comes from $p_1·p_2$:

$$S_{g_2}^{\left(1\right)}(p_1·p_2)=\left(-{\displaystyle \frac{{\delta }_{1g_2}}{s_{1g_2}}}+{\displaystyle \frac{{\delta }_{2g_2}}{s_{2g_2}}}\right)(p_1·F_2·p_2).$$

(64)

The terms in the first bracket are rewritten as soft factors:

$${\displaystyle \frac{{\delta }_{1g_2}}{s_{1g_2}}}-{\displaystyle \frac{{\delta }_{2g_2}}{s_{2g_2}}}={\displaystyle \frac{{\delta }_{1g_2}}{s_{1g_2}}}+{\displaystyle \frac{{\delta }_{g_2{g}_1}}{s_{g_2{g}_1}}}+{\displaystyle \frac{{\delta }_{g_1{g}_2}}{s_{g_1{g}_2}}}+{\displaystyle \frac{{\delta }_{g_{2}2}}{s_{g_{2}2}}}=\tau \left({\left.S_{g_2}^{\left(0\right)}\right|}_{1g_2{g}_1}+{\left.S_{g_2}^{\left(0\right)}\right|}_{g_1{g}_{2}2}\right).$$

(65)

Each soft factor is separately connected to the corresponding reduced YMS amplitude:

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })=& {\left.S_{g_2}^{\left(0\right)}\right|}_{1g_2{g}_1}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2),\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })=& {\left.S_{g_2}^{\left(0\right)}\right|}_{g_1{g}_{2}2}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3 | \mathit{\sigma }\setminus g_2).\hfill \end{array}$$

(67)

Thus, $B_{\mathrm{D}}$ becomes

$$B_{\mathrm{D}}=\tau {\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m_s^2}}{\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}\left(A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })+A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })\right).$$

(68)

Part of the obtained expression of $B_{\mathrm{D}}$ reproduces the soft behavior of a YMS amplitude with the expansion

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}(1,{\overline{g}}_2,2,3\parallel g_1 | \mathit{\sigma })=& {\displaystyle \frac{2(p_1+k_2)·F_1·p_2}{{(p_{12}+k_2)}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })\hfill \\ \hfill & +{\displaystyle \frac{2p_1·F_1·(k_2+p_2)}{{(p_{12}+k_2)}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma }).\hfill \end{array}$$

(69)

Specifically, the leading order $\mathcal{O}\left({\tau }^{-1}\right)$ of the soft limit takes the form of

$$A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,2,3\parallel g_1 | \mathit{\sigma })={\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}\left(A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })+A_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })\right),$$

(70)

which arises from the $\mathcal{O}\left({\tau }^0\right)$ term of coefficient and the $\mathcal{O}\left({\tau }^{-1}\right)$ terms of reduced YMS amplitudes. Therefore, $B_{\mathrm{D}}$ can be represented compactly as

$$B_{\mathrm{D}}=\tau {\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,2,3\parallel g_1 | \mathit{\sigma }).$$

(71)

Collecting the results for $B_{\mathrm{A}}$ (57), $B_{\mathrm{N}}$ (62), and $B_{\mathrm{D}}$ (71), the total soft behavior of the amplitude (56) is given by

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}^{{\left(1\right)}_{g_2}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })\hfill \\ \hfill =& {\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(1\right)}_{g_2}}(1,{\overline{g}}_1,2,3\parallel g_2 | \mathit{\sigma })+\tau {\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,2,3\parallel g_1 | \mathit{\sigma })\hfill \\ \hfill & -\tau \left[{\displaystyle \frac{2p_1·F_1·F_2·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })+{\displaystyle \frac{2p_1·F_2·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}^{{\left(0\right)}_{g_2}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma })\right].\hfill \end{array}$$

(72)

This result exactly matches the soft limit of the following proposed expansion:

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })\hfill \\ \hfill =& {\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,2,3\parallel g_2 | \mathit{\sigma })+{\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_2,2,3\parallel g_1 | \mathit{\sigma })\hfill \\ \hfill & -{\displaystyle \frac{2p_1·F_1·F_2·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_1,{\overline{g}}_2,2,3 | \mathit{\sigma })-{\displaystyle \frac{2p_1·F_2·F_1·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,{\overline{g}}_2,{\overline{g}}_1,2,3 | \mathit{\sigma }).\hfill \end{array}$$

(73)

This expression is identical to the expansion Formula (37):

$$A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })=\sum _{\mathit{\alpha }}-{\displaystyle \frac{{(-1)}^{|\mathit{\alpha }|}2p_1·F_{\mathit{\alpha }}·p_2}{p_{12}^2-m_s^2}}{A}_{\mathrm{YMS}}(1,\overline{\mathit{\alpha }},2,3\parallel \left\{g_i\right\}\setminus \mathit{\alpha } | \mathit{\sigma }).$$

(74)

which is constructed from the Hopf-algebra-based (HAB) formula. The validity of this expansion can be further verified by considering the soft limit of the gluon $g_1$.

Generalization of the example to amplitudes with more scalars and/or gluons: For any amplitude $A_{\mathrm{YMS}}(1,2,\dots ,n\parallel \left\{g_i\right\} | \mathit{\sigma })$ with $n>3$ massive scalars and/or more than two gluons, the soft behavior approach follows from a similar discussion. Specifically, when we consider $g_j$ as the soft gluon, i.e., $k_j\equiv \tau {\widehat{k}}_j$ ($\tau \to 0$), the subleading behavior of this amplitude is obtained by acting $S_{g_j}^{\left(1\right)}$ to the amplitude $A_{\mathrm{YMS}}(1,2,\dots ,n\parallel \left\{g_i\right\}\setminus g_j | \mathit{\sigma })$, which is already expanded in terms of amplitudes with fewer gluons according to (49). Analogous to the example in this section, this expression splits into three parts $B_{\mathrm{A}}$, $B_{\mathrm{N}}$, and $B_{\mathrm{D}}$, corresponding to those terms obtained from amplitudes with fewer gluons, the numerators and the denominators of coefficients:

$$\begin{array}{cc}\hfill B_{\mathrm{A}}& =\sum _{\mathit{\alpha } \mathrm{s}.\mathrm{t}. g_j\notin \mathit{\alpha }} \sum _{\mathit{\varrho }\left(\mathit{\alpha }\right)\in \mathit{\alpha }⧢\{2,\dots ,n-2\}}-{\displaystyle \frac{{(-1)}^{|\mathit{\alpha }|}2X_{\mathit{\alpha }}^{L\left(\mathit{\varrho }\right(\mathit{\alpha }\left)\right)}·F_{\mathit{\alpha }}·X_{\mathit{\alpha }}^{R\left(\mathit{\varrho }\right(\mathit{\alpha }\left)\right)}}{P^2-m_s^2}}\hfill \\ \hfill & \times A_{\mathrm{YMS}}^{{\left(1\right)}_{g_j}}(1,\mathit{\varrho }\left(\mathit{\alpha }\right),n-1,n\parallel \left\{g_i\right\}\setminus \mathit{\alpha } | \mathit{\sigma }),\hfill \\ \hfill B_{\mathrm{N}}& =\sum _{\mathit{\alpha } \mathrm{s}.\mathrm{t}. \mathit{\alpha }\setminus g_j\ne \varnothing }\sum _{\mathit{\varrho }\left(\mathit{\alpha }\right)\in \mathit{\alpha }⧢\{2,\dots ,n-2\}}-\tau {\displaystyle \frac{{(-1)}^{|\mathit{\alpha }|}2X_{\mathit{\alpha }}^{L\left(\mathit{\varrho }\right(\mathit{\alpha }\left)\right)}·F_{\mathit{\alpha }}·X_{\mathit{\alpha }}^{R\left(\mathit{\varrho }\right(\mathit{\alpha }\left)\right)}}{P^2-m_s^2}}\hfill \\ \hfill & \times A_{\mathrm{YMS}}^{{\left(0\right)}_{g_j}}(1,\mathit{\varrho }\left(\mathit{\alpha }\right),n-1,n\parallel \left\{g_i\right\}\setminus \mathit{\alpha } | \mathit{\sigma }),\hfill \\ \hfill B_{\mathrm{D}}& =\sum _{\mathit{\varrho }\left(g_j\right)\in {\overline{g}}_j⧢\{2,\dots ,n-2\}}-\tau {\displaystyle \frac{{(-1)}^{|\mathit{\alpha }|}2X_{\mathit{\alpha }}^{L\left(\mathit{\varrho }\left({\overline{g}}_j\right)\right)}·F_j·X_{\mathit{\alpha }}^{R\left(\mathit{\varrho }\left({\overline{g}}_j\right)\right)}}{P^2-m_s^2}}\hfill \\ \hfill & \times A_{\mathrm{YMS}}^{{\left(0\right)}_{g_j}}(1,\mathit{\varrho }\left(g_j\right),n-1,n\parallel \left\{g_i\right\}\setminus \mathit{\alpha } | \mathit{\sigma }).\hfill \end{array}$$

(75)

The sum of these three terms precisely matches with the soft behavior of the RHS of (49); it therefore induces the recursive expansion formula (49).

Comment on the boundary case (55) with only two scalars: The soft limit approach to the boundary case (55) follows from a slight adjustment of the above discussions on (49). When considering the soft behavior of gluon $g_j$ ($j\ne 1$) under the assumption that (55) holds for amplitudes with fewer gluons, one can verify (55) when $g_1$ plays the role of scalar $n-1$. Note that the starting point of this inductive procedure is the three-point amplitude when $\mathit{\beta }$ is empty in (55) that can also be induced by the soft behavior of gluon $g_1$.

5. The Relationship Between (1) and the HAB Formulas

In this section, we establish the relationship between the recursive expansion (1), and Formulas (49) and (55) derived from the HAB formula in the massless limit. To systematically study this relationship, we proceed from the simplest cases to more complicated ones, progressively. Particularly, we begin with the boundary case of two-scalar amplitudes. Next, we consider the base case of single-gluon amplitudes with an arbitrary number of scalars. Finally, we explicitly compute the five-point amplitude with two gluons, demonstrating that the recursive expansion (1) combined with the BCJ relations indeed reproduces the structure expected from the HAB formula. In each case, we derive a new expression from (1) and show that it matches the massless limit of the HAB formulas.

5.1. The Boundary Case: Amplitudes with Two Scalars

The YMS amplitudes involving massless scalars admit an expansion in terms of amplitudes with more scalars and fewer gluons, multiplied by kinematic coefficients. We show the relation between the two expansion Formulas (1) and (30) in the Appendix B. We now restrict the expression (A6), derived from (1), to YMS amplitudes containing exactly two scalars. In this section, the bar over a gluon $g_i$, if it acts as a scalar in the recursive formula, has been neglected, because it is straightforward to distinguish whether $g_i$ acts as a gluon or a scalar.

$$\begin{array}{cc}\hfill A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })=& {\displaystyle \frac{p_r·F_1·p_1}{p_r·k_1}}{A}_{\mathrm{YMS}}(1,g_1,2\parallel \left\{g_i\right\}\setminus g_1 | \mathit{\sigma })\hfill \\ \hfill & +\sum _{\mathit{\alpha }}{\displaystyle \frac{p_r·F_1·F_{{\mathit{\alpha }}^{\mathrm{T}}}·p_1}{p_r·k_1}}{A}_{\mathrm{YMS}}(1,\mathit{\alpha },g_1,2\parallel \left\{g_i\right\}\setminus (g_1\cup \mathit{\alpha }) | \mathit{\sigma }).\hfill \end{array}$$

(76)

Due to the gauge invariance, the reference momentum $p_r$ can be chosen arbitrarily. We choose $p_r=2p_1$, for which the coefficient of the first term in (76) vanishes. The factor of 2 is introduced for convenience, and it allows the denominator to be rewritten as a scalar propagator $2p_1·k_1={(p_1+k_1)}^2$. As a result, we obtain

$$A_{\mathrm{YMS}}(1,2\parallel \left\{g_i\right\} | \mathit{\sigma })=\sum _{\mathit{\alpha }}{\displaystyle \frac{2p_1·F_1·F_{{\mathit{\alpha }}^{\mathrm{T}}}·p_1}{{(p_1+k_1)}^2}}{A}_{\mathrm{YMS}}(1,\mathit{\alpha },g_1,2\parallel \left\{g_i\right\}\setminus (g_1\cup \mathit{\alpha }) | \mathit{\sigma }).$$

(77)

This precisely reproduces the massless limit (i.e., $m_s^2->0$) of the recursive expansion Formula (55) derived from the Hopf-algebra-based result, up to a reversal in the ordering of the field strength tensors

$$p_1·F_1·F_{{\mathit{\alpha }}^{\mathrm{T}}}·p_1={(-1)}^{|\mathit{\alpha }|+1}{p}_1·F_{\mathit{\alpha }}·F_1·p_1,$$

(78)

due to the antisymmetry property of the field strength tensors $F^{\mu \nu }=-F^{\nu \mu }$.

Having verified the equivalence for the boundary case, we now turn to amplitudes involving gluons. The simplest case involves a single gluon and multiple scalars, which serves as the base case for the recursive construction.

5.2. The Base Case: Amplitudes with One Gluon

In this subsection, we study YMS amplitudes involving at least three massless scalars and a single gluon.

Consider the general expansion Formula (30) restricted to amplitudes with only one gluon,

$$A_{\mathrm{YMS}}(1,2,\dots ,n-1,n\parallel g_1 | \mathit{\sigma })={\displaystyle \frac{p_r·F_1·X^L}{p_r·k_1}}{A}_{\mathrm{YMS}}(1,\{2,\dots ,n-1\}⧢g_1,n | \mathit{\sigma })$$

(79)

Exploiting gauge invariance, we choose the reference momentum to be $p_r=2p_n$. We define $X^R$ as the sum of the momenta of scalars to the right of the gluon $g_1$, excluding n. Then the numerators of the coefficients in the expansion (79) can be rewritten in a more symmetric form:

$$2p_n·F_1·X^L=-2(k_1+X^L+X^R)·F_1·X^L=-2X^R·F_1·X^L=2X^L·F_1·X^R.$$

(80)

Here, the momentum conservation $X^L+k_1+X^R+p_n=0$ and the antisymmetry property of the field strength tensor $F_1^{\mu \nu }$ are applied. Specifically, the coefficient associated with the YMS amplitude $A_{\mathrm{YMS}}(1,\{2,\dots ,n-1\},g_1,n | \mathit{\sigma })$ vanishes:

$$2p_n·F_1·(p_1+\dots +p_{n-1})=2p_n·F_1·(-k_1-p_n)=0,$$

(81)

or more directly, as a consequence of $X^R=0$. Hence, only permutations of the form $(1,\{2,\dots ,n-2\}⧢g_1,n-1,n)$ contribute to the expansion (79) of amplitudes, which implies that three scalars 1, $n-1$ and n are held fixed.

In parallel, the denominator of the coefficients in (79) can be rewritten as

$$2p_n·k_1={(p_n+k_1)}^2=P^2,$$

(82)

where $P\equiv p_1+\dots +p_{n-1}$ denotes the total momentum of all scalars except for n, namely $P=X^L+X^R$.

Collecting the above results, we obtain

$$A_{\mathrm{YMS}}(1,2,\dots ,n-1,n\parallel g_1 | \mathit{\sigma })={\displaystyle \frac{2X^L·F_1·X^R}{P^2}}{A}_{\mathrm{YMS}}(1,\{2,\dots ,n-2\}⧢g_1,n-1,n | \mathit{\sigma })$$

(83)

This result is consistent with the recursive expansion Formula (49) derived from the Hopf-algebra-based result. This serves as the base case for our construction of YMS amplitudes by induction.

5.3. Amplitudes with Two Gluons

We take the five-point amplitude $A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })$ with two gluons as our illustrative example, which can be straightforwardly generalized to amplitude with two gluons and more than three scalars. Amplitudes with more than two gluons can be calculated in a similar way, but will not be presented in the current work.

When applying (1) iteratively, we express $A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })$ by a combination of BS amplitudes as follows:

$$A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })=T_1+T_2,$$

(84)

where $T_1$ and $T_2$ refer to the terms involving ${({ϵ}_1·{ϵ}_2)}^0$ and ${({ϵ}_1·{ϵ}_2)}^1$, respectively. More explicitly, they are given as

$$\begin{array}{cc}\hfill T_1=& [({ϵ}_1·p_1)[{ϵ}_2·(p_1+k_1)]A_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })+({ϵ}_1·p_1)[{ϵ}_2·(p_{12}+k_1)]A_{\mathrm{BS}}(1,g_1,2,g_2,3 | \mathit{\sigma })\hfill \\ \hfill & +({ϵ}_1·p_{12})[{ϵ}_2·(p_{12}+k_1)]A_{\mathrm{BS}}(1,2,g_1,g_2,3 | \mathit{\sigma })]+(g_1\leftrightarrow g_2|\mathit{\sigma }),\hfill \\ \hfill T_2=& -{\displaystyle \frac{1}{2}}({ϵ}_1·{ϵ}_2)[(k_1·p_1)A_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })+(k_1·p_1)A_{\mathrm{BS}}(1,g_1,2,g_2,3 | \mathit{\sigma })\hfill \\ \hfill & +(k_1·p_{12})A_{\mathrm{BS}}(1,2,g_1,g_2,3 | \mathit{\sigma })]+(g_1\leftrightarrow g_2|\mathit{\sigma }).\hfill \end{array}$$

(85)

In the above expression, $(g_1\leftrightarrow g_2|\mathit{\sigma })$ means the term obtained by exchanging the roles of $g_1$ and $g_2$ in the coefficients and in the left permutation, keeping the right permutation as $\mathit{\sigma }$.

The $T_1$ sector. Now we focus on $T_1$. Using BCJ relations (A1) and (A2) [43,44] in Appendix A (an alternative approach to expanding YMS amplitudes in terms of BS ones on a BCJ basis can be found in [45]), one can transform all BS amplitudes in $T_1$ into a combination of $A_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })$ and $A_{\mathrm{BS}}(1,g_2,g_1,2,3 | \mathit{\sigma })$. Then $T_1$ turns into

$$T_1=\left(C_1+C_2+C_3+C_4+C_5\right)A_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })+(g_1\leftrightarrow g_2|\mathit{\sigma }),$$

(86)

where the coefficients are displayed as

$$\begin{array}{cc}\hfill C_1=& ({ϵ}_1·p_1) [{ϵ}_2·(p_1+k_1)]+({ϵ}_1·p_1) [{ϵ}_2·(p_{12}+k_1)] {\displaystyle \frac{2k_2·(p_1+k_1)}{{(p_{12}+k_1)}^2}}\hfill \\ \hfill C_2=& ({ϵ}_1·p_{12}) [{ϵ}_2·(p_{12}+k_1)] {\displaystyle \frac{2k_1·p_1}{p_{12}^2}}+({ϵ}_1·p_{12}) [{ϵ}_2·(p_{12}+k_1)] {\displaystyle \frac{2k_1·p_1}{p_{12}^2}}{\displaystyle \frac{2k_2·(p_1+k_1)}{{(p_{12}+k_1)}^2}}\hfill \\ \hfill C_3=& ({ϵ}_1·p_{12}) [{ϵ}_2·(p_{12}+k_1)] {\displaystyle \frac{2k_2·(p_1+k_1)}{p_{12}^2}}+({ϵ}_1·p_{12}) [{ϵ}_2·(p_{12}+k_1)] {\displaystyle \frac{-2k_2·(p_1+k_1)}{p_{12}^2}}\hfill \\ \hfill C_4=& ({ϵ}_2·p_1) [{ϵ}_1·(p_{12}+k_2)] {\displaystyle \frac{2k_1·p_1}{{(p_{12}+k_2)}^2}}+({ϵ}_2·p_{12}) [{ϵ}_1·(p_{12}+k_2)] {\displaystyle \frac{2k_2·p_1}{p_{12}^2}} {\displaystyle \frac{2k_1·p_1}{{(p_{12}+k_2)}^2}}\hfill \\ \hfill C_5=& ({ϵ}_2·p_{12}) [{ϵ}_1·(p_{12}+k_2)] {\displaystyle \frac{-2k_1·p_1}{p_{12}^2}}.\hfill \end{array}$$

(87)

We now calculate $C_1$ as a typical example

$$\begin{array}{cc}\hfill C_1=& ({ϵ}_1·p_1) {\displaystyle \frac{-2(p_{12}+k_1)·k_2 {ϵ}_2·(p_1+k_1)+2(p_{12}+k_1)·{ϵ}_2 k_2·(p_1+k_1)}{{(p_{12}+k_1)}^2}}\hfill \\ \hfill =& ({ϵ}_1·p_1) {\displaystyle \frac{-2p_2·F_2·(p_1+k_1)}{{(p_{12}+k_1)}^2}},\hfill \end{array}$$

(88)

where we have used the fact ${(p_{12}+k_1)}^2={(k_2+p_3)}^2=2k_2·p_3=-2k_2·(p_{12}+k_1)$ on the first line, due to momentum conservation and the massless condition. The antisymmetry of the field strength tensor has been considered on the second line. Using similar techniques, we simplify $C_2$, $C_3$ and $C_4$ as

$$\begin{array}{cc}\hfill C_2=& (-1) ({ϵ}_1·p_{12}) [{ϵ}_2·(p_{12}+k_1)] {\displaystyle \frac{2k_2·p_2}{{(p_{12}+k_1)}^2}} {\displaystyle \frac{2k_1·p_1}{p_{12}^2}}, C_3=0,\hfill \\ \hfill C_4=& (-1) {\displaystyle \frac{2p_2·F_2·p_1}{p_{12}^2}} {\displaystyle \frac{2{ϵ}_1·(p_{12}+k_2) k_1·p_1}{{(p_{12}+k_2)}^2}}+{\displaystyle \frac{2{ϵ}_2·p_1 {ϵ}_1·(p_{12}+k_2) k_1·p_1}{p_{12}^2}}.\hfill \end{array}$$

(89)

Further multiplying $1={\displaystyle \frac{p_{12}^2}{p_{12}^2}}=-{\displaystyle \frac{2k_1·p_{12}}{p_{12}^2}}+{\displaystyle \frac{2k_2·p_3}{p_{12}^2}}$ to $C_1$ and then taking the sum of $C_1,\dots ,C_5$, we finally get

$$\begin{array}{cc}\hfill C_1+C_2+C_3+C_4+C_5=& {\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2}} {\displaystyle \frac{2p_1·F_1·(p_2+k_2)}{{(p_{12}+k_2)}^2}}+{\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2}} {\displaystyle \frac{2(p_1+k_1)·F_2·p_2}{{(p_{12}+k_1)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{2(p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^0}}{p_{12}^2}}.\hfill \end{array}$$

(90)

Here $(p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^0}$ denotes the terms involving ${({ϵ}_1·{ϵ}_2)}^0$ as

$$\begin{array}{cc}\hfill (p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^0}& =p_1·k_1 {ϵ}_1·k_2 {ϵ}_2·p_2-p_1·{ϵ}_1 k_1·F_2·p_2\hfill \\ \hfill & =p_1·F_1·k_2 {ϵ}_2·p_2+p_1·{ϵ}_1 k_1·{ϵ}_2 k_2·p_2.\hfill \end{array}$$

(91)

With the above coefficients and the expression for amplitude with one gluon, we re-express $T_1$ in a convenient form:

$$\begin{array}{cc}\hfill & T_1={\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2}} A_{\mathrm{YMS}}(1,g_1,2,3\parallel \left\{g_2\right\} | \mathit{\sigma })+{\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2}} A_{\mathrm{YMS}}(1,g_2,2,3\parallel \left\{g_1\right\} | \mathit{\sigma })\hfill \\ \hfill & -{\displaystyle \frac{2(p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^0}}{p_{12}^2}}{A}_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })-{\displaystyle \frac{2(p_1·F_2·F_1·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^0}}{p_{12}^2}}{A}_{\mathrm{BS}}(1,g_2,g_1,2,3 | \mathit{\sigma }).\hfill \end{array}$$

(92)

The $T_2$ sector. When following an analogous discussion to the $T_1$ sector, by expressing the BS amplitudes in $T_2$ in to minimal basis $A_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })$ and $A_{\mathrm{BS}}(1,g_2,g_1,2,3 | \mathit{\sigma })$, we arrive at

$$T_2=-{\displaystyle \frac{2(p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^1}}{p_{12}^2}}{A}_{\mathrm{BS}}(1,g_1,g_2,2,3|\mathit{\sigma })-{\displaystyle \frac{2(p_1·F_2·F_1·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^1}}{p_{12}^2}}{A}_{\mathrm{BS}}(1,g_2,g_1,2,3|\mathit{\sigma }),$$

(93)

in which, $(p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^1}$ stands for the ${({ϵ}_1·{ϵ}_2)}^1$ sector of $p_1·F_1·F_2·p_2$:

$$(p_1·F_1·F_2·p_2){|}_{{({ϵ}_1·{ϵ}_2)}^1}=-p_1·k_1 {ϵ}_1·{ϵ}_2 k_2·p_2.$$

(94)

This result can also be straightforwardly obtained by considering gauge invariance condition:

$$\begin{array}{c}\hfill 0=A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\}{ | \mathit{\sigma })|}_{{ϵ}_2\to k_2}=T_1{{|}_{{ϵ}_2\to k_2}+T_2|}_{{ϵ}_2\to k_2}\end{array}$$

(95)

Noting that in the expression (97) of $T_1$, both amplitude $A_{\mathrm{YMS}}(1,g_1,2,3\parallel \left\{g_2\right\} | \mathit{\sigma })$ and coefficients containing the field strength tensor $F_2^{\mu \nu }$ are gauge-invariant objects; the only surviving terms in $T_1{|}_{{ϵ}_2\to k_2}$ are given by

$$T_1{|}_{{ϵ}_2\to k_2}=-{\displaystyle \frac{2p_1·k_1 {ϵ}_1·k_2 k_2·p_2}{p_{12}^2}} A_{\mathrm{BS}}(1,g_1,g_2,2,3|\mathit{\sigma })-{\displaystyle \frac{2p_1·k_2 k_2·{ϵ}_1 k_1·p_2}{p_{12}^2}} A_{\mathrm{BS}}(1,g_2,g_1,2,3|\mathit{\sigma }).$$

(96)

Substituting the above expression into (95), one can solve out the coefficient of ${ϵ}_1·k_2$ in $T_2{|}_{{ϵ}_2\to k_2}$ and verify (93).

Summing (97) and (93) together, we finally get

$$\begin{array}{cc}\hfill & A_{\mathrm{YMS}}(1,2,3\parallel \{g_1,g_2\} | \mathit{\sigma })\hfill \\ \hfill =& {\displaystyle \frac{2p_1·F_1·p_2}{p_{12}^2}} A_{\mathrm{YMS}}(1,g_1,2,3\parallel \left\{g_2\right\} | \mathit{\sigma })+{\displaystyle \frac{2p_1·F_2·p_2}{p_{12}^2}} A_{\mathrm{YMS}}(1,g_2,2,3\parallel \left\{g_1\right\} | \mathit{\sigma })\hfill \\ \hfill & -{\displaystyle \frac{2p_1·F_1·F_2·p_2}{p_{12}^2}}{A}_{\mathrm{BS}}(1,g_1,g_2,2,3 | \mathit{\sigma })-{\displaystyle \frac{2p_1·F_2·F_1·p_2}{p_{12}^2}}{A}_{\mathrm{BS}}(1,g_2,g_1,2,3 | \mathit{\sigma }),\hfill \end{array}$$

(97)

which is just the expected massless limit of the recursion Formula (49) based on the HAB formula.

Comments on the result: Since the coefficients of BCJ relations (A1) and (A2) are independent of the number of scalars, a straightforward generalization of the above discussions shows that amplitude with two gluons and more scalars obey (49) in the massless limit. However, the discussions on amplitudes with more than three gluons become much more complicated, because the complexity of this approach (precisely speaking, the application of BCJ relations) increases with the number of gluons. Nevertheless, one can always expect a general proof of (49) (in the massless limit) for amplitudes with more gluons via expressing the BS amplitudes in (4) by BCJ relations. This general proof deserves further study.

6. Conclusions

In this work, we established the relationship between the Hopf-algebra-based (HAB) formula for massive-scalar YMS amplitudes and the recursive expansion formula for massless-scalar YMS amplitudes. We proposed a convenient recursive formula for massive-scalar YMS amplitudes, and confirmed this formula with a soft behavior approach. This recursive formula iteratively results in the HAB formula, thus providing a convenient approach to the BCJ numerators for amplitudes with massive scalars. On the other hand, the massless limit of the HAB formula becomes a formula for massless-scalar YMS amplitudes that is further derived from the earlier proposed recursive formula [3]. We hope this work can provide a new insight into the study of matter coupling to gravitation.