1 Introduction

As is well known, soft theorems describe the universal behavior of scattering amplitudes when one or more external massless momenta are taken to near zero. Historically, soft theorems at the leading order for tree amplitudes were derived using Feynman rules [1, 2]. In 2014, new soft theorems at higher orders were proposed for gravity (GR) [3] and Yang–Mills (YM) theory [4] at the tree level, by applying the Britto–Cachazo–Feng–Witten (BCFW) recursion relation [5, 6]. Subsequently, these new results were generalized to arbitrary spacetime dimensions [7, 8], using Cachazo–He–Yuan (CHY) formulas [9,10,11,12,13]. Quite interestingly, the soft theorems can be understood as the consequence of asymptotic symmetries, and are related to memory effects [14,15,16,17,18,19,20,21,22]. Furthermore, the soft theorems and asymptotic symmetries for a wide range of other theories, including string theory, and the soft theorems at the loop level have been investigated in [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38].

On the other hand, the soft theorems have been exploited in the construction of tree-level amplitudes, such as using another type of soft behavior called Adler’s zero to construct amplitudes of various effective theories, the inverse soft theorem program, and so on [39,40,41,42,43,44,45]. Using methods reported in the above literature, one can bootstrap amplitudes by also assuming the explicit form of soft factors or soft operators. Recently, in [46], it was shown that the tree amplitudes can be fixed by imposing only the factorization property of soft theorems and the universality of soft operators, without knowledge of the explicit form of such operators. The results mentioned above indicate that the tree amplitudes are completely determined by soft theorems. Thus it is natural to expect that the relations among tree amplitudes can also be determined and understood via soft theorems. This is the main motivation for the current report.

Herein, we use the soft theorems to derive the well-known fundamental Bern–Carrasco–Johansson (BCJ) relation among color-ordered amplitudes [47,48,49,50]. More explicitly, we use the leading-order soft theorem for external scalars to obtain the fundamental BCJ relation among double-color-ordered bi-adjoint scalar (BAS) amplitudes at the tree level. Furthermore, we generalize the fundamental BCJ relation to the 1-loop level via the forward limit method. By expanding tree amplitudes to BAS amplitudes, we also use the fundamental BCJ relation to understand the Adler zero which describes the soft behavior of the nonlinear sigma model (NLSM) and Born–Infeld (BI) amplitudes at the tree level.

The remainder of this paper is organized as follows. In Sect. 2, we briefly introduce the tree BAS amplitudes, as well as their soft behavior at the leading order. In Sect. 3, we derive the fundamental BCJ relation by using the leading soft theorem for external scalars. Then we generalize the tree-level fundamental BCJ relation to the 1-loop level in Sect. 4. In Sect. 5, we use the fundamental BCJ relation, as well as the expanded formula of tree NLSM and BI amplitudes, to understand Adler’s zero. Finally, we end with a summary in Sect. 6.

2 Tree BAS amplitudes

For the reader’s convenience, in this section we give a brief review of the necessary background. In Sect. 2.1, we introduce the tree-level amplitudes of bi-adjoint scalar (BAS) theory. Various notations and techniques which will be used in subsequent sections are also mentioned. In Sect. 2.2, we discuss the soft behavior of tree BAS amplitudes, including the leading soft factor, as well as the statement that the leading soft behaviors of all external scalars fully determine the tree BAS amplitudes.

2.1 Tree-level BAS amplitudes

The BAS theory describes the bi-adjoint scalar field \(\phi _{a\bar{a}}\) with the Lagrangian

$$\begin{aligned} \mathcal{L}_\mathrm{{BAS}}={1\over 2}\,\partial _\mu \phi ^{a\bar{a}}\,\partial ^{\mu }\phi ^{a\bar{a}}-{\lambda \over 3!}\,f^{abc}f^{\bar{a}\bar{b}\bar{c}}\, \phi ^{a\bar{a}}\phi ^{b\bar{b}}\phi ^{c\bar{c}}, \end{aligned}$$
(1)

where the structure constant \(f^{abc}\) and generator \(T^a\) satisfy

$$\begin{aligned}{}[T^a,T^b]=if^{abc}T^c, \end{aligned}$$
(2)

and the dual-color algebra encoded by \(f^{\bar{a}\bar{b}\bar{c}}\) and \(T^{\bar{a}}\) is analogous. The tree-level amplitudes of this theory contain only propagators, and can be decomposed into double-color-ordered partial amplitudes via the standard technique. Each double-color-ordered partial amplitude is simultaneously planar with respect to two-color orderings, and arise from expanding the full n-point amplitude to \(\mathop \textrm{Tr}(T^{a_{\sigma _1}}\cdots T^{a_{\sigma _n}})\) and \(\mathop \textrm{Tr}(T^{\bar{a}_{\bar{\sigma }_1}}\cdots T^{\bar{a}_{\bar{\sigma }_n}})\), respectively, where \(\sigma _i\) and \(\bar{\sigma }_i\) denote permutations among all external scalars.

To calculate the double-color-ordered partial amplitudes, it is convenient to employ the diagrammatic method proposed by Cachazo et al. [11]. For a BAS amplitude whose double-color orderings are given, this method directly provides the corresponding Feynman diagrams as well as the overall sign. Let us consider the 5-point example \(\mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)\). In Fig. 1, the first diagram satisfies both two-color orderings (1, 2, 3, 4, 5) and (1, 4, 2, 3, 5), while the second one satisfies the ordering (1, 2, 3, 4, 5) but not (1, 4, 2, 3, 5). Thus, the first diagram is allowed by the double-color orderings (1, 2, 3, 4, 5|1, 4, 2, 3, 5), while the second one is not. It is easy to see that other diagrams are also forbidden by the ordering (1, 4, 2, 3, 5); thus the first diagram in Fig. 1 is the only diagram that contributes to the amplitude \(\mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)\).

Fig. 1
figure 1

Two 5-point diagrams

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The Feynman diagrams that contribute to a given BAS amplitude can be obtained via systematic diagrammatic rules. For the above example, one can draw a disk diagram as follows.

  • Draw points on the boundary of the disk according to the first ordering (1, 2, 3, 4, 5).

  • Draw a loop of line segments which connecting the points according to the second ordering (1, 4, 2, 3, 5).

The obtained disk diagram is shown in the first diagram in Fig. 2. One can see that two orderings share the boundaries \(\{1,5\}\) and \(\{2,3\}\). These co-boundaries indicate channels \({1/s_{15}}\) and \({1/s_{23}}\), therefore the first Feynman diagram in Fig. 1. Then the BAS amplitude \(\mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)\) can be computed as

$$\begin{aligned} \mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)={1\over s_{23}}{1\over s_{51}}, \end{aligned}$$
(3)

up to an overall sign. The Mandelstam variable \(s_{i\cdots j}\) is defined as

$$\begin{aligned} s_{i\cdots j}\equiv K_{ij}^2,~~~~K_{ij}\equiv \sum _{a=i}^j\,k_a,~~~~ \end{aligned}$$
(4)

where \(k_a\) is the momentum carried by the external leg a.

As another example, let us consider the BAS amplitude \(\mathcal{A}_S(1,2,3,4,5|1,2,4,3,5)\). The corresponding disk diagram is shown in the second configuration in Fig. 2, and one can see two orderings having co-boundaries \(\{3,4\}\) and \(\{5,1,2\}\). The co-boundary \(\{3,4\}\) indicates the channel \({1/ s_{34}}\). The co-boundary \(\{5,1,2\}\) indicates the channel \({1/s_{512}}\) which is equivalent to \(1/s_{34}\), as well as sub-channels \({1/ s_{12}}\) and \({1/ s_{51}}\). Using the above decomposition, one can calculate \(\mathcal{A}_S(1,2,3,4,5|1,2,4,3,5)\) as

$$\begin{aligned} \mathcal{A}_S(1,2,3,4,5|1,2,4,3,5)={1\over s_{34}}\Bigg ({1\over s_{12}}+{1\over s_{51}}\Bigg ), \end{aligned}$$
(5)

up to an overall sign.

The overall sign, determined by the color algebra, can be fixed by the following rules.

  • Each polygon with an odd number of vertices contributes a plus sign if its orientation is the same as that of the disk and a minus sign if it is opposite.

  • Each polygon with an even number of vertices always contributes a minus sign.

  • Each intersection point contributes a minus sign.

We now apply these rules to previous examples. In the first diagram in Fig. 2, the polygons are three triangles, namely 51A, A4B and B23, which contribute \(+\), −, \(+\), respectively, while two intersection points A and B contribute two −. In the second one in Fig. 2, the polygons are 512A and A43, which contribute two −, while the intersection point A contributes −. Then we arrive at the full results

$$\begin{aligned} \mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)= & {} -{1\over s_{23}}{1\over s_{51}},\nonumber \\ \mathcal{A}_S(1,2,3,4,5|1,2,4,3,5)= & {} -{1\over s_{34}}\Bigg ({1\over s_{12}}+{1\over s_{51}}\Bigg ). \end{aligned}$$
(6)

In the remainder of this paper, we adopt another convention for the overall sign. If the line segments form a convex polygon, and the orientation of the convex polygon is the same as that of the disk, then the overall sign is \(+\). For instance, the disk diagram in Fig. 3 indicates the overall sign \(+\) under the new convention, while the old convention gives − according to the square formed by four line segments. Note that the diagrammatic rules described previously still give the related sign between different disk diagrams. For example, two disk diagrams in Fig. 2 show that the relative sign between \(\mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)\) and \(\mathcal{A}_S(1,2,3,4,5|1,2,4,3,5)\) is \(+\).

Fig. 2
figure 2

Diagram for \(\mathcal{A}_S(1,2,3,4,5|1,4,2,3,5)\) and \(\mathcal{A}_S(1,2,3,4,5|1,2,4,3,5)\)

Full size image
Fig. 3
figure 3

The overall sign \(+\) under the new convention

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When considering the soft limit, the 2-point channels play a central role. Since the partial BAS amplitude carries two color orderings, if the 2-point channel contributes \(1/s_{ab}\) to the amplitude, external legs a and b must be adjacent to each other in both two orderings. Suppose the first color ordering is \((\cdots ,a,b,\cdots )\); then \(1/s_{ab}\) is allowed by this ordering. To denote whether it is allowed by another one, we introduce the symbol \(\delta _{ab}\), whose ordering of two subscripts a and b is determined by the first color ordering.Footnote 1 The value of \(\delta _{ab}\) is \(\delta _{ab}=1\) if another color ordering is \((\cdots ,a,b,\cdots )\), \(\delta _{ab}=-1\) if another color ordering is \((\cdots ,b,a,\cdots )\), due to the anti-symmetry of the structure constant, i.e., \(f^{abc}=-f^{bac}\), and \(\delta _{ab}=0\) otherwise. From the definition, it is straightforward to see that \(\delta _{ab}=-\delta _{ba}\). The symbol \(\delta _{ab}\) will be frequently used later.

2.2 Leading soft behavior of tree BAS amplitudes

In this subsection, we first derive the leading-order soft factor for the BAS scalar, then explain that the leading soft behaviors of all external scalars uniquely determine the tree BAS amplitudes.

The soft limit can be achieved by rescaling the massless momenta via a soft parameter as \(k^\mu \rightarrow \tau k^\mu \), and taking the limit \(\tau \rightarrow 0\). Consider the double-color-ordered BAS amplitude \(\mathcal{A}_S(1,\ldots ,n|\sigma )\), which carries two color orderings \((1,\ldots ,n)\) and \(\sigma \). We rescale \(k_i\) as \(k_i\rightarrow \tau k_i\), and expand the amplitude in \(\tau \). The leading-order contribution clearly comes from 2-point channels \(1/s_{i(i+1)}\) and \(1/ s_{(i-1)i}\) which provide the \(1/\tau \) order contributions, namely,

$$\begin{aligned}{} & {} \mathcal{A}^{(0)}_S(1,\ldots ,n|\sigma )\nonumber \\{} & {} \quad ={1\over \tau }\Bigg ({\delta _{i(i+1)}\over s_{i(i+1)}}+{\delta _{(i-1)i}\over s_{(i-1)i}}\Bigg ) \nonumber \\{} & {} \qquad \times \mathcal{A}_S(1,\ldots ,i-1,,i+1,\ldots ,n|\sigma \setminus i)\nonumber \\{} & {} \quad =S^{(0)}_s(i)\,\mathcal{A}_S(1,\ldots ,i-1,,i+1,\ldots ,n|\sigma \setminus i),~~~ \end{aligned}$$
(7)

where \(i\!\!\!/\) indicates the elimination of leg i, and \(\sigma \setminus i\) represents the color ordering generated from \(\sigma \) by eliminating i. The leading soft factor \(S^{(0)}_s(i)\) for the scalar i is extracted as

$$\begin{aligned} S^{(0)}_s(i)={1\over \tau }\,\Bigg ({\delta _{i(i+1)}\over s_{i(i+1)}}+{\delta _{(i-1)i}\over s_{(i-1)i}}\Bigg ),~~~~ \end{aligned}$$
(8)

which acts on external scalars that are adjacent to i in two color orderings. Note that in our notation, the parameter \(\tau \) is absorbed into the soft factor \(S^{(0)}_s(i)\).

If the factorization in (7) is satisfied for arbitrary external leg \(i\in \{1,\ldots ,n\}\) when \(k_i\rightarrow \tau k_i\), the tree BAS amplitude is completely determined, as can be understood as follows. The tree BAS amplitudes consist only of propagators, and thus are determined by correct channels. Since the soft factor (8) determines the 2-point channels, factors \(S^{(0)}_s(i)\) with \(i\in \{1,2,3,4\}\) clearly fix the 4-point BAS amplitudes. Assume that all \((n-1)\)-point BAS amplitudes are already obtained, which means that all amplitudes \(\mathcal{A}_S(1,\ldots ,i-1,i\!\!\!/,i+1,\ldots ,n|\sigma {\setminus } i)\) at the r.h.s. of (7) are provided. The soft theorem in (7) provides all correct 2-point poles of each n-point amplitude when i is around \(\{1,\ldots ,n\}\). For higher-point channels, we observe that \(1/s_{\pmb {{\alpha }}}=1/s_{\bar{\pmb {{\alpha }}}}\), due to the momentum conservation. Here, \(\pmb {{\alpha }}\) is a subset of \(\{1,\ldots ,n\}\), and \(\bar{\pmb {{\alpha }}}\) is the complement, and we assume that \(i\in \pmb {{\alpha }}\), where i is the soft external leg. Since the pole \(1/s_{\bar{\pmb {{\alpha }}}}\) is already included in the \((n-1)\)-point sub-amplitude, it also contributes to the n-point amplitude. Thus, correct poles of n-point amplitude for all channels are determined, and these poles uniquely fix the n-point BAS amplitude. Therefore, one can start from the 4-point amplitudes which are determined by the leading soft theorem, and use the leading soft theorem to generate higher-point amplitudes recursively. As a result, we conclude that the leading soft behaviors of all external scalars fix the tree BAS amplitudes. This conclusion is equivalent to the following statement: if a relation among tree BAS amplitudes is satisfied when taking arbitrary external leg \(i\in \{1,\ldots ,n\}\) to be soft, then this relation is satisfied by BAS amplitudes themselves. This inference will play an important role when deriving the fundamental BCJ relation in the next section.

3 Fundamental BCJ relation at the tree level

In this section, we use the soft theorem in (7) to derive the fundamental BCJ relation among double-color-ordered tree BAS amplitudes. In Sect. 3.1, we derive the fundamental BCJ relation at the leading order when one of the external scalars is taken to be soft. In Sect. 3.2, we show that this relation is satisfied at any order, by verifying the soft behaviors of other external scalars.

3.1 Derivation

Consider the color-ordered \((n+1)\)-point BAS amplitudes whose external legs are denoted as \(i\in \{1,\ldots ,n\}\) and s. The definition of \(\delta _{is}\) in Sect. 2.1 indicates

$$\begin{aligned} 0=\sum _{i=1}^n\,\delta _{is},~~ \end{aligned}$$
(9)

therefore,

$$\begin{aligned} 0= & {} \sum _{i=1}^n\,(k_s\cdot k_i)\,{\delta _{is}\over s_{is}}\nonumber \\= & {} -(k_s\cdot K_{1(n-1)})\,{\delta _{ns}\over s_{ns}}+\sum _{i=1}^{n-1}\,(k_s\cdot k_i)\,{\delta _{is}\over s_{is}}\nonumber \\= & {} \sum _{i=1}^{n-1}\,(k_s\cdot k_i)\,\Bigg ({\delta _{is}\over s_{is}}+{\delta _{sn}\over s_{sn}}\Bigg )\nonumber \\= & {} \sum _{i=1}^{n-1}\,\sum _{j=i}^{n-1}\,(k_s\cdot k_i)\,\Bigg ({\delta _{js}\over s_{js}}+{\delta _{s(j+1)}\over s_{s(j+1)}}\Bigg )\nonumber \\= & {} \sum _{j=1}^{n-1}\,(k_s\cdot K_{1j})\,\Bigg ({\delta _{js}\over s_{js}}+{\delta _{s(j+1)}\over s_{s(j+1)}}\Bigg ),~~~~ \end{aligned}$$
(10)

where the combinatorial momentum \(K_{ab}\) is defined as \(K_{ab}\equiv \sum _{i=a}^b\,k_i\), and the Mandelstam variable \(s_{ij}\) is defined as \(s_{ij}=2k_i\cdot k_j\). The second equality uses the momentum conservation. The third and fourth equalities are obtained by employing \(\delta _{ab}=-\delta _{ba}\). Using (10) and the leading-order soft theorem (7) for the scalar s, we find

(11)

which suggests the full fundamental BCJ relation

(12)

Here, the combinatorial momentum \(X_s\) is defined by summing over momenta carried by external scalars at the l.h.s. of s in the color ordering. The symbol means summing over all possible shuffles of two ordered sets \(\pmb {{\beta }}_1\) and \(\pmb {{\beta }}_2\), i.e., all permutations in the set \(\pmb {{\beta }}_1\cup \pmb {{\beta }}_2\), while preserving the orderings of \(\pmb {{\beta }}_1\) and \(\pmb {{\beta }}_2\). For instance, suppose that \(\pmb {{\beta }}_1=\{1,2\}\) and \(\pmb {{\beta }}_2=\{3,4\}\). Then

(13)

The derivation in this subsection only gives (11), which is the fundamental BCJ relation at the leading order, and the full one (12) should be regarded as a conjecture at the current step. To make this work self-contained, it seems that we need to verify the relation (12) at all orders. However, we have another more efficient choice. In Sect. 2.2, we pointed out that the tree BAS amplitudes are fully determined by leading-order soft behaviors of all external scalars. The derivation of (11) only uses the leading soft theorem for the scalar s. Thus, we can verify the conjectured relation (12) by checking leading soft behaviors for other external particles, as we will do in the next subsection. As will be seen, all these soft behaviors are finally reduced to the algebraic relation in (9).

The fundamental BCJ relation (12) can be generalized to color-ordered YM amplitudes via the well-known double-copy structure. More explicitly, the double-copy structure indicates the following expansion [51,52,53,54,55,56]

$$\begin{aligned} \mathcal{A}_\textrm{YM}(1,\ldots ,n)=\sum _{\sigma '\in S_{n-2}}\,C(\sigma ')\,\mathcal{A}_\textrm{S}(1,\ldots ,n|1,\sigma ',n).~~\nonumber \\ \end{aligned}$$
(14)

Here, \(S_{n-2}\) stands for permutations among \((n-2)\) legs in \(\{2,\ldots ,n-1\}\). The coefficients \(C(\sigma ')\) serve as the BCJ numerators, and depend on permutations \(\sigma '\), external momenta, and the polarization of external gluons. Combining (12) and (14), we arrive at the relation

(15)

This is the standard BCJ relation in the literature [47]. One can also generate the general BCJ relations from the fundamental one, see in [57].

3.2 Verification

In this subsection, we prove the fundamental BCJ relation (12) by considering the leading-order soft behavior of each external scalar. More explicitly, we will show that under the rescaling \(k_i\rightarrow \tau k_i\) with \(i\in \{1,\ldots ,n\}\), the leading-order contribution of (12) gives the fundamental BCJ relation for BAS amplitudes with fewer external legs. Such reduction procedure is terminated at the simplest fundamental BCJ relation for 4-point BAS amplitudes, whose correct soft behaviors are ensured by the definition of \(\delta _{ab}\).

Under the rescaling \(k_1\rightarrow \tau k_1\), the leading-order contribution of (12) is given by

(16)

where the amplitude \(\mathcal{A}(s,2,\ldots ,n||\sigma \setminus 1)\) does not contribute, since \(k_s\cdot X_s\) is proportional to \(\tau \) when \(X_s=k_1\). The relation (16) requires

(17)

which is nothing but the fundamental BCJ relation among n-point BAS amplitudes .

We can use the momentum conservation to rewrite the fundamental BCJ relation (12) as

(18)

where \(X^\textrm{R}_s\) is defined as the summation of momenta of legs at the r.h.s. of s in the color ordering. Comparing (18) with (12), we see the manifest symmetry between external legs 1 and n. This symmetry, together with the result (16), indicates that for \(k_n\rightarrow \tau k_n\) we have

(19)

which includes the fundamental BCJ relation among n-point BAS amplitudes .

For \(k_i\rightarrow \tau k_i\) with \(i\in \{2,\ldots ,n-1\}\), the leading-order contribution of (12) can be separated as

(20)

One can add the last two lines at the r.h.s. of (20) to obtain

$$\begin{aligned}{} & {} {1\over \tau }\,\Bigg ({\delta _{si}\over s_{si}}+{\delta _{i(i+1)}\over s_{i(i+1)}}\Bigg )\,(k_s\cdot X_s)\nonumber \\{} & {} \quad \times \mathcal{A}_\textrm{S}(1,\ldots ,i-1,s,,i+1,\ldots ,n|\sigma \setminus i)\nonumber \\{} & {} \quad +{1\over \tau }\,\Bigg ({\delta _{(i-1)i}\over s_{(i-1)i}}+{\delta _{is}\over s_{is}}\Bigg )\,(k_s\cdot X_s)\nonumber \\{} & {} \quad \times \mathcal{A}_\textrm{S}(1,\ldots ,i-1,,s,i+1,\ldots ,n|\sigma \setminus i)\nonumber \\{} & {} \qquad ={1\over \tau }\,\Bigg ({\delta _{(i-1)i}\over s_{(i-1)i}}+{\delta _{i(i+1)}\over s_{i(i+1)}}\Bigg )\,(k_s\cdot X_s)\nonumber \\{} & {} \quad \times \mathcal{A}_\textrm{S}(1,\ldots ,i-1,s,i+1,\ldots ,n|\sigma \setminus i).~~~ \end{aligned}$$
(21)

Substituting (21) into (20), we arrive at

(22)

which can be recognized as the fundamental BCJ relations among n-point BAS amplitudes .

From the calculations above, we see that taking \(k_i\rightarrow \tau k_i\) for each \(i\in \{1,\ldots ,n\}\) reduces the fundamental BCJ relation (12) among \((n+1)\)-point BAS amplitudes to the same relation among n-point BAS ones. This reduction can be repeated recursively. Thus, to prove the leading-order soft behavior for the conjectured relation (12), we only need to check this relation among 4-point BAS amplitudes , namely,

(23)

The 4-point case is quite special, since all Mandelstam variables vanish at the \(\tau ^0\) order when one of external legs is soft. In other words, the leading-order contribution of (23) is at the \(\tau ^0\) order rather than the \(\tau ^{-1}\) order. To see this, we use the momentum conservation to rewrite (23) as

$$\begin{aligned} 0=(k_s\cdot k_1)\,\mathcal{A}_\textrm{S}(1,s,2,3|\sigma )-(k_2\cdot k_1)\,\mathcal{A}_\textrm{S}(1,2,s,3|\sigma ),\nonumber \\ \end{aligned}$$
(24)

then rescale \(k_1\) as \(k_1\rightarrow \tau k_1\), and obtain the leading-order contribution as

$$\begin{aligned} 0= & {} (k_s\cdot k_1)\,\Bigg ({\delta _{31}\over s_{31}}+{\delta _{1s}\over s_{1s}}\Bigg )\,\mathcal{A}_\textrm{S}(s,2,3|\sigma \setminus 1)\nonumber \\ {}{} & {} -(k_1\cdot k_2)\,\Bigg ({\delta _{31}\over s_{31}}+{\delta _{12}\over s_{12}}\Bigg )\nonumber \\{} & {} \times \mathcal{A}_\textrm{S}(2,s,3|\sigma \setminus 1).~~~ \end{aligned}$$
(25)

To verify (25), we observe that \(\mathcal{A}_\textrm{S}(s,2,3|\sigma \setminus 1)=-\mathcal{A}_\textrm{S}(2,s,3|\sigma \setminus 1)\), which can be understood via either the anti-symmetry of the structure constant of the Lie group, or the well-known Kleiss–Kuijf relation [64]. This observation turns (25) into

$$\begin{aligned} 0= & {} \Bigg (\delta _{12}+\delta _{13}+\delta _{1s}\Bigg )\,\mathcal{A}_\textrm{S}(s,2,3|\sigma \setminus 1),~~~ \end{aligned}$$
(26)

where we have used the momentum conservation to obtain the coefficient of \(\delta _{13}\). The relation (26) is guaranteed by \(0=\delta _{12}+\delta _{13}+\delta _{1s}\), due to the definition of \(\delta _{ab}\). The correct soft behavior for \(k_3\rightarrow \tau k_3\) is ensured by the symmetry between legs 1 and 3, as discussed around (18).

Next, we use the momentum conservation to rewrite (23) as

$$\begin{aligned} 0=(k_2\cdot k_3)\,\mathcal{A}_\textrm{S}(1,s,2,3|\sigma )-(k_2\cdot k_1)\,\mathcal{A}_\textrm{S}(1,2,s,3|\sigma ),\nonumber \\ \end{aligned}$$
(27)

and consider \(k_2\rightarrow \tau k_2\). The leading-order contribution is

$$\begin{aligned} 0= & {} (k_2\cdot k_3)\,\Bigg ({\delta _{s2}\over s_{s2}}+{\delta _{23}\over s_{23}}\Bigg )\,\mathcal{A}_\textrm{S}(1,s,3|\sigma \setminus 2)\nonumber \\{} & {} -(k_2\cdot k_1)\,\Bigg ({\delta _{12}\over s_{12}}+{\delta _{2s}\over s_{2s}}\Bigg )\,\mathcal{A}_\textrm{S}(1,s,3|\sigma \setminus 2)\nonumber \\= & {} \Bigg (\delta _{21}+\delta _{23}+\delta _{2s}\Bigg )\,\mathcal{A}_\textrm{S}(1,s,3|\sigma \setminus 2),~~~ \end{aligned}$$
(28)

where the momentum conservation is used to obtain the coefficient of \(\delta _{2s}\). Again, the relation (28) is ensured by the definition of \(\delta _{ab}\).

Thus, we conclude that the relation (12) among double-color-ordered BAS amplitudes is satisfied when the arbitrary external scalar is taken to be soft, and therefore is correct.

4 Fundamental BCJ relation at the 1-loop level

The purpose of this section is to generalize the tree-level fundamental BCJ relation to the 1-loop level. To realize the goal, we first generalize the fundamental BCJ relation to tree BAS amplitudes with two massive external scalars in Sect. 4.1. In Sect. 4.2, we review the forward limit method which generates the 1-loop Feynman integrands from tree amplitudes. Then, in Sect. 4.3, we use the result obtained in Sect. 4.1, together with the forward limit method, to obtain the fundamental BCJ relation among Feynman integrands at the 1-loop level.

4.1 Tree-level fundamental BCJ relation with two massive external states

In the manipulation (10), supposing that all external momenta except \(k_s\) are massive, with equal mass \(k_i^2=m^2\) where \(i\in \{1,\ldots ,n\}\), and all \(s_{is}\) are replaced by \(s_{is}-m^2=2k_i\cdot k_s\), we see that the result still holds. Furthermore, the soft factor (8) is also valid under the replacement \(s_{is}\rightarrow s_{is}-m^2\). This observation yields the relation (11) at the leading order, and thus leads us to guess that the fundamental BCJ relation holds for the above massive case. However, in the current situation, one cannot conclude that the tree BAS amplitude is completely fixed by soft behaviors of external scalars. First, there is only one massless scalar s that can be taken to be soft, and thus is not sufficient to fix poles for all channels. Secondly, since we have not chosen the explicit structure of interaction vertices for the new theory which includes massive scalars, in principle each internal scalar can be either massless or massive, and thus it is impossible to determine the corresponding poles. Consequently, the fundamental BCJ relation does not hold for this case, as can be verified directly.

However, we can restrict ourselves to a quite special case, where the external legs 1 and n are massive, with \(k_1^2=k_n^2=m^2\), while all other external legs are massless. Furthermore, we assume that a massive scalar propagates through the amplitude from leg 1 to leg n. Let us call this path from 1 to n the massive scalar line. One can think of the full amplitude as a variety of massless scalars coupled to each other and finally coupled to the massive scalar line. For this special case, the following simple argument can convince us that the fundamental BCJ relation (12) still holds. Let us use the Mandelstam variable \(s_{\pmb {{\alpha }}}\) to denote the corresponding channel, where \(\pmb {{\alpha }}\) is a set of external legs satisfying \(\pmb {{\alpha }}\subset \{1,\ldots ,n\}\cup s\). We separate all channels into two classes; one is \(1\in \pmb {{\alpha }}\), \(n\in \bar{\pmb {{\alpha }}}\), and the other one is \(1,n\in \pmb {{\alpha }}\). For the first case, \(s_{\pmb {{\alpha }}}\) corresponds to the internal line which belong to the massive line; thus, the propagator takes the form \(1/(s_{\pmb {{\alpha }}}-m^2)\). Since there is only one massive scalar 1 in \(\pmb {{\alpha }}\), we have \(s_{\pmb {{\alpha }}}-m^2=\sum _{i,j\in \pmb {{\alpha }}}2k_i\cdot k_j\), and this form is the same as that for the massless case. For the second case, we can use \(s_{\bar{\pmb {{\alpha }}}}\) instead of \(s_{\pmb {{\alpha }}}\), due to the momentum conservation. In other words, for the second case, one can always choose an expression that cannot “see” the massive legs 1 and 2. To summarize, we find that the expression of the BAS amplitude is not affected when legs 1 and n turn out to be massive. Furthermore, in the fundamental BCJ relation (12), it seems that there is no chance for \(k_1^2\) or \(k_n^2\) to play any role, even if we use the momentum conservation to rewrite any \(k_i\cdot k_s\). The above argument provides strong evidence for the validity of the fundamental BCJ relation for the case \(k_1^2=k_n^2=m^2\), and we numerically verified this statement up to the 8-point.

Starting from the BAS amplitudes with \(k_1^2=k_n^2=m^2\), one can generate the 1-loop Feynman integrand by sewing two massive legs together via the so-called forward limit method. In this procedure, one needs to allow \(m^2<0\); thus, it is more suitable to interpret 1 and n as off-shell legs with \(k_1^2=k_2^2\). Applying this sewing manipulation, one can generalize the tree-level fundamental BCJ relation to the 1-loop level, as will be seen in Sect. 4.3.

4.2 Forward limit method

The 1-loop Feynman integrands can be generated from the corresponding tree amplitudes, via the so called forward limit procedure. For instance, the 1-loop CHY formulas can be obtained by applying this operation, as studied in [58,59,60,61]. For the reader’s convenience, in this subsection we introduce the general idea and features of the forward limit.

The forward limit is reached as follows:

  • Consider a \((n+2)\)-point tree amplitude \(\mathcal{A}_{n+2}(k_+,k_-)\) including n massless legs with momenta in \(\{k_1,\ldots ,k_n\}\) and two off-shell legs with \(k_+^2=k_-^2\ne 0\).

  • Take the limit \(k_{\pm }\rightarrow \pm \ell \), and glue the two corresponding legs together. We denote this manipulation as \(\mathcal{L}\). Performing \(\mathcal{L}\) on the tree amplitude leads to a special tree amplitude with \(k_+=-k_-=\ell \), rather than a 1-loop-level object.

  • Sum over all allowed internal states of the internal particle with loop momentum \(\ell \), such as polarization vectors or tensors, colors, flavors, and so on,Footnote 2 we denote this manipulation as \(\mathcal{E}\).

Roughly speaking, the obtained object, times the factor \(1/\ell ^2\) as

$$\begin{aligned} {1\over \ell ^2}\,\mathcal{F}\,\mathcal{A}_{n+2}(k_+^{h_+},k_-^{h_-})={1\over \ell ^2}\,\sum _{h}\,\mathcal{A}_{n+2}(\ell ^{h},-\ell ^{\bar{h}}),~~~~ \end{aligned}$$
(29)

contributes to the n-point 1-loop Feynman integrand \(\textbf{I}_n\). Here we introduce the forward limit operator

$$\begin{aligned} \mathcal{F}\equiv \mathcal{E}\,\mathcal{L}, \end{aligned}$$
(30)

to denote the operation of taking forward limit. In this paper, we denote the 1-loop Feynman integrands by \(\textbf{I}\). From now on, we will neglect the subscript n of \(\textbf{I}\), since we will use other methods to denote the number of external legs.

For the individual Feynman diagram, the manipulation in (29) obviously transforms the tree diagram to the 1-loop one. However, the full 1-loop Feynman integrand is obtained by summing over all appropriate diagrams. Thus, let us consider what requirement should be satisfied if the resulting object of the manipulation in (29) can be interpreted as the correct 1-loop Feynman integrand. It is easy to see that after summing over all allowed tree-level diagrams, each 1-loop diagram receives contributions from tree diagrams corresponding to cutting each propagator in the loop once (cutting is understood as the inverse operation of gluing legs \(+\) and − together, where \(+\), − denote external legs carrying \(k_+\) and \(k_-\), respectively), as can be seen in Fig. 4. Thus, the statement that the operation in (29) generates the correct Feynman integrand holds if and only if the term for an individual 1-loop diagram can be decomposed to terms for related tree diagrams, as shown in Fig. 4. Such decomposition can be realized via the so-called partial fraction identity [58, 62]:

$$\begin{aligned} {1\over D_1\cdots D_m}=\sum _{i=1}^m\,{1\over D_i}\Bigg [\prod _{j\ne i}\,{1\over D_j-D_i}\Bigg ], \end{aligned}$$
(31)

which implies

$$\begin{aligned}{} & {} {1\over \ell ^2(\ell +K_1)^2(\ell +K_{12})^2\cdots (\ell +K_{1(m-1)})^2}\nonumber \\{} & {} \quad \simeq {1\over \ell ^2}\,\sum _{i=1}^m\,\Bigg [\prod _{j= i}^{i+m-2}\,{1\over (\ell +K_{ij})^2-\ell ^2}\Bigg ].~~~~ \end{aligned}$$
(32)

For each individual term at the r.h.s. of the above relation, the loop momentum is shifted while the result of the Feynman integral is not altered. Here, \(\simeq \) means that the l.h.s. and r.h.s. are not equivalent at the integrand level, but are equivalent at the integration level. At the r.h.s. of (32), we have seen the propagators with the denominators \((\ell +K_{ij})^2-\ell ^2\), which are different from the standard ones \((\ell +K_{ij})^2\). This feature of propagators is the condition which should be satisfied if the manipulation in (29) provides the correct 1-loop Feynman integrand. In CHY formulas, this requirement is satisfied via the 1-loop-level scattering equations. From the viewpoint of Feynman diagrams, one can assume that each propagator in the loop is massive, with \(m^2=\ell ^2\). Thus, one can assume that the 1-loop Feynman integrand \(\textbf{I}_\circ \) is obtained from the manipulation in (29). To distinguish the full and the partial Feynman integrands obtained by decomposing the full integrands via the partial fraction identity, from now on, we use \(\textbf{I}_\circ \) to denote the former and \(\textbf{I}\) to denote the latter.

Fig. 4
figure 4

Decomposition of 1-loop Feynman integrand

Full size image

The above discussion concerns the full Feynman integrands without any color ordering, and now we turn to the color-ordered Feynman integrands. Since we have made sure that the full 1-loop Feynman integrand can be generated from the full tree amplitude via the forward limit operation, let us start with the full tree amplitude. Consider a theory in which external particles are in the adjoint representation of the U(N) group; the full tree amplitude can be expanded using the standard color decomposition as a sum over \((n+1)!\) terms

$$\begin{aligned} \mathcal{A}_{n+2}= & {} \sum _{\sigma _1\in S_{n+2}/\mathbb {Z}_{n+2}}\,\textrm{Tr}(T^{a_{\sigma _+}}T^{a_{\sigma _1}}\cdots T^{a_{\sigma _n}}T^{a_{\sigma _-}})\nonumber \\{} & {} \times \mathcal{A}(\sigma _+,\sigma _1,\ldots ,\sigma _n,\sigma _-).~~~~ \end{aligned}$$
(33)

Note that at the r.h.s., it is not necessary to add the subscript \(n+2\) to \(\mathcal{A}\), since the color ordering \(\sigma _+,\sigma _1,\ldots ,\sigma _n,\sigma _-\) already reflects the number of external legs. Taking the forward limit of external legs requires summing over the U(N) degrees of freedom of the two internal particles. This gives rise to two kinds of terms. The first comes from permutations such that legs \(+\) and − are adjacent, the corresponding color factors are given as

$$\begin{aligned}{} & {} \sum _{a_+=a_-=1}^{N^2}\,\delta _{a_+a_-}\,\textrm{Tr}(T^{a_{+}}T^{a_{\sigma _1}}\cdots T^{a_{\sigma _n}}T^{a_{-}})\nonumber \\{} & {} \quad \qquad =N\textrm{Tr}(T^{a_{\sigma _1}}\cdots T^{a_{\sigma _n}}), \end{aligned}$$
(34)

thus contributing to the n-point color-ordered Feynman integrand \(\textbf{I}_\circ (\sigma _1,\ldots ,\sigma _n)\). The second case wherein \(+\) and − are not adjacent gives rise to double-trace terms. In this paper, we only consider the single-trace terms, since they determine the double-trace terms [63], as can be proved by employing the tree-level Kleiss–Kuijf relation together with the forward limit operation [64]. For the single-trace case, the above discussion shows that the partial integrand obtained from taking the forward limit for \(\mathcal{A}(+,\sigma _1,\ldots ,\sigma _n,-)\) contributes to \(\textbf{I}_\circ (\sigma _1,\ldots ,\sigma _n)\). To find the full decomposition of \(\textbf{I}_\circ (\sigma _1,\ldots ,\sigma _n)\), we use the clear observation that several original color orderings give rise to the same trace factor after summing over \(a_+\) and \(a_-\), due to the cyclic symmetry of the trace factors. Collecting theses color orderings, one finds that after taking the forward limit, the decomposition (33) can be organized as

$$\begin{aligned}{} & {} \mathcal{F}\,\mathcal{A}_{n+2}=\sum _{\sigma _1\in S_{n+2}/\mathbb {Z}_{n+2}}\,\textrm{Tr}(T^{a_{\sigma _1}} \cdots T^{a_{\sigma _n}})\nonumber \\{} & {} \quad \times \sum _{j=0}^{n-1}\,\mathcal{F}\, \mathcal{A}(+,\sigma _{1+j},\ldots ,\sigma _{n+j},-)+(\mathrm{double-trace}).\nonumber \\ \end{aligned}$$
(35)

Consequently, the full color-ordered Feynman integrand can be expanded as the following cyclic summation

$$\begin{aligned}{} & {} \textbf{I}_\circ (\sigma _1,\ldots ,\sigma _n)=\sum _{j=0}^{n-1}\, \textbf{I}(+,\sigma _{1+j},\ldots ,\sigma _{n+j},-),~~~~ \end{aligned}$$
(36)

where the partial color-ordered integrands \(\textbf{I}(+,\sigma _{1+j},\ldots ,\sigma _{n+j},-)\) are obtained from the color-ordered tree amplitudes via the standard forward limit procedure in (29), namely,

$$\begin{aligned}{} & {} \textbf{I}(+,\sigma _{1+j},\ldots ,\sigma _{n+j},-)={1\over \ell ^2}\,\mathcal{F}\,\nonumber \\{} & {} \quad \mathcal{A}(+,\sigma _{1+j},\ldots ,\sigma _{n+j},-).~~ \end{aligned}$$
(37)

The cyclic summation in (36) indicates that each propagator in the loop has been cut once; thus \(\textbf{I}_\circ (\sigma _1,\ldots ,\sigma _n)\) and \( \textbf{I}(+,\sigma _{1+j},\ldots ,\sigma _{n+j},-)\) are also related via the partial fraction identity.

The forward limit is well defined for the \(\mathcal{N}=4\) SYM theory. For other theories, a quite general feature is that the obtained Feynman integrand suffers from divergence in the forward limit. Fortunately, the singular parts are found to be physically irrelevant, at least for theories under consideration in this paper. From the viewpoint of Feynman diagrams, the singular parts generated by the forward limit correspond to tadpole diagrams, and external legs correspond to bubble diagrams, which do not contribute to the S-matrix. From the CHY point of view, the singular parts can be bypassed by employing the following observation [59]: as long as the CHY integrand is homogeneous in \(\ell ^\mu \), the singular solutions contribute to the scaleless integrals which vanish under the dimensional regularization. The homogeneity in \(\ell ^\mu \) is satisfied by the BAS Feynman integrands under consideration in this note. Thus, in this section, we simply assume that the singular parts generated by the forward limit are excluded in an appropriate manner.

4.3 Generalizing the fundamental BCJ relation to the 1-loop level

With the forward limit method introduced in Sect. 4.2, we are now ready to generalize the fundamental BCJ relation to the 1-loop level. Using (36) and (37), we see that the single-trace BAS Feynman integrand can be obtained via

(38)

with

(39)

where \(+\) and − are two off-shell external legs for tree amplitudes , satisfying \(k_+=-k_-=\ell \). Using the tree-level fundamental BCJ relation (12), we know that

(40)

where \(X'_s\) is defined as the summation over the loop momentum \(\ell \) and momenta carried by external legs at the l.h.s. of s in the color ordering. Since \(k_s\cdot X'_s\) is commutable with the forward limit operator \(\mathcal{F}\), the partial integrands satisfy

(41)

Substituting (41) into (38), we arrive at

(42)

which serves as the fundamental BCJ relation among BAS Feynman integrands at the 1-loop level.

Using the double-copy structure, the relation (42) can be straightforwardly generalized to YM Feynman integrands,

(43)

which is the same as the result found in [65].

5 Understanding Adler’s zero

This section is devoted to understanding Adler’s zero for tree NLSM and BI amplitudes. In Sect. 5.1, we introduce the expanded formulas of tree NLSM and BI amplitudes. Then, in Sect. 5.2, we show that Adler’s zero can be manifested by applying the fundamental BCJ relation to the expanded formulas in Sect. 5.1.

5.1 Expanded NLSM and BI amplitudes

Among recent investigations of scattering amplitudes, one area of remarkable progress is in the expansions of amplitudes, which implies that the amplitudes of different theories can be unified, and provides a new tool for calculating them. These expansions have been studied from various angles [46, 51,52,53,54,55,56]. Among all of these angles, the double-copy structure plays a central role. In this section, we will use the following two expansions.

The n-point tree NLSM amplitude \(\mathcal{A}_\textrm{N}(\sigma )\) with arbitrary color ordering \(\sigma \) can be expanded to BAS amplitudes as follows [55, 56]

(44)

The analogous expansion holds for the BI amplitudes, i.e., the n-point BI amplitude \(\mathcal{A}_\textrm{B}(\{1,\ldots ,n\})\) can be expanded to YM amplitudes as [55, 56]

(45)

Obviously, these two expressions of NLSM and BI amplitudes allow us to apply the fundamental BCJ relation directly.

5.2 Adler’s zero

The NLSM and BI amplitudes satisfy the so-called Adler’s zero condition, i.e., amplitudes vanish when one of the external legs is soft. Such a phenomenon can be easily understood via the BCJ relation and expanded formulas in (44) and (45).

We first show that the NLSM amplitude \(\mathcal{A}_\textrm{N}(\sigma )\) vanishes when the external leg 2 is taken to be soft. The generalization to the other leg i with \(i\in \{2,\ldots ,n-1\}\) is straightforward. Let us consider a subset of BAS amplitudes at the r.h.s of (44),

(46)

We define \(X^{(2)}_{{\alpha }_i}\) as

$$\begin{aligned} X^{(2)}_{{\alpha }_i}=\lim _{k_2\rightarrow 0}\,X_{{\alpha }_i},~~ \end{aligned}$$
(47)

which is a constant for . Then, the BCJ relation (12) imposes

(48)

since all \(k_{{\alpha }_i}\cdot X^{(2)}_{{\alpha }_i}\) are constants. Thus, the effective part of (46) is

(49)

Since \(X_{{\alpha }_i}-X^{(2)}_{{\alpha }_i}=0~\textrm{or}~k_2\), we have

$$\begin{aligned}{} & {} \Bigg (\prod _{i=3}^{n-1}\,k_{{\alpha }_i}\cdot X_{{\alpha }_i}\Bigg )-\Bigg (\prod _{i=3}^{n-1}\,k_{{\alpha }_i}\cdot X^{(2)}_{{\alpha }_i}\Bigg )\nonumber \\{} & {} \quad =c_1\,(k_2\cdot R_{11})+c_2\,(k_2\cdot R_{21})\,(k_2\cdot R_{22})+\cdots ,~~~ \end{aligned}$$
(50)

where \(R_{ij}\) denotes proper Lorentz vectors.

Now we rescale \(k_2\) as \(k_2\rightarrow \tau k_2\), and expand (46) by \(\tau \). The leading-order behavior of is at the \(\tau ^{-1}\) order. On the other hand, the observation (50) shows that the effective coefficient \((k_2\cdot X_2)\,\Bigg [\Bigg (\prod _{i=3}^{n-1}\,k_{{\alpha }_i}\cdot X_{{\alpha }_i}\Bigg )-\Bigg (\prod _{i=3}^{n-1}\,k_{{\alpha }_i}\cdot X^{(2)}_{{\alpha }_i}\Bigg )\Bigg ]\) is vanishing or at the \(\tau ^{q}\) order, where q is an integer satisfying \(q\ge 2\). Consequently, we have

(51)

Therefore,

(52)

The consideration for \(k_2\rightarrow \tau k_2\) can be generalized to \(k_i\rightarrow \tau k_i\) directly, with \(i\in \{2,\ldots ,n-1\}\). Thus, all these legs satisfy Adler’s zero condition. The remaining task is to understand Adler’s zero for external legs 1 and n. To deal with the case \(k_1\rightarrow \tau k_1\), we consider the following subset of the r.h.s where the leg at the r.h.s of 1 in the color ordering is fixed, for instance

(53)

Taking \(k_1\rightarrow \tau k_1\) and expanding (53) by \(\tau \), we obtain the leading-order contribution

(54)

which vanishes due to the BCJ relation

(55)

Here, the definition of \(X^{(1)}_i\) is analogous to the definition of \(X^{(2)}_{{\alpha }_i}\) in (47), i.e., \(X^{(1)}_i=\lim _{k_1\rightarrow 0} X_i\). Thus, the combination (53) satisfies Adler’s zero condition. In the above discussion, we fixed the leg at the r.h.s. of 1 to be 2. The analogous argument holds when replacing 2 by any \(i\in \{2,\ldots ,n-1\}\). Thus, we conclude that the expanded formula (44) vanishes when the external leg 1 is soft. Adler’s zero for the leg n being soft can be understood via a similar manipulation by employing the equivalent representation (18) for the fundamental BCJ relation.

Adler’s zero for BI amplitudes can be understood via the parallel procedure by using the fundamental BCJ relation for YM amplitudes (15).

6 Summary

In this note, we used the soft theorem for external scalars to derive the fundamental BCJ relation among double-color-ordered tree BAS amplitudes. Then we generalized this relation to the case wherein two external legs are massive. Using the fundamental BCJ relation for such special tree BAS amplitudes with massive external legs, along with the forward limit method, we obtained the fundamental BCJ relation among BAS Feynman integrands at the 1-loop level. We also used the fundamental BCJ relation to understand Adler’s zero, which describes the soft behavior of tree NLSM and BI amplitudes.

In [57], one can see that the general BCJ relations can be generated from the fundamental one. It is interesting to ask whether such generalization can be realized via the soft theorems. In this note, all soft limits under consideration are those where only one external leg is soft, which is referred to as single soft behavior. In order to obtain the general BCJ relations, it seems that we need to consider multiple soft behaviors, which will be an interesting future direction.

In this work, we used the expanded formulas and the fundamental BCJ relation to understand the soft behavior of NLSM and BI amplitudes. In [46], it was shown that the expansions of single-trace tree Yang–Mills-scalar and Yang–Mills amplitudes can be uniquely determined by assuming only the existence of soft theorems and the universality of soft factors, without knowledge of the details of soft factors. It is natural to ask whether the expansions of tree NLSM and BI amplitudes can be determined by the general consideration of soft behavior. This question will be answered in our next work [66].