Tree level amplitudes from soft theorems

The BAS theory describes the bi-adjoint scalar field $\phi_{a\bar{a}}$ with the Lagrangian

where the structure constant $f^{abc}$ and generator $T^{a}$ satisfy

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 technic. Each double color ordered partial amplitude is simultaneously planar with respect to two color orderings, arise from expanding the full $n$-point amplitude to $\mathop{\rm Tr}(T^{a_{\sigma_{1}}}\cdots T^{a_{\sigma_{n}}})$ and $\mathop{\rm Tr}(T^{\bar{a}_{\bar{\sigma}_{1}}}\cdots T^{\bar{a}_{\bar{\sigma}_{n}}})$ respectively, where $\sigma_{i}$ and $\bar{\sigma}_{i}$ denote permutations among all external scalars.

To calculate double color ordered partial amplitudes, it is convenient to employ the diagrammatical method proposed by Cachazo, He and Yuan in Cachazo:2013iea . For a BAS amplitude whose double color-orderings are given, this method provides the corresponding Feynman diagrams as well as the overall sign directly. Let us consider the $5$-point example ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$. In Figure.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 other diagrams are also forbidden by the ordering $(1,4,2,3,5)$, thus the first diagram in Figure.1 is the only diagram contributes to the amplitude ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$.

The Feynman diagrams contribute to a given BAS amplitude can be obtained via systematic diagrammatical 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 Figure.2. From the diagram, 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 Figure.1. Then the BAS amplitude ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$ can be computed as

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

where $k_{a}$ is the momentum carried by the external leg $a$.

As another example, let us consider the BAS amplitude ${\cal A}_{S}(1,2,3,4,5|1,2,4,3,5)$. The corresponding disk diagram is shown in the second configuration in Figure.2, and one can see two orderings have 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 ${\cal A}_{S}(1,2,3,4,5|1,2,4,3,5)$ as

up to an overall sign.

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

• •

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

• •

  Each polygon with 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 Figure.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 Figure.2, the polygons are $512A$ and $A43$, which contribute two $-$, while the intersection point $A$ contributes $-$. Then we arrive at the full results

In the reminder 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 Figure.3 indicates the overall sign $+$ under the new convention, while the old convention gives $-$ according to the square formed by four line segments. Notice that the diagrammatical rules described previously still give the related sign between different disk diagrams. For example, two disk diagrams in Figure.2 shows that the relative sign between ${\cal A}_{S}(1,2,3,4,5|1,4,2,3,5)$ and ${\cal A}_{S}(1,2,3,4,5|1,2,4,3,5)$ is $+$.

One application of the new convention, which is crucial in subsequent sections, is as follows. For the double color ordered BAS amplitude ${\cal A}_{S}(\cdots,a,p,b,\cdots|\cdots,a,p,b,\cdots)$, suppose we remove the external scalar $p$, the overall sign for the resulted amplitude ${\cal A}_{S}(\cdots,a,b,\cdots|\cdots,a,b,\cdots)$ is $+$. Notice that in the notations above we do not require the full color orderings at the l.h.s and r.h.s of $|$ to be the same. On the other hand, if we remove $p$ from ${\cal A}_{S}(\cdots,a,p,b,\cdots,c,d,\cdots|\cdots,a,b,\cdots,c,p,d,\cdots)$, whose color orderings are the same as those for ${\cal A}_{S}(\cdots,a,p,b,\cdots|\cdots,a,p,b,\cdots)$ except the positions of $p$, the obtained amplitude ${\cal A}_{S}(\cdots,a,b,\cdots|\cdots,a,b,\cdots)$ carries an overall $-$. We now give the interpretation to the above observation. One can change the color orderings for ${\cal A}_{S}(\cdots,a,p,b,\cdots|\cdots,a,p,b,\cdots)$ and ${\cal A}_{S}(\cdots,a,b,\cdots|\cdots,a,b,\cdots)$ simultaneously, to arrive at two convex polygons in disk diagrams in Figure.4. This procedure will not alter the relative sign. Each diagram in Figure.4 corresponds to an overall $+$, due to our new convention. Thus, generating ${\cal A}_{S}(\cdots,a,b,\cdots|\cdots,a,b,\cdots)$ from ${\cal A}_{S}(\cdots,a,p,b,\cdots|\cdots,a,p,b,\cdots)$ will not create the overall $-$. Then, we change the color orderings for ${\cal A}_{S}(\cdots,a,p,b,\cdots|\cdots,a,p,b,\cdots)$ and ${\cal A}_{S}(\cdots,a,p,b,\cdots,c,d,\cdots|\cdots,a,b,\cdots,c,p,d,\cdots)$ simultaneously to get two configurations in Figure.5, where the line segments in first one again form a convex polygon. The relative sign between these two configurations in Figure.5 can be figured out via the diagrammatical rules. Suppose the number of points on the boundary is odd, the old convention indicates that the polygon carries a $+$. In the second configuration, two intersection points contribute two $-$, the triangle carries the orientation opposite to the disk thus contributes $-$, and two convex polygons whose numbers of vertices are both odd or both even contribute $+$. Consequently, there is a relative $-$ between two configurations. If the number of points on the boundary is even, the first configuration carries $-$. The second configuration contains two convex polygons which carry the same orientations as the disk, and the numbers of their vertices are even for one polygon and odd for another one, thus the total number of $-$ for the second configuration is $4$. Again, there is a relative $-$ between two configurations. For the first configuration, the new convention for overall sign implies that removing $p$ will not create $-$. Thus, for the second one, removing $p$ provides a $-$ to the resulted amplitude.

Then, we discuss which propagators can contribute to a double color ordered BAS amplitude with fixed color orderings. In the diagrammatical rules, this question is answered by the co-boundaries of the disk diagram. One can also deduce the following new rule which will be used latter. For the set $\alpha$ of successive points on the boundary of disk, let us call line segments connecting two points in $\alpha$ as internal lines, and line segments connecting one point in $\alpha$ and another one in $\bar{\alpha}$ as external lines, where $\bar{\alpha}$ is the complementary set of $\alpha$. The new rule is, the set $\alpha$ contributes the propagator $1/s_{\alpha}$ to the BAS partial amplitude if and only if it has two external lines. This rule is manifestly equivalent to requiring the co-boundary.

When considering the soft limit, the $2$-point channels play the 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 if 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 ordering¹¹1The Kronecker symbol will not appear in this paper except in the third footnote, thus we hope the notation $\delta_{ab}$ will not confuse the readers.. 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 ani-symmetry of the structure constant, i.e., $f^{abc}=-f^{bac}$, and $\delta_{ab}=0$ otherwise. The value of $\delta_{ab}$ is consistent with our new convention of the overall sign, as can be seen in Figure.6. The first configuration in Figure.6 gives $\delta_{ab}=1$, due to the convention that the overall sign corresponds to a convex polygon is $+$. The second configuration carries a relative sign $-$ when comparing with the first one, as can be figured out via the diagrammatical rules, thus implies $\delta_{ab}=-1$. From the definition, it is straightforward to see $\delta_{ab}=-\delta_{ba}$. The symbol $\delta_{ab}$ will be used frequently latter.

Before ending this subsection, we determine the leading order soft factor for the BAS scalar. Consider the double color ordered BAS amplitude ${\cal A}_{S}(1,\cdots,n|\sigma)$, which carries two color orderings $(1,\cdots,n)$ and $\sigma$. We re-scale $k_{i}$ as $k_{i}\to\tau k_{i}$, and expand the amplitude in $\tau$. The leading order contribution manifestly aries from $2$-point channels $1/s_{1(i+1)}$ and $1/s_{(i-1)i}$ which provide the $1/\tau$ order contributions, namely,

where $\not{i}$ stands for removing the leg $i$, $\sigma\setminus 1$ means the color ordering generated from $\sigma$ by eliminating $i$. The leading soft operator $S^{(0)}_{s}(i)$ for the scalar $i$ is extracted as

which acts on external scalars which are adjacent to $i$ in two color orderings. This operator serves as one of starting points in subsection.3.1.

It is worth to take (9) as the instance, to give further explanations for the universality of soft operator. The operator (9) acts on external BAS scalars which are adjacent to $i$ in either of two color orderings, we assume such manner is universal. For amplitudes contain other types of particles, such as scalars coupled to gluons, the leading soft factor $S^{(0)}_{s}(i)$ for scalar still acts on external scalars in the manner described by (9), i.e., it always acts on external BAS scalars which are adjacent to $i$ in both of two color orderings. On the other hand, it is possible $S^{(0)}_{s}(i)$ also acts on other types of external particles such as gluons. This possibility can not be studied in the current case, since the pure BAS amplitudes only include external BAS scalars. One may wonder the action of soft operator should depend on theories defined by Lagrangians, rather than types of external particles. For amplitudes under consideration in the current paper, this puzzle can be solved via the following argument. Let us take the single trace YMS amplitudes as the example. One can regard the pure BAS and pure YM amplitudes as special cases of general single trace YMS amplitudes²²2For the multiple trace YMS amplitudes, such point of view is not valid, since the BAS amplitude is single trace., thus the soft behavior of pure BAS amplitude when one external scalar being soft can be obtained by acting the soft operator for the single trace YMS amplitudes to the pure BAS one. The soft operator can be split into two parts, one acts on external scalars, another one acts on external gluons. The second part annihilates the pure BAS amplitude which does not include any external gluon, thus the effective part is only the first one. The first part must be equal to (9), otherwise the action of the first part can not reproduce the result (8). Therefore, for the single trace YMS case, the leading soft operator for the scalar always acts on external scalars in the manner showed in (9), without regarding whether the amplitude contains external gluons. Similar discussion shows that the soft factors for the gluon for the single trace YMS amplitudes can be applied to pure YM ones, by removing the part which acts on external scalars. One can also regard the YM and GR amplitudes as special cases of single trace EYM amplitudes and obtain the similar conclusion, the discussion is analogous.

Tree level amplitudes which contain only massless particles can be expanded to double color ordered BAS amplitudes, due to the observation that each Feynman diagram for pure propagators can be mapped to at least one disk diagram whose polygons are all triangles. An illustrative example is given in Figure.7. Since each tree amplitude can be expanded by tree Feynman diagrams, and each Feynman diagram contributes propagators together with a numerator without any pole, one can conclude that each tree amplitude can be expanded to double color ordered BAS amplitudes, with coefficients which are polynomials depend on Lorentz invariants created by external kinematical variables. To realize the expansion, one need to find the basis consists of BAS amplitudes. Such basis can be determined by the well known Kleiss-Kuijf (KK) relation Kleiss:1988ne

which is based on the color algebra. Here $\alpha$ and $\beta$ are two ordered subsets of external scalars, and $\beta^{T}$ stands for the ordered set generated from $\beta$ by reversing the original ordering. The BAS amplitude ${\cal A}_{S}(1,\alpha,n,\beta|\sigma)$ at the l.h.s of (10) carries two color orderings, one is $(1,\alpha,n,\beta)$, another one is denoted by $\sigma$. The symbol $\shuffle$ means summing over all possible shuffles of two ordered subsets ${\beta}_{1}$ and ${\beta}_{2}$, i.e., all permutations in the set ${\beta}_{1}\cup{\beta}_{2}$ while preserving the orderings of ${\beta}_{1}$ and ${\beta}_{2}$. For instance, suppose ${\beta}_{1}=\{1,2\}$ and ${\beta}_{2}=\{3,4\}$, then

The analogous KK relation holds for another color ordering $\sigma$. The KK relation implies that different double color ordered BAS amplitudes are not independent, thus the basis can be chosen as BAS amplitudes ${\cal A}_{S}(1,\sigma_{1},n|1,\sigma_{2},n)$, with $1$ and $n$ are fixed at two ends in each color ordering. We call such basis the KK BAS basis. Based on the discussion above, the KK BAS basis can provide any structure of propagators, thus any amplitude can be expanded to this basis, with coefficients which contain no pole³³3The well known Bern-Carrasco-Johansson (BCJ) relation Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia ; Johansson:2019dnu implies the relations among BAS amplitudes in the KK basis, and the independent BAS amplitudes can be obtained by fixing three legs at three particular positions in the color orderings. However, in the BCJ relation, coefficients of BAS amplitudes depend on Mandelstam variables, this character leads to poles in coefficients when expanding to BCJ basis. We hope all poles are contributed by propagators in amplitudes, and the coefficients contain no pole, thus chose the KK basis..

In this paper, we will consider the expansions of single trace YMS amplitudes, YM amplitudes, single trace EYM amplitudes, and GR amplitudes. We now discuss them one by one. In the single trace YMS amplitude ${\cal A}_{YS}(1,\sigma_{1},n;p_{1},\cdots,p_{m}|1,\sigma_{2},n)$, the external scalars encoded by $i\in\{1,\cdots,n\}$ are included in two color orderings, one is $(1,\sigma_{1},n)$ among only scalars, another one is $(1,\sigma_{2},n)$ among all external legs. The external gluons labeled by $p_{j}$ with $j\in\{1,\cdots,m\}$ belong to the color ordering $(1,\sigma_{2},n)$, while $p_{1},\cdots,p_{m}$ at the l.h.s of $|$ and r.h.s of ; are un-ordered. Here we fixed $1$ and $n$ at two ends in each color ordering, due to the KK relation. The amplitude ${\cal A}_{YS}(1,\sigma_{1},n;p_{1},\cdots,p_{m}|1,\sigma_{2},n)$ can be expanded to KK BAS basis as

where $\sigma_{3}$ are permutations among external legs in $\{2,\cdots,n-1\}\cup\{p_{1},\cdots,p_{m}\}$. The double copy structure Kawai:1985xq ; Bern:2008qj ; Chiodaroli:2014xia ; Johansson:2015oia ; Johansson:2019dnu indicates that the coefficient ${\cal C}(\sigma_{1},\sigma_{3},\epsilon,k)$ depends on polarization vectors $\epsilon_{i}$ of external gluons, momenta $k_{i}$ of all external particles, permutations $\sigma_{3}$ and $\sigma_{1}$, but is independent of the permutation $\sigma_{2}$⁴⁴4Originally, the double copy means the GR amplitude can be factorized as ${\cal A}_{G}={\cal A}_{Y}\times{\cal S}\times{\cal A}_{Y}$, where the kernel ${\cal S}$ is obtained by inverting propagators. Our assumption that the coefficients depend on only one color ordering is equivalent to the original version, as can be observed from (13) and (15).. Thus, suppose we replace $(1,\sigma_{2},n)$ by the more general ordering $\sigma$ among all external legs, without fixing $1$ and $n$ at any position, the expansion in (12) still holds.

The $n$-point YM amplitude ${\cal A}_{Y}(\sigma)$, where $\sigma$ is the color ordering among $n$ external gluons, can be expanded to BAS KK basis as

Again, $\sigma$ is the color ordering among all external legs, without fixing any one at particular position. The coefficient $\hat{\cal C}(\sigma_{1},\epsilon_{i},k_{i})$ depends on polarization vectors and momenta of external gluons, as well as the permutation $\sigma_{1}$, but is independent of the color ordering $\sigma$, as implied by the double copy structure.

It is natural to image another type of coefficients $\hat{\cal C}(\sigma,\epsilon_{i},k_{i})$, which depend on the color ordering $\sigma$ rather than $\sigma_{1}$, thus one can generalize the expansions (12) and (13) to

and

The external particle $i$ which carries both polarization vectors $\epsilon_{i}$ and $\widetilde{\epsilon}_{i}$ is interpreted as the graviton, whose polarization tensor $\varepsilon_{i}$ can be decomposed as $\varepsilon_{i}^{\mu\nu}=\epsilon_{i}^{\mu}\widetilde{\epsilon}_{i}^{\nu}$. Thus, (14) is the expansion of the EYM amplitude ${\cal A}_{EY}(1,\sigma_{1},n;p_{1},\cdots,p_{m})$, which contains $n$ external gluons encoded by $i\in\{1,\cdots,n\}$, and $m$ external gravitons encoded by $p_{j}$ with $j\in\{1,\cdots,m\}$. The expansion (15) is the expansion of the $n$-point GR amplitude ${\cal A}_{G}(1,\cdots,n)$. Notice that in this paper we focus on gravitons of Einstein gravity. In such case, polarization vectors $\widetilde{\epsilon}_{i}$ are the same as $\epsilon_{i}$. However, we still use notations $\epsilon_{i}$ and $\widetilde{\epsilon}_{i}$ to manifest the double copy structure.

One can sum over $\sigma_{2}$ in (14) and (15) via the expanded formulas of YM amplitudes in (13), resulted in

and

These are the expansions of EYM and GR amplitudes to YM KK basis, which bears strong similarity with expansions (12) and (13), respectively.

The explicit formulas of ${\cal C}(\sigma_{1},\sigma_{3},\epsilon_{i},k_{i})$ and $\hat{\cal C}(\sigma_{1},\epsilon_{i},k_{i})$ were computed via different methods in Fu:2017uzt ; Teng:2017tbo ; Du:2017kpo ; Du:2017gnh ; Feng:2019tvb . In this paper, we shall reconstruct them by using the constraints from the soft theorems and the universality of soft factors.

We start by considering the single trace YMS amplitude ${\cal A}_{YS}(1,\cdots,n;p|\sigma)$, which contains $n$ external scalars labeled by $i\in\{1,\cdots,n\}$, and a gluon labeled by $p$. Here $\sigma$ denotes an arbitrary color ordering among all external particles including the gluon, while another color ordering $(1,\cdots,n)$ includes only scalars. Based on the discussions in subsection.2.2, we know that the amplitude ${\cal A}_{YS}(1,\cdots,n;p|\sigma)$ can be expanded to BAS amplitudes as

We express the coefficients as $\epsilon_{p}\cdot P_{i}$, since the amplitude ${\cal A}_{YS}(1,\cdots,n;p|\sigma)$ is a Lorentz invariant which is linear in the polarization vector $\epsilon_{p}$. The double copy structure indicates that $P_{i}$ are independent of the color ordering $\sigma$. In $d$ dimensional space-time, the coupling constants of BAS and YM theories have mass dimensions $3-d/2$ and $2-d/2$ respectively, thus the mass dimension of $P_{i}$ must be $1$. It means $P_{i}$ are combinations of external momenta. Our aim is to determine the combinatory momenta $P_{i}$ via the soft theorem.

Let us re-scale the external momentum of the leg $1$ as $k_{1}\to\tau k_{1},$ and expand ${\cal A}_{YS}(1,\cdots,n;p|\sigma)$ in $\tau$. The leading order term is given by

where $P^{(0)}_{i}$ are leading order contributions of $P_{i}$. The leading order terms of ${\cal A}_{S}(1,\cdots,i,p,i+1,\cdots,n|\sigma)$ are determined by the leading soft factor for the BAS scalar in (9), namely,

Here $\not{1}$ means removing the particle $1$, and $\sigma\setminus 1$ stands for the color ordering obtained from $\sigma$ by removing $1$. Substituting (20) into (19), we get

Since the color ordering $\sigma$ is general, non of $\delta_{n1}$, $\delta_{12}$ and $\delta_{1p}$ can be fixed to be $0$.

The soft theorem requires the factorization

where $S^{(0)}_{s}(1)$ is the universal leading soft factor for the scalar $1$. In subsection.2.1, we have derived this factor, which is given in (9). However, as discussed previously, since the operator (9) is derived from the pure BAS amplitude, it can not tell us whther the operator $S^{(0)}_{s}(1)$ acts on external gluons. The universality of soft factor requires that $S^{(0)}_{s}(1)$ always acts on external BAS scalars in the manner described by (9), therefore the form of $S^{(0)}_{s}(1)$ in (22) should be

where ${\cal S}_{s}(1;p)$ stands for the operator acts on external gluon $p$.

To determine ${\cal S}_{s}(1;p)$, we use the KK relation to expand ${\cal A}_{S}(p,2,\cdots,n|\sigma\setminus 1)$ in the first line of (21) as

this manipulation turns (21) to

From (25), we see that the condition (23) can be satisfied if and only if $P^{(0)}_{1}=0$, therefore ${\cal S}_{s}(1;p)=0$. Hence, we get the soft factor

which is the generalization of (9) to the current case. Since ${\cal S}_{s}(1;p)=0$, the contribution from external gluon is excluded. Consequently, $P_{1}$ is the combination of external momenta which vanishes at the leading order, thus is proportional to $k_{1}$. We can take $P_{1}=k_{1}$ via an overall re-scaling of the amplitude.

The phenomenon that the soft factor for the BAS scalar does not act on the gluon can be understood from the Feynman diagram point of view. If one removes the soft scalar from the gluon-scalar-scalar interaction vertex to get the lower-point amplitude, the remaining diagram means the gluon can be turned to a scalar without any interaction, as can be seen in Figure.8. This picture is physically unacceptable. Thus, such diagram can never contribute to the soft factor.

The vanishing of $P^{(0)}_{1}$ indicates that ${\cal A}^{(0)}_{YS}(1,\cdots,n;p|\sigma)$ only comes from the second line at the r.h.s of (21), thus the soft theorem (22) together with the soft factor in (26) impose

which implies the expansion

Comparing the expansion in (28) with that in (18), we see that for general $n$ they are totally the same, up to a relabeling. This observation is based on the condition that the coefficients are independent of the color ordering $\sigma$, which arises from the double copy structure. Thus, the solution $P_{1}=k_{1}$ indicates $P^{(0)}_{2}=k_{2}$ in (28), therefore

To fix the parameter $\alpha$, we consider the soft behavior of the the external scalar $2$. After taking $k_{2}\to\tau k_{2}$ and expanding (18) in $\tau$, the leading order term is given as

The soft theorem (22) together with the universality of leading order soft factor in (9) indicate that

Expanding ${\cal A}_{YS}(1,3,\cdots,n;p|\sigma\setminus 2)$ in (31) via the expansion (18) (with a relabeling), and using the solution $P_{1}=k_{1}$, one can observe that the combination of first two lines at the r.h.s of (30) gives

which means

For $k_{2}\to\tau k_{2}$, we have $P_{1}^{(0)}=k_{1}$ and $P_{2}^{(0)}=\alpha k_{1}$. Then, we find the solution to (33) is $\alpha=1$. Here we have used the property $\delta_{ab}=-\delta_{ba}$.

Until now, we have found $P_{1}=k_{1}$ and $P_{2}=k_{1}+k_{2}$. Taking the soft limit of other external scalars successively, and applying the same method, we get

Consequently, the YMS amplitude with one external gluon can be expanded as

The combinatory momentum $X_{p}$ is defined as the summation of momenta of legs at the l.h.s of $p$ in the color ordering. The shuffle $\shuffle$ is explained in (11).

Form the expansion in (35), one can determine the leading and sub-leading soft factors for the gluon. Let us take $k_{p}\to\tau k_{p}$, and expand (35) in $\tau$. The leading order term is given as