Expanding single trace YMS amplitudes with gauge invariant coefficients

In this note, we use the new bottom up method based on soft theorems to construct the expansion of single-trace Yang-Mills-scalar amplitudes recursively. The resulted expansion manifests the gauge invariance for any polarization carried by external gluons, as well as the permutation symmetry among external gluons. Our result is equivalent to that found by Clifford Cheung and James Mangan via the so called covariant color-kinematic duality approach.


I. INTRODUCTION
The investigations of S-matrix in the past decade revealed deep connections among amplitudes of various different theories, which are invisible upon inspecting traditional Feynman rules.For instance, tree level gravitational (GR) and Yang-Mills (YM) amplitudes are related by the so called Kawai-Lewellen-Tye (KLT) relation [1], and Bern-Carrasco-Johansson (BCJ) color-kinematic duality [2][3][4][5].In the well known CHY formalism [6][7][8][9][10], tree amplitudes for a large variety of theories can be generated from tree GR amplitudes through the compactifying, squeezing and the generalized dimensional reduction procedures.Similar unifying relations were proposed by introducing appropriate differential operators, which transmute tree GR amplitudes to others [11][12][13].At the same time, another type of relations also caused attentions, which can be called the expansions of amplitudes, namely, tree amplitudes of one theory can be expanded to tree amplitudes of other theories [14][15][16][17][18][19][20].Indeed, the expansions serve as the dual picture of differential operators, as interpreted in [20].In the dual web, tree amplitudes for a wide range of theories can be expanded to tree bi-adjoint scalar (BAS) amplitudes.
In this note, we focus on the expansion of the tree single-trace Yang-Mills-scalar (YMS) amplitudes [3,[21][22][23][24][25][26], due to the special role of these amplitudes.Both tree YM and BAS amplitudes can be regarded as special cases of tree single-trace YMS amplitudes.When performing the squeezing procedure in CHY framework, or acting differential operators, the YM amplitudes are transmuted to single-trace YMS amplitudes, then end with pure BAS ones.These connections can be extended to tree GR amplitudes, tree YM amplitudes, and single-trace tree EYM amplitudes directly [8-10, 12, 13].Furthermore, single-trace tree YMS amplitudes exhibit double copy structure and color-kinematic duality in an elegant manner [3, 8-10, 23, 27].The single-trace YMS amplitudes which describe the scattering of massless gluons and scalars, can be expanded to YMS ones with less external gluons and more external scalars [14][15][16][17][18]28].Such recursive expansion can be applied iteratively, end with the expansion to pure BAS amplitudes.
In the recursive expansion given in [14,15,18], a fiducial external gluon is required.This special gluon breaks the manifest permutation symmetry among external gluons: if one rechose the fiducial gluon, we expect that the new expansion is equivalent to the old one, but such equivalence is hard to prove.Another disadvantage of the expansion in [14,15,18] is related to the gauge invariance.In the expansion in [14,15,18], the gauge invariance for the polarization vector carried by the fiducial gluon is obscure, namely, if one replace the polarization by the corresponding momentum, the vanishing of amplitude is not manifest.suppose one use this expansion iteratively to expand the YMS amplitude to BAS ones, in the resulting expansion, non of polarizations has the manifest gauge invariance, since each external gluon will play the role of the fiducial gluon once in the iterative process.As well known, the gauge invariance plays the crucial role in the modern S-matrix program, and always hints new understanding and new mathematical structure for scattering amplitudes.For instance, tree amplitudes in general relativity and Yang-Mills theory, turned out to be completely determined by gauge invariance and singularity structures [29,30].Another well known example is that the Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion relation expresses YM amplitudes in gauge invariant formulas which include spurious poles [31,32].These new formulas motivated elegant new constructions of amplitudes, such as Grassmannia representation and Amplituhedron [33][34][35].These experiences force us to seek the new expansion with the explicit gauge invariance.After expanding to BAS amplitudes, the BAS basis contributes only poles, thus the gauge invariance for each polarization is completely determined by coefficients.Thus, it is natural to expect a formula of coefficients which manifests the gauge invariance.The above consideration leads to an important question, how to achieve the new expansion which have manifest permutation symmetry among external gluons and the gauge invariance for each polarization of external gluons?
Such new expansion was first found by Clifford Cheung and James Mangan in [27], via the so called covariant color-kinematic duality method.The approach in [27] is based on the traditional Lagrangian and equations of motion.The very premise of the modern S-matrix program is to bootstrap scattering dynamics without the aid of an action or equation of motion (see for reviews in [36,37]).Thus, it is natural to ask whether the new expansion in [27] can be obtained through the bottom up construction which uses only on-shell information?This question is the main motivation for the current short note.
In this note, we reconstruct the expansion in [27], from a totally different perspective based on universal soft behaviors of massless particles.We first bootstrap the lowest 3point tree single-trace YMS amplitudes with only one external gluon, by imposing general principles such as the appropriate mass dimension and Lorentz invariance.Then, we invert the soft theorem for external scalars to construct the expanded single-trace YMS amplitudes with more external scalars, while keeping the number of external gluons to be un-altered.After such construction, we use the well known Bern-Carrasco-Johansson (BCJ) relation [2][3][4][5] to turn the resulted expansion to the new formula which manifests the gauge invariance.Next, we invert the sub-leading soft theorem for external gluons to generate the expanded single-trace YMS amplitudes with more external gluons.Such procedure inserts external gluons to the original amplitude in a manifestly gauge invariant pattern, thus the explicit gauge invariance will be kept if the starting point is expressed in a gauge invariant form.The manifest permutation symmetry among external gluons are also kept at each step.
The remainder of this note is organized as follows.In section.II, we rapidly introduce the necessary background including the expansion of tree massless amplitudes to BAS amplitudes, the recursive expansion of single-trace YMS amplitudes, as well as soft theorems for external scalars and gluons.In section.III, we construct the expansion in [27] by using our recursive method based on inverting soft theorems.Then, we end with a brief summery in section.IV.

II. BACKGROUND
For readers' convenience, in this section we give a brief review of necessary background.In subsection.II A, we introduce the tree level amplitudes of bi-adjoint scalar (BAS) theory, as well as expansions of tree amplitudes to BAS amplitudes.In subsection.II B, we give the recursive expansion of Yang-Mills-scalar (YMS) amplitudes.In subsection.II C, we review the soft theorems for external scalars and gluons.

A. Expanding tree level amplitudes to BAS basis
The bi-adjoint scalar (BAS) theory describes the massless bi-adjoint scalar fields ϕ Aa with cubic interaction, the Lagrangian is given as where the structure constant F ABC and generator and the dual algebra encoded by f abc and T a is analogous.Tree amplitudes of this theory only contain propagators for massless scalars.Decomposing the group factors via the standard procedure gives where A n denotes the kinematic part of n-point amplitude in which the coupling constants are dropped, S n and S n are non-cyclic permutations of external legs.In this note, partial amplitudes up to an overall sign.The Mandelstam variable s i•••j is defined as where k a is the momentum carried by the external leg a.
Each double color ordered partial BAS amplitude carries an overall ± sign, arises from swapping two lines at the common vertex, due to the anti-symmetry of structure constants F ABC and f abc .In this note, we choose the convention that the overall sign is + if two orderings carried by the BAS amplitude are the same.For instance, the amplitude A S (1, 2, 3, 4|1, 2, 3, 4) carries the overall sign + under the above convention.Notice that this convention is different from that in [8].The overall signs for other BAS amplitudes with general orderings can be determined by counting the number of flipping [8].
The double color ordered partial amplitudes can be systematically evaluated by applying the diagrammatical rules proposed by Cachazo, He and Yuan in [8].We do not introduce this method in the current note, the reader can see this interesting and useful approach in [8].
Tree level amplitudes for massless particles and cubic interactions can be expanded to double color ordered partial BAS amplitudes, since each associated Feynman diagram can be included in at least one partial BAS amplitude.For higher-point vertices, one can transmute them to cubic ones by inserting the propagator 1/D and the numerator D simultaneously, as shown in Figure .2.This manipulation expands any tree amplitude to tree Feynman diagrams with only cubic vertices.Since each Feynman diagram contributes propagators which can be provided by partial BAS amplitudes, accompanied with a numerator, one can conclude that each tree amplitude for massless particles can be expanded to double color ordered partial BAS amplitudes.In such expansions, coefficients are polynomials depend on Lorentz invariants created by external kinematical variables, without any pole.
The expansion requires appropriate basis.Such basis can be obtained by employing the well known Kleiss-Kuijf (KK) relation [38] A The analogous KK relation holds for the ordering σ n .The KK relation among partial BAS amplitudes implies that the basis can be chosen as amplitudes A S (1, σ 1 , n|1, σ 2 , n), with 1 and n are fixed at two ends in each ordering.Such basis is called the KK BAS basis.Consequently, any tree amplitude for massless particles can be expanded to this basis.The basis provides poles, while the coefficients contribute numerators.The double color ordered partial BAS amplitudes also satisfy well known Bern-Carrasco-Johansson (BCJ) relations [2][3][4][5].Here we give the explicit formula of the fundamental BCJ relation, since it will be used subsequently.The combinatory momentum Y s is defined as the summation of external momenta carried by legs at the l.h.s of s in the ordering (1, s ¡ {2, • • • , n − 1}, n), the ordering σ n+1 is defined for n + 1 legs in {1, • • • , n} ∪ s.BCJ relations imply the independence of BAS amplitudes in KK basis, thus the so called BCJ basis can be chosen as BAS amplitudes with three fixed legs in color orderings.However, in BCJ relations, coefficients of BAS amplitudes depend on Mandelstam variables, this character leads to poles in coefficients when expanding to BCJ basis.On the other hand, when expanding to KK basis, coefficients contain no pole.In this note, we choose the KK basis since we expect that all poles are included in basis, and coefficients only contribute numerators.

B. Recursive expansion of single-trace YMS amplitudes
The YMS theory under consideration is the massless YM ⊕ BAS theory with Lagrangian [23] The indices a, b, c and d run over the adjoint representation of the gauge group.Scalar fields carry additional flavor indices A, B, C. The field strength and covariant derivative are defined in the usual way The general tree amplitude of this theory includes both massless external scalars and massless external gluons.Through the standard technic, one can decompose the gauge group factors to obtain where s n+m denotes non-cyclic permutations among n external scalars and m external gluons, T aσ i encodes the generator of the gauge group.Here A n+m is the kinematic part of the amplitude without coupling constants.Meanwhile, decomposition of the flavor group factors leads to the structure of tree amplitudes which is similar to that of loop amplitudes in that, unlike pure gauge theories, it is not restricted to have only single-trace term [3,23]: where S n i again stands for the set of non-cyclic permutations.The single-trace and doubletrace terms are collected in the first and second lines respectively, and • • • in the third line denotes the remaining multi-trace terms.External scalars for the partial amplitudes . Analogously, partial amplitudes in multi-trace terms in the third line include more orderings among external scalars.
In this note, we focus on single-trace partial tree amplitudes A n+m,1 (σ 1 , • • • , σ n ) in the first line of ( 12), due to the following reasons.First, there are special connections between tree YM amplitudes, tree BAS amplitudes, and single-trace tree YMS amplitudes [8,10].These relations can be extended to tree GR amplitudes, tree YM amplitudes, and single-trace tree EYM amplitudes, via the well known double copy structure [8-10, 12, 13].Secondly, single-trace tree YMS amplitudes play the crucial role when studying color-kinematic duality [3, 8-10, 23, 27].By definition, single-trace YMS amplitudes under consideration are partial amplitudes A YS (σ n ; {p i } m |σ n+m ), obtained by decomposing flavor and gauge group factors simultaneously.Here the color ordering in (11) among all external legs is denoted as σ n+m , while the flavor ordering among external scalars, for single-trace terms in the first line in (12), is labeled as σ n .The un-ordered set of m external gluons is denoted by In other words, gluons belong to only one ordering σ n+m .Notice that the single-trace sector of YMS theory is equivalent to dropping the ϕ 4 terms in the Lagrangian (9), see in [3,23,27].
The discussion for expansions of tree level amplitudes in the previous subsection.II A indicates that the single-trace tree YMS amplitude where The double copy structure [1][2][3][4][5]8] indicates that the coefficient C( σ n+m−2 , ϵ i , k j ) depends on polarization vectors ϵ i carried by external gluons, momenta k i carried by either gluons or scalars, and orderings σ n+m−2 , but is independent of the ordering σ n+m−21 .The independence of the ordering σ n+m−2 leads to the more general ansatz where σ n+m stands for the general ordering among all external legs, without fixing any one at any particular position.The expansion in ( 14) can be achieved by applying the following recursive expansion iteratively [14,15,18], where p is the fiducial gluon which can be chosen as any element in {p i } m , and α α α are subsets of {p i } m \ p which is allowed to be empty.When α α α = {p i } m \ p, the YMS amplitudes in the second line of ( 15) are reduced to pure BAS ones.The ordered set ⃗ α α α is generated from α α α by endowing an order among elements in α α α.The tensor F µν ⃗ α α α is defined as for where each anti-symmetric strength tensor f i is given as The combinatory momentum Y ⃗ α α α is the summation of momenta carried by external scalars at the l.h.s of α 1 in the color ordering where α 1 is the first element in the ordered set ⃗ α α α.The symbol ¡ means summing over permutations of p} which preserve the orderings of two ordered sets {2, • • • , n − 1} and {⃗ α α α, p}, as explained around (7).The summation in ( 15) is over all un-equivalent ordered sets ⃗ α α α.In the recursive expansion (15), the YMS amplitude is expanded to YMS amplitudes with less gluons and more scalars.Repeating such expansion, one can finally expand any YMS amplitude to pure BAS ones.
In the recursive expansion (15), the gauge invariance for each gluon in {p i } m \ p is manifest, since the tensor f µν vanishes automatically under the replacement ϵ i → k i , due to the definition.However, the gauge invariance for the fiducial gluon p has not been manifested.When applying (15) iteratively, a fiducial gluon will be required at every step.Consequently, in the resulted expansion to pure BAS amplitudes, the gauge invariance for each polarization will be spoiled.Furthermore, the manifest permutation invariance among external gluons are also broken.Notice that the breaking of manifest permutation invariance among external scalars can not be avoided, since the KK basis requires fixing two legs at two special positions in orderings.However, a special external gluon is not necessary.To obtain the expansion which manifests the gauge invariance and permutation invariance for external gluons simultaneously, one should employ another recursive expansion where k r is a reference massless momentum.Here the notations are parallel to those for (15).The formula (17) was found by Clifford Cheung and James Mangan, in the so called covariant color-kinematic duality framework [27].The expansion (17) does not require any fiducial gluon, the gauge invariance for any polarization and the permutation symmetry among external gluons are manifested.Using (15) iteratively, one finally obtains the new expansion of YMS amplitudes to KK BAS basis, with explicit gauge and permutation invariance for gluons.Reconstruct the expansion (17) through a bottom up method is the main purpose of this note.

C. Soft theorems for external scalars and gluons
In this subsection, we rapidly review the soft theorems for external scalars and gluons, which are crucial for subsequent constructions in next section.
For the double color ordered BAS amplitude A S (1, • • • , n|σ n ), we re-scale k i as k i → τ k i , and expand the amplitude by τ .The leading order contribution aries from propagators 1/s 1(i+1) and 1/s (i−1)i which are at the τ −1 order, where ̸ i stands for removing the leg i, σ n \ i means the color ordering generated from σ n by eliminating i.The superscript (0) i is introduced for denoting the leading order when considering the soft behaviour of k i .The symbol δ ab is defined as follows2 [28].In an ordering, if two legs a and b are adjacent, then δ ab = 1 if a precedes b, and If a and b are not adjacent, δ ab = 0. From the definition, it is straightforward to observe δ ab = −δ ba .In (18), δ i(i+1) and δ (i−1)i are defined for the ordering σ n , and the leading soft operator S (0) i s for the scalar i is given as By requiring the universality of soft behavior, the above result can be generated to the YMS amplitude as [28] A where the soft factor S (0) i s is the same as in (19).In other words, the soft operator S (0) i s does not act on any external gluon.
The soft theorems for external gluons at leading and sub-leading orders can be obtained via various approaches [39,40].Notice that one of these methods is to use the expanded formula of YMS amplitude in (15), as can be seen in [28].Such soft theorems are given as and where the external momentum k p j is re-scaled as k p j → τ k p j .The soft factors at leading and sub-leading orders are given by and respectively.In ( 23) and ( 24), one should sum over all external legs a, i.e., these soft operators for external gluon act on both external scalars and gluons.
The sub-leading soft operator (24) for external gluon plays the central role in the next section.Here we list some useful results for the action of this operator.The angular momentum operator J µν a acts on Lorentz vector k ρ a with the orbital part of the generator, and on ϵ ρ a with the spin part of the generator in the vector representation, Then the action of S (1)p g can be re-expressed as due to the observation that the amplitude is linear in each polarization vector.In (26), the summation over V a is among all Lorentz vectors including both momenta and polarizations.The operator (26) is a differential operator which satisfies Leibnitz's rule.Using (26), we immediately get where V is an arbitrary Lorentz vector, and for two arbitrary Lorentz vectors V 1 and V 2 , where the anti-symmetric tensor f i is defined as , as introduced previously.

III. EXPANSION OF SINGLE-TRACE YMS AMPLITUDE
In this section, we study the expansion of single-trace tree YMS amplitudes.Our purpose is to reconstruct the expansion in (17) through a purely bottom up approach.For simplicity, we will chose the reference momentum k r in (17) as k r = k n .We first bootstrap the 3-point YMS amplitudes with only one external gluon.Then, we invert the soft theorem for BAS scalars to construct YMS amplitudes with more external scalars, while keeping the number of external gluon to be fixed.After this construction, we use the BCJ relation to transmute the resulted expansion of such special BAS amplitudes with only one external gluon to a manifestly gauge invariant form.Next, we invert the sub-leading soft theorem for external gluons, to give a recursive pattern which leads to the general expansion of single-trace YMS amplitudes with arbitrary number of external gluons, which is manifestly gauge invariant for any polarization.Through the whole process, the manifest permutation symmetry among external gluons is kept at each step.

A. YMS amplitude with one external gluon
In this subsection, we construct the single-trace YMS amplitude A YS (1, • • • , n; p|σ n+1 ) which contains external scalars i ∈ {1, • • • , n} and only one external gluon p, by using the purely bottom up method.To start, we first bootstrap 3-point amplitudes A YS (1, 2; p|σ 3 ).In d-dimensional space-time, any n-point amplitude has the mass dimension d − d−2 2 n, thus the 3-point one has the mass dimension 3 − d 2 .On the other hand, the coupling constant g in Lagrangian (9) has mass dimension 2 − d 2 , thus the kinematic part A YS (1, 2; p|σ 3 ) has mass dimension 1.Meanwhile, the amplitude A YS (1, 2; p|σ 3 ) with one external gluon p should be linear in ϵ p , where ϵ p is the polarization vector carried by p. Finally, the 3-point amplitude does not include any pole since it can never be factorized into lower-point amplitudes.The above constraints uniquely fix A YS (1, 2; p|σ 3 ) to be ϵ p • k 1 , up to an overall sign 3 .Notice that ϵ p • k 2 is equivalent to ϵ p • k 1 , due to the momentum conservation and the on-shell condition ϵ p • k p = 0. Using the observation A S (1, p, 2|1, p, 2) = 1, we arrive at the following expansion coincides with the general ansatz in (14).Here the overall sign of A YS (1, 2; p|1, p, 2) is chosen to be 1, which forces the overall sign of A YS (1, 2; p|2, p, 1) to be −1.Notice that σ 3 has only two inequivalent choices which are (1, p, 2) and (2, p, 1), due to the cyclic symmetry of ordering.
Repeating the above manipulation, one can determine the general YMS amplitude A YS (1, • • • , n; p|σ n+1 ) with arbitrary number of external scalars and one external gluon p, in the expanded formula where the combinatory momentum Y p was defined in analogously.In the above expansion, it is direct to observe the gauge invariance for the polarization ϵ p , since the replacement ϵ µ p → k µ p yields the BCJ relation (8), However, if one use the formula (37) and the soft theorem for external gluons to construct YMS amplitudes with more external gluons, the resulting expansion is in the formula (15) [28], and the gauge invariance for the polarization of the fiducial gluon is obscure.The proof of gauge invariance for such fiducial polarization requires the application of BCJ relations in a complicated way, and the complexity increases rapidly as the number of external gluons increases.To avoid this disadvantage, we want to rewrite (37) so that the manifest gauge invariance is carried by coefficients.To realize the goal, one can use the BCJ relation to modify (37) as In the last line at the r.h.s of (39), the coefficients vanish under the replacement ϵ µ p → k µ p , due to the definition f µν p ≡ k µ p ϵ ν p − ϵ µ p k ν p .This is the desired new expansion for the YMS with one external gluon, trivialized the gauge invariance of polarization ϵ µ p , with the cost that a spurious pole k n • k p is introduced.As will be seen, by applying the recursive technic based on the soft theorem for external gluons, this new expansion leads to general expansion of YMS amplitudes with arbitrary number of external gluons, which manifests the gauge invariance for each polarization.

B. YMS amplitude with two external gluons
Now we turn to the single-trace YMS amplitude A YS (1, • • • , n; {p i } 2 |σ n+2 ) with two external gluons labeled as p 1 and p 2 .Let us consider the soft behavior of gluon p 2 , i.e., we re-scale k p 2 as k p 2 → τ k p 2 , and expand A YS (1, • • • , n; {p i } 2 |σ n+2 ) in τ .The soft theorem (22) indicates that the contribution at sub-leading order should be where and The second equality of ( 40) is obtained by substituting the expansion (39).Three parts B 1 , B 2 and B 3 arise from acting S and k n • k p 1 respectively, due to Leibnitz's rule.The expression for the first part B 1 in (41) is obtained via the soft theorem The second part B 2 can be calculated as follows.Using relations ( 27) and ( 28), we have where with Y p 1 = j i=1 k i .We use δ ab = −δ ba to reorganize H 1 as The reason for expressing H 1 in the above manner is that we want to interpret as coefficients times leading or sub-leading terms of amplitudes, as can be seen in ( 42).Combining (47) and the leading soft factor (19) for the scalar, we find We emphasize that ) which is at the τ −1 order, while f p 2 in the coefficients is accompanied with τ .Combining them together leads to the τ 0 order contribution which satisfies the order of A Based on the reason similar as that for rewriting H 1 , we reorganize H 2 as where δ q(j+1) and s q(j+1) in the last line should be understood as j + 1 = p 1 .Thus, Putting ( 48) and (50) together yields the expression of B 2 in (42).The third part B 3 can be calculated by using the relation (27), therefore we get the expression of B 3 in (43).We will use B 1 , B 2 and B 3 to reconstruct the complete expansion of Mathematically, it is impossible to restore the full amplitude from the sub-leading contribution, thus further physical constrains are needed.The most important one is the symmetry under the permutation p 1 ↔ p 2 .The physical amplitude ) should be invariant under such re-labeling.In the expansion (15), since a fiducial gluon p 1 is required, such symmetry is not manifest.However, one can use the BCJ relations to prove that the resulting expansions obtained by choosing different fiducial gluons are equivalent to each other.On the other hand, the purpose of the current work is to seek the new expansion which manifests the gauge invariance for all polarizations, which implies the democracy among all external gluons, thus it is natural to expect the manifested symmetry under With the additional condition of symmetry p 1 ↔ p 2 , let us try to find the expansion of the amplitude It is straightforward to recognize that B 2 arises as the leading term of which is at the τ 0 order.The reason for turning There is another way to keep such symmetry, which is adding terms with k n • k p 2 in the denominator.However, A YS (1, • • • , n; p 1 , p 2 |σ n+2 ) shall receive contributions from such terms, for example the coefficients carry 1/(τ k n • k p 2 ) under the re-scaling k p 2 → τ k p 2 , and the amplitudes contribute τ 1 , which combine to terms at the τ 0 order.Since such contributions do not exist in A (40), the 1/k n • k p 2 terms are excluded.Then the symmetry p 1 ↔ p 2 also excludes the 1/k n • k p 1 terms, and the remaining choice is only One can also observe that B 1 comes from the sub-leading term of which is also at the τ 0 order.Again, the denominator is set to be k n • k p 1 p 2 due to the symmetry.However, the sub-leading contribution of P 2 receives additional contribution τ 1 order in the expansion of the denominator of P 2 .Thus, we conclude that One can verify that taking k p 1 → τ k p 1 and expanding (61) in τ give the correct behavior at sub-leading order, namely, Notice that in the formula (61), both the symmetry among two gluons and the gauge invariance for each polarization are manifested.

C. YMS amplitude with three external gluons
The next example is the single-trace YMS amplitude ) which includes three external gluons p 1 , p 2 and p 3 .The treatment is parallel to that in the previous subsection for the 2-gluon case, and such similarity exhibits the recursive pattern.As in the previous case, we consider k p 3 → τ k p 3 , and expand A YS (1, • • • , n; {p i } 3 |σ n+3 ) in τ .The soft theorem requires the sub-leading term to be where the expanded formula (61) for YMS amplitude with two external gluons is used.
Applying the Leibnitz's rule, we get where The first and second parts where The result B 8 in (73) can be obtained as follows.Using the expansion of since all other terms are at the τ 1 order.Then we use the expansion of Notice that for pure BAS amplitudes Y p 3 is equivalent to will not alter Y p 2 in (76) at the leading order, since k p 3 is accompanied with τ thus does not contribute.Substituting (77) into (76), we find

D. General case
Comparing the constructions in subsections.III B and III C, we see the processes bear strong similarity.This similarity implies the recursive pattern which yields the general expansion for single trace YMS amplitudes with arbitrary number of external gluons.Such general expansion is the goal of this subsection.
In general, the single-trace YMS amplitude A YS (1, • • • , n; {p i } m |σ n+m ) with m external gluons can be expanded as in (17).When α α α = {p i } m , the YMS amplitudes at the second line of ( 17) are reduced to pure BAS ones.As mentioned previously, in this note we restrict ourselves to the special choice of the reference momentum k r in (17), which is k r = k n .
Obviously, expansions in ( 39), ( 61), (84) satisfy the general formula in (17) with k r = k n .To achieve the general formula, let us prove that suppose ( 17) is satisfied for a particular m, then it is also satisfied for m + 1.Consider k p m+1 → τ k p m+1 , and expand where with ⃗ α α α ′ = ⃗ α α α ¡ p m+1 , and The calculation are paralleled to those in subsections III B and III C. Again, the notation the second line of (87), we used the property where ⃗ a a a, ⃗ b b b and ⃗ c c c are three ordered sets.In (88), K p m+1 stands for the summation of momenta carried by external gluons p i at the l.h.s of p m+1 in the color ordering.We emphasize that the external leg p m+1 is not included in the set α α α appear in B 1 , B 2 and B 3 .Using K p m+1 = Y p m+1 − X p m+1 , one can split B 3 as where The computations of B 4 and B 5 are analogous to those for obtaining B 7 , B 8 , B 9 and B 10 in subsection.III C. It is worth to give a brief discussion for how to generalize the manipulation from (76) to (78) to the current general case.We first observe that Here β β β is a subset of {p i } m \ α α α which does not include p m+1 , since otherwise k p m+1 will occur in F ⃗ β β β and contributes τ under the re-scaling k p m+1 → τ k p m+1 .The combinatory momentum k {p i }m\α α α is the summation of momenta for external legs in the set {p i } m \ α α α.Applying the above procedure iteratively, one can expand the leading terms of YMS amplitudes in the second line of (93) further, until there is only one remaining gluon p m+1 .In other words, A ).Then, we expand them further by employing and reorganize the full expansion as based on the observation that k p m+1 which is accompanied with τ does not contribute to any Y at the leading order.The relation (95) is the generalization of the previous result (78).Using (95), we arrive at the expression of B 5 in (92).Now we observe that B 1 in (86) together with B 5 in (92) give the sub-leading term of B 2 in (87) can be interpreted as the leading term of where α α α ′ includes the external leg p m+1 and α α α ′ \ p m+1 ̸ = ∅, and B 4 in (91) corresponds to the leading terms of Combining P 1 , P 2 and P 3 together, we see that the general formula ( 17) is correct for singletrace YMS amplitude with (m + 1) external gluons, thus the proof is completed.

IV. SUMMERY
In this note, we reconstructed the expansion of single-trace YMS amplitudes proposed by Clifford Cheung and James Mangan in [27], via the bottom up method based on soft theorems, with out the aid of a Lagrangian or equations of motion.The whole process is onshell, without using any off-shell ansatz.The new recursive method inserts gluons into the original amplitude in the manifestly gauge invariant manner, leads to the resulting expansion which manifests the gauge invariance for all polarizations carried by external gluons, and the permutation symmetry among external gluons, with the cost of sacrificing the manifest locality.One can use such expansion iteratively to expand the single-trace YMS amplitudes to pure BAS ones without breaking the manifest gauge invariance for any polarization, but a verity of spurious poles will be created.
Through the double copy structure, our result can be extended to the expansion of singletrace Einstein-Yang-Mills amplitudes directly, where 1, • • • , n are color ordered gluons, and elements in {p i } m are gravitons without any ordering.It is straightforward to verify the expansion (99) is equivalent to that found by Clifford Cheung and James Mangan in [27].After some modifications of notations, the expansion in [27] with the choice of reference momentum k r = k n is given as up to an overall sign.Here k i contributes when i is at the l.h.s of the first element in ⃗ α α α in the ordering, therefore the definition of Y ⃗ α α α is satisfied.On the other hand, the momentum conservation gives k 1•••n = −k p 1 •••pm .Thus the equivalence between (99) and (100) is proved.
The sub-leading soft theorem for external gluon plays the crucial role in our recursive method.One may ask the reason for choosing sub-leading soft theorem rather than the leading order one.The answer is, the leading order soft theorem can not detect all terms in the expansion.Suppose we re-scale k p j as k p j → τ k p j , as can be seen in the general expanded formula (17), various terms include f µν p j in corresponding coefficients therefore are accompanied with τ , these terms have no contribution to the leading order.Another question is whether the sub-leading soft theorem is sufficient to detect all terms in the complete expansion?For example, if the coefficient of one term behaviors as τ 2 , then this coefficient can not be detected at both leading and sub-leading orders.This question was answered in [28], by employing the universality of soft factor for external scalars, as well as the counting of mass dimensions.The argument in [28] at least shows that the old expansion (15) can be fully detected by the sub-leading soft theorem for external gluon.Since any correct expansion must be equivalent to (15), we claim that the new expansion, which manifests gauge invariance for all polarizations, can also be detected completely.
A related interesting future direction is seeking the expanded formula for pure YM amplitudes which manifests the gauge invariance for each polarization.Such expansion leads to the manifestly gauge invariant BCJ numerators.We expect the recursive method in this note can be applied to solve this challenge.