Exploring the Landscape for Soft Theorems of Nonlinear Sigma Models

We generalize soft theorems of the nonlinear sigma model beyond the $\mathcal{O} (p^2)$ amplitudes and the coset of $\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $. We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known $\mathcal{O} (p^2)$ single soft theorem for $\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $ in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of $\text{SO} (N)$, where a special flavor ordering of the"pair basis"is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to $\mathcal{O} (p^4)$, where for at least two specific choices of the $\mathcal{O} (p^4)$ operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at $\mathcal{O} (p^2)$. Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the $\mathcal{O}(p^2)$ Lagrangian, while any possible corrections to the subleading part are determined by the $\mathcal{O}(p^4)$ Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.

On the other hand, the past decade has seen much activity in studying on-shell properties of the NLSM [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. The bulk of these endeavors focus on the NLSM amplitudes at the leading O(p 2 ) in the derivative expansion of the EFT, as well as of the coset SU(N) × SU(N)/SU(N) seen in the chiral perturbation theory. This appears as a disconnection to the IR universality of the EFT Lagrangian. There are notable exceptions [32,33], such as interesting higher derivative terms connected to the Z-theory [34,35], which involves a very specific set of operators starting at O(p 6 ). However, the general amplitudes at the subleading O(p 4 ), as well as other kinds of flavor symmetries, are less understood.
A particularly important aspect of the NLSM amplitudes is the soft theorems, which dictate the on-shell behaviors when there exists a hierarchy among the energy of the external states. Interesting soft theorems are seen in a variety of quantum field theories, and have received much attention in the recent years for their connections to other subjects such as asymptotic symmetries and memory effects [36]. For NLSM, in the single soft limit, i.e. when one of the external momenta is taken to zero, the amplitudes vanish 1 , a behavior known as the Adler zero [38]. Acting as a defining property of the EFT by enforcing a nonlinear shift symmetry, the Adler zero is a key ingredient in the IR construction of the universal Lagrangian. An on-shell equivalent of such a construction is the soft bootstrap [39][40][41][42][43], which utilizes recursion relations that are only valid because of the Adler zero, and has been used to explore higher derivative corrections [44]. Other constructions in similar spirit have been realized as well [19,[45][46][47].
The leading non-vanishing term in the single soft theorem of the tree level O(p 2 ) NLSM amplitudes involves an extended theory with additional field content of bi-adjoint scalars, which was first discovered [48] using the Cachazo-He-Yuan (CHY) formalism [49][50][51][52]. The appearance of the extended theory was later understood as a consequence of the Ward identity corresponding to the shift symmetry, which enables a direct calculation of the relevant Feynman rules [53][54][55]. These previous results have all been presented in the context of a symmetry breaking pattern of SU(N) × SU(N)/SU(N). We argue here that such a restriction is unnecessary, because the flavor ordering at this derivative order is universal.
The extended theory emerging from the soft theorem thus also can be interpreted in more general group representations.
A specific application of such a generalization is for NGB's in the fundamental representation N of SO(N), as seen in composite Higgs models, where the Higgs doublet is treated as pseudo NGB's furnishing 4 of the custodial SO (4). Apart from the single trace ordering for a general group representation, a special "pair basis" is available in this case. An extended theory amplitude in such a basis has been shown to exist in the corresponding single soft theorem [56], for which we provide detailed derivations, uncovering interesting properties of the pair basis amplitudes along the way.
By using the Ward identity, there is no obstruction to calculate the leading non-vanishing piece in the single soft theorem to higher derivative orders. What is less clear is whether there still exist the interpretations of extended theories. Recent work by the authors [57] asked a similar question in the context of the double copy [58]. It was demonstrated that at least one O(p 4 ) operator, which results in a theory dubbed NLSM d 2 , admits a double copy construction, through novel color-kinematic numerators [56,59] which do not necessarily imply Bern-Carrasco-Johansson (BCJ) relations [60].
Naturally, one would expect an extended theory emerging from the single soft theorem of NLSM d 2 , as previously its appearance has been known to be intricately related to the existence of double copy structures and CHY representations of the amplitudes. We find that this is indeed the case. However, we are also able to discover that at least another O(p 4 ) operator gives rise to an extended theory as well. This is surprising, as the corresponding amplitudes have no known double copy structures or CHY representations, marking the first instance when an extended theory emerging in the single soft theorem is uncovered using the Ward identity alone.
The double soft theorem for the NLSM has also been well established, when two of the external states take much less energy than the rest. Unlike the single soft case, the double soft limits of NLSM do not generate extended theory amplitudes with new field content, but just the usual lower point amplitudes as in the well-known single and double soft limits of gauge theory and gravity [36]. The double soft theorem has been known to the leading and subleading orders in the soft expansion [61,62], and has been studied in a completely general group representation as well [63], but still only for O(p 2 ) in the EFT expansion.
It is natural then to extend it to higher derivative orders. We find that at tree level, the double soft theorems can actually be computed to all orders in the derivative expansion, by judiciously applying various single soft limits in the spirit of Ref. [63]. Interestingly, they are fully determined by the O(p 2 ) and O(p 4 ) order Lagrangians, implying that higher derivative corrections to NLSM all satisfy the same soft theorems.
The paper is organized as follows. In Section II we review the usual trace decomposition of the NLSM amplitudes governed by the linear flavor symmetry, as well as the single soft theorem as a consequence of the nonlinear shift symmetry and the corresponding Ward identity. The universality of the flavor ordering for a general symmetry group representation enables us to reinterpret the known extended theory in the single soft limit. In Section III we inspect the specific case of NGB's furnishing N of SO(N), where we study the alternative flavor decomposition in the pair basis. We present new amplitude relations and a CHY formula for the amplitudes in such a basis, then derive the corresponding single soft theorem at O(p 2 ) and the associated extended theory amplitudes. In Section IV we compute the single soft theorems for O(p 4 ) amplitudes, and study the possible existence of the extended theories, first for the NLSM d 2 and then for the general case. In Section V we compute the double-soft theorems valid to arbitrary derivative order. We conclude in Section VI. We also provide supporting materials in the appendices: useful derivations in Appendix A, as well as examples related to the single soft theorems in Appendix B.

II. SYMMETRIES AND THE AMPLITUDES OF THE NLSM
As an EFT, the NLSM is valid below a high energy scale Λ, and its Lagrangian admits a derivative expansion of ∂/Λ. Up to the 4-derivative level, we can write with L (2) = O(1/Λ 0 ) and L (4) = O(1/Λ 2 ).
Let us consider a general NLSM of NGB's furnishing the representation R of a linearly realized flavor group H, with associated generators (T i ) ab 2 , and the NGB π a carries the flavor index a. The basic building blocks for the Lagrangian are where X a and T i are generators of some group G containing the subgroup H. In the traditional coset construction of Callan, Coleman, Wess and Zumino [2,3], the NGB's are known to be coming from the spontaneous symmetry breaking of group G to the unbroken group H, T i are the "unbroken generators" of the subgroup H, while X a are the "broken generators" associated with the coset G/H. On the other hand, the same Lagrangian can also be constructed using entirely IR information of the linearly realized group H and its representation R [9,10], where T i and X a are constructed using the generators (T i ) ab of R and the structure constants f ijk of H [43]. We are also assuming that the generators T i satisfy the "closure condition": which guarantees that R of H can be embedded into a symmetric coset G/H, so that one can identify T i ab = −if iab as the structure constants for G, whose Lie algebra is given by In general, d a µ and E i µ can be expressed as 2 We choose a totally imaginary and anti-symmetric basis for any group generators throughout this work, and the normalization is given by tr(T i T j ) = δ ij , tr(X a X b ) = δ ab , etc. where Notice that both d µ and E µ are linear in ∂, while being a series expansion of π/f , where f is the coupling constant with the same mass dimension as π. One can construct a "covariant derivative" ∇ µ so that Then the Lagrangian can be expressed as: whereL is a dimensionless function. Up to O(p 4 ), the Lagrangian is given by where C i and C − are dimensionless Wilson coefficients, are the parity (P) even generators at O(p 4 ), and is the P-odd Wess-Zumino-Witten (WZW) term [64,65] that can exist when the spacetime dimension d = 4, and can be expressed using d µ by compactifying a 5-dimensional spacetime: The form of L NLSM is dictated by the linearly realized symmetry of H, as well as a nonlinearly realized shift symmetry, which we will discuss in detail in Section II B. Notice that up to O(p 4 ), the Lagrangian can be expressed entirely using d µ without involving ∇ µ .
One may write down other operators for the Lagrangian, e.g. tr(d µ d ν ∇ µ d ν ), but they will not be independent of the operators in Eq. (11): they are related by total derivatives, symmetry transformations or the equation of motion (EoM) [43]. If we specify the R of H, the basis of independent operators may further be reduced. For example, in the chiral perturbation theory, the coset is The amplitudes for the NLSM also exhibit a derivative expansion, and at tree level one can write where n is the multiplicity, with which is controlled by the Lagrangian up to O(p m ). Below we discuss the consequences of the symmetries in NLSM at the amplitude level. In Section II A we review the flavor decomposition of the amplitudes, while in Section II B we review the single soft theorem resulting from the nonlinearly realized shift symmetry.

A. Flavor symmetry and flavor ordering
The existence of the linearly realized flavor symmetry of H leads to a convenient separation of flavor and kinematics for the tree level amplitudes. We will see that up to O(p 4 ), the general NLSM can be expressed in a single or double trace basis. An additional "pair basis" is available when we specify R of H to be N of SO(N), which will be discussed later in Section III.
Let us consider the leading order Lagrangian and denote the corresponding theory NLSM (2) . Using the relations in Eq. (4) we can rewrite where The interactions given by Eq. (18) are even powers of π a contracted with a single trace of generators X a .
Therefore, it is convenient to consider flavor-ordered partial amplitudes, as used in the soft bootstrap as the on-shell construction of the NLSM [40][41][42][43]. These partial amplitudes are similar to the color-ordered amplitudes of the Yang-Mills (YM) theory [66], where the interactions involve the structure constant f ijk , which corresponds to T i ab in Eq. (17). From the perspective of just the group H, T i ab is a group generator in some general representation; however, from the perspective of the broken group G and coset G/H, T i ab = −if iab is the structure constant of G, i.e. the generator of G in the adjoint representation. Similarly, the gauge bosons in YM theories furnish the adjoint representation as well.
The color-decomposition of the YM theory can thus be directly applied to general NLSM (2) . The flavor structure of the full amplitude can be expanded in the trace basis as where α is a permutation of {2, 3, · · · , n} and M (2) n (1, α) is the single-trace flavor-ordered partial amplitude. The right-hand side (RHS) of Eq. (20) is a sum of (n − 1)! terms. The lesson we learn from YM theories is that the flavor expansion in Eq. (20) is over-complete, and can be further reduced to the Del Duca-Dixon-Maltoni (DDM) basis [67] as a sum of (n − 2)! terms: where α is a permutation of {2, 3, · · · , n − 1}. The ordered amplitudes thus need to satisfy the Kleiss-Kuijf (KK) relations [68]. If we further take flavor-kinematics duality into account, using the BCJ relations [60] we can reduce the number of independent ordered amplitudes to (n − 3)!.  A common practice of calculating the ordered amplitude is working out the ordered Feynman vertices, and then summing up all distinct planar Feynman diagrams [66]. It is important to note that in principle, such operation does not work for a general group representation of the NLSM, as it relies on the correct factorization of traces, which is only valid in some cases such as the adjoint of SU(N) [19]. This applies to recursion relations as in the soft bootstrap as well [43]. However, in practice, we can still always calculate the on-shell amplitudes correctly using these methods, as the flavor ordering is universal, thus the partial amplitudes for the adjoint of SU(N) are not different from the partial amplitudes of any other group representations 3 . This is similar to the case of gauge theory: the ordered amplitudes are universal whatever the gauge group is.

B. Shift symmetry and the single soft theorem
The NLSM effective Lagrangian is determined by a non-linear shift symmetry. From the UV perspective of spontaneous symmetry breaking, this shift symmetry is the non-linear realization of the broken symmetry associated with the coset G/H. However, the shift symmetry can also be fixed without knowing the UV information of the broken group G.
This is directly related to the fact that the amplitudes of NLSM satisfy the Adler zero condition: for an on-shell amplitude M a 1 ···a n−1 a n (p 1 , · · · , p n−1 , q) 4 , if we take the soft limit of q, i.e. replace q with τ q and take the limit of τ → 0, the amplitude vanishes linearly in τ : M a 1 ···a n−1 a n (p 1 , · · · , p n−1 , τ q) = O(τ ).
Such a condition can be treated as the defining property of the NLSM, and is the most basic of the soft theorems of NLSM amplitudes. Upon recognizing the Adler zero condition, the non-linear shift symmetry can be derived without the UV information of the broken group G. The associated transformation of the shift symmetry for the NGB's is [9,10,53,54] where and ε a are constants that parameterize the shift. Starting at O(p 4 ), there are both P-even and P-odd parts in the Lagrangian, the latter of which are the WZW terms that capture the effects of the anomalies. Under the shift transformation, the P-even parts of the Lagrangian are invariant, while the WZW terms change by a total derivative.
We can then calculate the current associated with the shift symmetry: we promote the shift parameter to a local one: ε a → ε a (x), and find out that up to total derivatives, the variation of the Lagrangian is given by where Classically, the current is conserved because the action should not change: As we have taken care of the quantum anomalies with the WZW term in the Lagrangian, the current remains conserved at the quantum level, leading to the Ward identity for the correlation functions: Performing the LSZ reduction and taking the on-shell limit, the RHS of the above vanishes, and we arrive at a single soft theorem of the on-shell amplitude: M a 1 ···ana n (p 1 , · · · , p n , q) = q · R a 1 ···ana (p 1 , · · · , p n ; q), · · · · · · q · · · where the momentum q is carried by the current. As shown in Fig. 3, the left-hand side (LHS) of the above comes from the one particle pole in J , while the rest of the current enter the remainder function R a 1 ···ana µ : where Z is the field strength renormalization factor, with Z = 1 at tree level, which is what we will assume in the rest of the work, and As there are no cubic interactions in the Lagrangian, which is the case when the closure condition of Eq. (3) is satisfied, we haveJ = O(π 3 ). Therefore, the only divergence possible when we take q → 0, given by the "pole diagrams" shown in Fig. (4), cannot exist, so that the remainder function R is finite. This leads to the Adler zero condition given by Eq. (24).
Notice that in the flavor decomposition, the flavor factors are linearly independent, thus each of the ordered amplitudes also need to satisfy the Adler zero condition, similar to the YM theory where the ordered amplitudes are also gauge invariant. single soft theorem, which is already known. The current from L (2) is: As we see from Eq. (32), each term inJ (2),a µ inserts the following single-trace vertex into R µ , after we strip the flavor factors: where I n ≡ {1, 2, · · · , n} is the identity permutation for n labels. As shown in Fig. 5, the legs of the above vertex are connected to semi-on-shell amplitudes, i.e. the Berends-Giele currents [69] J a 1 ···an,a (p 1 , · · · , p n ) ≡ 0|π a (0)|π a 1 (p 1 ) · · · π an (p n ) .
These objects have one uncut off-shell leg of momentum − i p i , which is connected to V in the single soft theorem, while all the other legs are on-shell. Just like the on-shell amplitudes, at O(p 2 ) the off-shell amplitudes can be ordered in a single trace basis, so that · · · · · · · · · · · · · · · the subleading single soft theorem for NLSM (2) is where we have taken p n+1 to be soft. In the above l is a way to split {1, 2, · · · , n} into 2k + 1 disjoint, ordered subsets {l m−1 + 1, · · · , l m }, with l 0 = 0, l 2k+1 = n and q l j+1 = l j+1 i=l j +1 p i . It was first discovered in Ref. [48] using the CHY formalism that an extended theory with the NGB's interacting with the bi-adjoint scalars emerges in the soft theorem. Originally, NGB's generated by the coset of SU(N) × SU(N)/SU(N) were considered, so that R is the adjoint representation of the unbroken SU(N) group, and the coset space is isomorphic to the unbroken SU(N). Then the generator T i ab in the Lagrangian can be exchanged with the structure constant −if iab of SU(N): the difference between the broken and unbroken indices becomes non-existent. The bi-adjoint scalars φ aã , as the additional field content in the extended theory, transform under both the original flavor group SU(N) and another flavor group SU(Ñ), and has the following cubic self-interaction: characterized by the coupling constant λ. For the (n + 1)-point (pt) amplitude, taking p n+1 to be soft, we have where s i,j ≡ (p i + p j ) 2 , and NLSM + φ 3 denotes the extended theory 5 . In the RHS of the above, "||" separates flavor structures of different flavor groups, which should not be confused with "|" in the multi-trace amplitudes in Eq. (22) which separates traces of the generators of the same group. In M NLSM+φ 3 n (I n ||1, n, i), the external states 1, n and i are apparently bi-adjoint as they have two separate orderings, the left for SU(N) and the right for SU(Ñ); other external legs, which only have left orderings, then belong to the NGB's.
In the DDM basis, the flavor factor for M NLSM+φ 3 n (I n ||1, n, i) in Eq. (40) is where f abc andfãbc are structure constants of SU(N) and SU(Ñ ), respectively.
Comparing with Eq. (38), one can identify all the new Feynman vertices in the extended theory that are relevant in Eq. (40), in addition to ones already in NLSM (2) [53-55]: • Type I vertices with two φ and an even number of π, which has to take exactly the same value as the vertices in NLSM (2) with the same left ordering, i.e.
• Type II vertices with three φ, two of whose left orderings are adjacent, and an even number of π. These vertices are generated by the current J , and to match Eq. (40) we need We know that V (2) is linear in q, but for the above to hold the coefficients of q · p 1 and q · p 2k+1 need to vanish at the same time. Applying total momentum conservation to 5 It should be understood that whenever it appears in the name of an mixed theory like NLSM + φ 3 , "NLSM" means NLSM (2) .
Eq. (36), as well as the fact that whenever we use V (2) we have the on-shell condition of q 2 = 0, we see that which matches the φ 3 interaction given by Eq. (39).
As discussed in Section II A, the flavor ordering works equally well for NLSM (2)  will need to be where T i ab andT˜ĩ ab are generators of H andH in some representation R andR, respectively. The flavor factor in Eq. (46) can be presented graphically as in Fig. 6. Notice that external states 1, 2, · · · n − 1 carry indices a k that furnish some representation R, while particle n has indices in the adjoint. In other words, we have two different kinds of bi-index scalars: ψ aã in R andR, and the bi-adjoint scalars φ jj . We will denote such an extended theory as NLSM + φ + ψ. An example useful in the following will be H = SO(N) and R is the fundamental representation, so that the amplitudes can also be expressed in the pair basis.

III. THE SINGLE SOFT THEOREM FOR N OF SO(N )
In this section we consider NGB's furnishing N of SO(N), focusing on the leading O(p 2 ) in the EFT expansion. There are N flavors in such a theory, and the minimal coset that realizes this is SO(N + 1)/SO(N). The generators T i ab satisfy the following completeness relation: where we have adopted the bra-ket notation (|π ) a ≡ π a , and r ≡ π|π /(2f 2 ). We see that in the vertices given by the above, π a are pair-wise contracted, which implies that the flavor factor for the amplitudes are products of Kronecker deltas: we have where P n is all the distinct partitions of non-ordered set {1, 2, · · · , n} into n/2 subsets of two Eq. (49) contains n/2 non-ordered pairs of external particle indices. The RHS of Eq. (49) is a sum of (n − 1)!! terms, and as the flavor factors in front of each term are completely independent of each other, they form a basis which we call the pair basis. As δ ab = tr X a X b , the pair basis can also be understood as a multi-trace basis. The amplitude in the basis is invariant under exchanging the positions of different traces, as well as exchanging two labels in each trace.
When the multiplicity n is large, we have (n − 1)!! ≪ (n − 3)!, thus the pair basis is much smaller than the minimal BCJ basis of a general NLSM. A comparison of the size of the different bases is given in Table I  In the following, we explore the amplitude relations for the pair basis in Section III A, which will be useful when we derive the subleading single soft theorem for the pair basis in Section III B.

A. Amplitude relations for the pair basis
It turns out that the relation between partial amplitudes in the pair basis and the singletrace amplitudes is quite straightforward. Let us first look at the DDM basis given by Eq.
n (1, n|2, 3|4, 5| · · · |n − 2, n − 1). By definition, it is the coefficient of the flavor factor δ a 1 an in the full amplitude. As shown in Fig. 8, the indices a 1 and a n are always on the two ends of the half-ladder. As explained in Figs. 7 and 8, each dotted circle can be either a "×" or an "=", but for a 1 and a n to be contracted to get δ a 1 an , all of the dotted circles must contain an "=". Therefore, the upper two indices in each of the dotted circles must be contracted as well. Then the coefficient of the flavor factor in Eq. (50) is  In hindsight, the form of the RHS of Eq. (51) is very natural: it has the correct pole structure, it satisfies the Adler zero condition, and it also has the correct mass dimension and permutation symmetry. The only thing non-trivial is that the RHS of Eq. (51) must also factorize correctly.
One should also recognize that Eq. (51) is not the unique way to write the multi-trace partial amplitudes in terms of single-trace ones: other representations can be easily generated by using the KK relations among the single-trace amplitudes.
An immediate consequence of Eq. (51) is that we can easily write down the CHY formula for the partial amplitudes in the pair basis. In the CHY representation, the tree-level amplitude for a scalar theory is in general written in the following form: where {p} are the on-shell external momenta, while {σ} are dimensionless variables satisfying the scattering equation with σ ij ≡ σ i − σ j . This is enforced by the measure dµ n of the integral: Choosing {i, j, k} and {p, q, r} is called "fixing the gauge", and the measure dµ n is actually gauge invariant, i.e. independent of the choice of {i, j, k} and {p, q, r}. The integrands I L and I R are different among different theories.
For the general NLSM (2) , the single-trace partial amplitudes are given by, up to coupling constants, [52] where C n (α) is the Parke-Taylor factor given by and the anti-symmetric matrix A n is given by The reduced Pfaffian Pf ′ is defined as n is the matrix A n with rows and columns of labels a and b removed. It turns out such a definition does not depend on the choices of {a, b}.
An important observation is that in Eq. (57), both the measure dµ n and the reduced Pfaffian Pf ′ A n are independent of the ordering α: the ordering information is only contained in the Parke-Taylor factor C n (α). Then using Eq. (51) we can easily arrive at the CHY formula for the partial amplitude in the pair basis: In other words, we have which remains the same as the single-trace amplitudes, while Another relation between the partial amplitudes in the pair basis can be easily proved using Eq. (51). Firstly, we know the U(1)-decoupling relation between the single-trace partial amplitudes, which is the simplest kind of the KK relations: n (1, 2, 3, · · · , n − 1, n) + M (2) n (1, 3, 4, · · · , n, 2) + · · · + M (2) n (1, n, 2, · · · , n − 2, n − 1) = 0. (62) Then as the LHS of the above can be expressed as (n − 2)! sums like the LHS of Eq. (62). The relation given by Eq. (63) can also be easily understood from a physical perspective: we see from the definition of the pair basis in Eq. (49), that (3),α(4)| · · · |α(2n − 1),α(2n)).
Namely, the LHS of Eq. (63) is actually the two-derivative full amplitude when all external particles are of a single flavor. The shift symmetry in NLSM forbids any 2-derivative interactions for a single kind of scalar, and the corresponding leading order amplitude must vanish [9]. Therefore, Eq. (63) just states the fact that at the two-derivative level, the NLSM amplitude between scalars of a single flavor vanishes.
where j p ≡ j + (−1) j , Ar n−3 are permutations of {2, 3, · · · , n} \ {j, j p } so that for all the pairs {m, m p } in the set, the two indices in the pair are adjacent to each other. Note that although j = k, the situation when j p = k, i.e. j and k form a pair in the symmetrization of the LHS in the above, can still happen.
Next, we need to express the amplitudes of the extended theory in the pair basis as well.
Using the procedure similar to Section III A, we can derive a relation between the singletrace and the pair basis in the extended theory, the details and examples of which are shown in Appendix A 1. We have where we have identified the different bi-index scalars φ and ψ in the right ordering. Plugging the above into Eq. (65), we arrive at [56] M (2) n+1 (n + 1, 1|2, 3|4, 5| · · · |n − 1, n) whereα/j is the partition {2, 3|4, 5| · · · |n − 1, n} with the pair {j, j p } removed. Examples of Eq. (67) are given in Appendix B 1.
Similarly, we can also work out the vertices in the pair basis of NLSM + φ + ψ: the Type I vertices are given by while the Type II vertices are Plugging in the vertices in the single-trace basis given by Eq. (44), we arrive at with the 3-pt vertex given by Note that although V NLSM+φ+ψ We can work out the operators in the Lagrangian that give the vertices in Eq. (71): where the cubic operator in the above is (λ/2)T i ab T˜ĩ ab φ iĩ ψ aã ψ bb . The results of Eqs. (67) and (71) can also be confirmed by a direct calculation from the Ward identity. As we are still at O(p 2 ), the current is still given by Eq. (34), though it can be simplified using the completeness relations of the SO(N) fundamental generators given by Eq. (47): Such a current inserts the following flavor-ordered vertices to the soft theorem, i.e. the RHS of Eq. (31): The second equality in the above utilizes total momentum conservation as well as the onshell condition of q, while the last equality is a consequence of the vanishing contributions of vertices given by Eq. (70). This directly leads to the soft theorem given by Eq. (67).

IV. THE SINGLE SOFT THEOREM AT O(p 4 )
Now let us work out the subleading single soft theorem of NLSM at O(p 4 ) for a general group representation. We will use the universal trace basis and focus on the 4 P-even operators given by Eq. (12), which will always exist for a general spacetime dimension d.
The next question to ask is: can we interpret the RHS of Eqs. (76) and (77) as given by the amplitudes of some extended theory? We will first focus on a special case where we have a definite answer, and then proceed to the more general case. To avoid complicated flavor labels, we will assume that the NGB's furnish the adjoint representation, so that the extended theory of NLSM (2) is just NLSM + φ 3 , though the results can be straightforwardly reinterpreted for a general group representation.

A. The d 2 case
As demonstrated in Ref. [48], the extended theory can be identified in a concrete manner when the amplitudes of the original theory have a CHY representation and admit a double copy structure. In Ref. [57] a special case of the NLSM up to O(p 4 ), dubbed NLSM d 2 , was observed to demonstrate these properties. In such a theory the Wilson coefficients are fixed to be The corresponding O(p 4 ) amplitude M (4),d 2 always has a double trace ordering, and is a component of an EFT named the extended Dirac-Born-Infeld (DBI) theory [52], also called DBI + NLSM [70], which is a double copy of NLSM (2) and a gauged version of the bi-adjoint scalar theory called YM + φ 3 [71], or generalized YM scalar [52]. We can write where KLT ⊗ indicates a Kawai-Lewellen-Tye (KLT) relation [72] between the ordered amplitudes of two theories, which we will use extensively below to show that in the single soft limit of NLSM d 2 there is indeed an extended theory.

From the double copy
The amplitudes for NLSM d 2 can be expressed using the KLT formula as where l is odd and 1 ≤ l ≤ n − 3, M YM+φ 3 is the amplitude for YM + φ 3 where all external states are scalars, and K n is the KLT kernel for the n-pt amplitude satisfying [51] with M φ 3 n being the doubly ordered amplitudes for the bi-adjoint scalar theory. In Eq. (98) K n and M φ 3 n are understood as (n − 3)! × (n − 3)! square matrices, whose rows and columns correspond to (n − 3)! ways of the left and right ordering, respectively. For example, in Eq. so that the KLT kernel in Eq. (97) is given by Similar to the φ 3 amplitudes, we have M YM+φ 3 n = O(τ −1 ), but the pole diagrams here not only contain ones shared with the φ 3 theory as shown in Fig. 10, but more diagrams where the internal leg giving the 1/τ pole belongs to a gauge boson, as shown in Fig. 11.
However, these additional diagrams are proportional to where ǫ µ i M YM+φ 3 µ,n−1 (i g ) is the on-shell amplitude of YM + φ 3 with all external states being scalar except for the state i, which is a gauge boson with polarization vector ǫ i . Then In the above, the amplitudes of the extended theory d 2 +φ 3 are given by the following double copy formula: Just like the theory NLSM d 2 is part of DBI + NLSM, the extended theory d 2 + φ 3 in Eq.
(104) can be identified as part of a theory called DBI + YM + NLSM + φ 3 [70], which is the double copy of NLSM + φ 3 and YM + φ 3 : Actually, by the same argument as above one can easily calculate the single soft theorem for a general n-pt amplitude of DBI + NLSM, taking p n to be soft: where j L and j R are the legs left/right adjacent to the leg n in the ordering of the original amplitude, i.e. the last and first element in the sequence α m .

Matching to the Ward identity
The first thing one may want to check is the special case of Eq. (106), where the O(τ ) term vanishes. In such a case, Eq. (77) is reduced to with p n being soft, where {γ ′ 1 , · · · , γ ′ i } are partitions of any of the cyclic permutations of α. It is clear that vertices V (4),d 2 (j|α) with only one leg j in one of their traces appear in all the O(τ ) terms in the above. Therefore, to satisfy Eq. (106) we need total cancellations of all terms involving V (4),d 2 (j|α). This, however, does not necessarily imply that V (4),d 2 (j|α) = 0. It is easy to show that which does not vanish, but its effect can be moved to higher-pt vertices. For example, if p 1 in V (4),d 2 (1, 2|3) is an internal momentum in a Feynman diagram, the term with p 2 1 will cancel the propagator and effectively resulting in a higher-pt vertex. This suggests that the EoM may be needed here. Indeed, let us directly look at the current: the vertices V (4),d 2 (j|α) come from In the second line of the above we select all terms that generate V (4),d 2 (j|α), in the third line we drop total derivatives, which give a contribution of O(τ 2 ) in the soft theorem, while in the last line we apply an identity: The vertex that we need to consider is then V (4),d 2 2m+2n+1 (I 2m |2m + 1, · · · , 2m + 2n + 1) Similar to what we see in Eq. (43), the above dictates that terms involving q · p 2m+1 and q · p 2m+2n+1 in V (4),d 2 need to be eliminated at the same time. The final form of the vertices for the extended theory would be V d 2 +φ 3 2m+2n+1 (I 2m |2m + 1, · · · , 2m + 2n + 1||2m + 1, 2m + 2n + 1, j) for 1 ≤ j ≤ 2m, and V d 2 +φ 3 2m+2n+1 (I 2m |2m + 1, · · · , 2m + 2n + 1||2m + 1, 2m + 2n + 1, j) Examples of the vertices and amplitudes for d 2 + φ 3 are presented in Appendix B 3.

B. The general case
It is still unknown whether the most general O(p 4 ) amplitudes of NLSM, with arbitrary Wilson coefficients C i , have a double copy structure or a CHY representation. Without such input it is hard to answer definitely whether the soft theorems of Eqs. (76) and (77) have an interpretation of extended theories. If they do, as they are for amplitudes of the higher derivative corrections to NLSM (2) , it is natural to assume that they are higher derivative corrections to the NLSM (2) soft theorem, given by Eq. (40). However, the correction can either enter the extended theory or the soft factor. Schematically, we can have where S (2) is the known soft factor in the soft theorem of NLSM (2) , (4)+ is some higher derivative corrections to the extended theory NLSM + φ 3 , and S (4) a new soft factor at O(p 4 ). One needs to determine which part of the RHS of Eqs. (76) and (77)  Here we make the first steps to answer these questions. We observe that in the special case of NLSM d 2 , the form of the soft theorem is very similar to that of NLSM (2) : In the NLSM d 2 case the result is somewhat expected, given that these amplitudes can be built by a double copy with NLSM (2) amplitudes, which themselves satisfy this type of single soft theorem. The single trace case on the other hand is surprising, as these O(p 4 ) amplitudes are not known to have any direct connection to the O(p 2 ) amplitudes. We will call the NLSM with O(p 4 ) Wilson coefficients C 3 ′ = 1, C 4 ′ = C 1 = C 2 = 0 as NLSM C 3 ′ , and the corresponding extended theory C 3 ′ + φ 3 . The soft theorem for the O(p 4 ) amplitudes of Again, as in the case of d 2 + φ 3 , there should be Type I and Type II vertices in C 3 ′ + φ 3 , Type I taking the same value the NLSM C 3 ′ vertices with identical left orderings, while Type II vertices should be given by We see from Eq. (86) that V (4),3 ′ 2k+1 (I 2k+1 ) can indeed be put in a form without 2q · p 1 and 2q · p 2k+1 . However, when k = 1, j has to be 2, and the coefficient of q · p 2 is proportional to which cannot work as V (1, 2, 3||1, 3, 2), because the 3-pt φ 3 vertex need to be invariant under cyclic permutations. This problem persists in higher-pt vertices as well: we need (1, 2k + 1, 2k, 2k − 1, · · · , 2||1, 2, 2k + 1), (123) the two sides of which are related by a reversion of both the left and the right orderings.
At 3-pt, we see that In other words, the difference between Eq. (122) and a cyclic form is associated with p 2 i . Similar to what we have seen in V (4),d 2 (1, 2|3) as in Eq. (111), this amounts to corrections to higher-pt vertices. Similar to Section IV A 2, it would be easier to fix the problem at the level of the current using the EoM. The suitably modified vertices given by the current iŝ (q · p l 3 +1;l 3 +l 2 p l 3 +1;l 3 +l 2 · p l 1 +l 2 +l 3 +1;2k+1 +q · p l 1 +l 2 +1;l 1 +l 2 +l 3 p l 1 +1;l 1 +l 2 · p l 1 +l 2 +1;2k+1 ) and the details of the derivation of the above are presented in Appendix A 3. Then the Type II vertices are given by (1, 2, · · · , 2k + 1||1, 2k + 1, j) Let us consider the (n + 2)-pt tree amplitude M n+2 , and take p n+1 and p n+2 to be soft.
The leading non-vanishing term in the double soft limit is O(τ 0 ), instead of O(τ ) which the Adler zero in the single soft limit may suggest. This is because when two of the external legs are soft, there are Feynman diagrams where a pole in the soft parameter τ develops: this happens when both of the soft legs are attached to a single 4-pt vertex, and a third external leg with momentum p i is attached to the vertex as well, as shown in Fig. (12). The pole in FIG. 12: The pole diagram in the double soft limit.
τ then appears in the propagator attached to the vertex: Therefore, we are able to classify all Feynman diagrams into two groups: the pole diagrams with such a pole in τ , summed to M pole ; all the other diagrams, which are called "gut diagrams" and sum to M gut . It is straightforward to calculate M pole in the double soft limit, while the contributions of the gut diagrams are then fixed by applying symmetry constraints on the full amplitude M = M pole + M gut , including flavor symmetry as well as the shift symmetry, which manifests as the Adler zero condition in the single soft limit. We will be able to calculate the double soft limit up to O(τ ) in this manner.
The pole diagrams for M n+2 are given by M a 1 ···a n+2 n+2, pole (p 1 , · · · , p n , τ p n+1 , τ p n+2 ) where V 4 is the 4-pt vertex attached by the two soft legs as shown in Fig. (12), whilẽ M a 1 ···a i−1 ba i+1 ···an n (p 1 , · · · , p i−1 ,p i , p i+1 , · · · , p n ) is the n-pt amplitude with leg i off-shell, with momentump i = p i + τ (p n+1 + p n+2 ). Herẽ M is clearly equivalent to the Berends-Giele current given by Eq. (37), except that the off-shell leg is also cut inM but not in J. Of course, in the fully on-shell limit we havẽ M n (p i ) = M n (p i ). However, for a general off-shell momentump i ,M n (p i ) = M n (p i ) is not guaranteed. Therefore, to relateM and M in a concrete manner, it is necessary to express M n (p i ) asM a 1 ···b···an n (· · · ,p i , · · · ) = M a 1 ···b···an n (· · · ,p i , · · · ) +p i 2 X a 1 ···b···an (p 1 , · · · ,p i , · · · , p n ), where X(p i ) is some unknown function of the momenta and must be non-singular in the limit ofp i → p i . The X piece is usually neglected in the literature, but should be included in a rigorous derivation. As it clearly contains off-shell data, it should vary under local redefinitions of the field variables, but its contributions need to vanish in any on-shell results.
As we would like to calculate the double soft limit up to O(τ ), from Eq. (129) we know that we need V 4 up to O(τ 2 ), which turns out to be completely fixed by the Lagrangian up to O(p 4 ). This is because for a general, off-shell 4-pt scalar vertex V 4 with momenta p 1 , · · · , p 4 satisfying total momentum conservation, it can only be a polynomial of 6 independent momentum invariants, for example s 12 , s 13 , s 23 , p 2 1 , p 2 2 , p 2 3 .
In particular, for NGB's furnishing N of SO(N), the alternative flavor decomposition of the pair basis is convenient, and we derive the corresponding single soft theorem of NLSM (2) in Eq. (67) where the extended theory is presented in this basis as well. To achieve this, we have studied the pair basis in detail, uncovering novel amplitude relations given in Eqs. (51), (63) and (66), as well as the CHY formula Eq. (59) for the pair basis NLSM (2) amplitudes.
As the pair basis can be regarded as a multi-trace basis where each trace only contains two external states, it appears in amplitudes of other theories as well, such as the multiflavor DBI scalars as well as the YM scalar theory of a single flavor but multiple colors.
There may be connections between these theories and the NLSM (2) for N of SO(N), such as in the web of theories of Ref. [25]. Amplitude relations for these theories are also worth exploring. Moreover, the DBI scalar amplitudes start at O(τ 2 ) in the single soft limit, which can be seen as a special case of the DBI + NLSM soft theorem given by Eq. (108). The leading non-vanishing terms at O(τ 2 ) have been calculated using the Ward identity [55], and it remains to be seen whether the double copy structure of the DBI amplitudes leads to extended theories in this context.
Using the Ward identity corresponding to the shift symmetry, we have also extended the subleading single soft theorem to O(p 4 ), given by Eqs. (76) and (77). We found that for at least two specific choices of the Wilson coefficients for the O(p 4 ) operators, which result in NLSM d 2 and NLSM C 3 ′ , the single soft theorems can be interpreted as generating extended theory amplitudes, similar to the NLSM (2) . For the NLSM d 2 case, as seen in Eq. (104), this fact is expected as the theory is known to have a double copy construction in terms of NLSM (2) , but the emergence of C 3 ′ + φ 3 in Eq. (120) can only be understood from the perspective of the Ward identity. This may give us a handle to explore the possible double copy structure or CHY representation of NLSM C 3 ′ . where only a limited number of EFT operators modify soft theorems, are known in gauge theory and gravity [73]. It would be interesting to see how these patterns extend to higher orders in the soft expansion, as well as the multi-soft limits studied in Refs. [54,74]. The investigation of the multi-soft limits has previously relied heavily on manipulating Feynman diagrams in the Cayley parameterization, which is only available for the adjoint U(N) NLSM [19]. Although the resulting amplitudes in the trace basis are universal, it would still be beneficial to develop other techniques that are more physically intuitive.
We have seen clearly that the higher derivative operators at O(p 4 ) generate amplitudes with properties similar to what we have seen in NLSM (2) , but only for different specific choices of the Wilson coefficients. These include: • A double copy construction and a CHY formula for NLSM d 2 .
• The same subleading single soft theorem in NLSM C 3 ′ and NLSM d 2 as in NLSM (2) , with emergent extended theories.
• The double soft theorem of NLSM (2) does not change when we only add the WZW term.
• The double soft theorem, after flavor ordering and with two adjacent legs taken to be soft, is not modified by higher derivative operators if C 3 = −3C 4 , C 1 = C 2 = 0.
• The KK relation is satisfied when On the other hand, no combination of the Wilson coefficients generates amplitudes that satisfy BCJ relations or appear in the Z-theory. It is evident that the higher derivative amplitudes in NLSM exhibit a wide variety of behaviors, and is an ideal testing ground to study the origin of the numerous properties known in NLSM (2) as well as in gauge theory and gravity. It is also well-known that causality enforces positivity constraints on the Wilson coefficients of the chiral Lagrangian [75,76]. This suggests a potential positive geometry which exists in the EFT expansion of NLSM that may be worth investigating as well, using the approach recently developed in [77], including for the closely related Z-theory [78].
As for the WZW term, which is not featured prominently in this work, although it does not modify the double soft theorem of NLSM (2) at all, it definitely will contribute to the subleading single soft theorem. Calculating its contribution using the Ward identity would be straightforward, but the result is more useful if we have a closed-form expression of the WZW operator at the beginning.
For applications to phenomenological models such as the chiral perturbation theory or the composite Higgs models, more features need to be added to the idealized NLSM, including explicit symmetry breaking which gives the pseudo NGB's masses, as well as interactions with fermions and vector bosons. As long as the explicit symmetry breaking is soft, the Ward identity, which is our main tool for deriving the soft theorems, will still hold at the leading order. It is certainly desirable to have purely on-shell techniques such as the CHY formalism generalized to these realistic cases as well. Further investigations in this direction are left for future work.

Useful relations involving the NGB field operators
Here we briefly prove relations used in this work that involve the NGB field operator π.
The building blocks of the NLSM Lagrangian, d µ and E µ , come from the Maurer-Cartan where The Lie algebra of a symmetric coset, given by Eq. (4), leads to an automorphism Aut under which the broken generators change sign, namely Aut(X a ) = −X a .
As Aut(ξ) = ξ † , we have while the covariant derivative is given by and Now let us work out the EoM of the NLSM [16]. From the completely fixed form of L (2) given in Eq. (11), we calculate the EoM as D a,(2) An important observation is where F (Π) can be any function of Π. The above implies Combining Eqs. (A16), (A17) and (A18), one can show that The explicit form of d µ , given by Eq. (5), tells us that We know that F 1 (T ) is invertible, thus 3. Preparing the current for the theory C 3 ′ + φ 3 The current associated with the coefficient C 3 ′ , as we originally derive it in Eq. (84), is We need to rewrite the above so that the resulting vertices of the extended theory C 3 ′ + φ 3 satisfy the correct symmetry condition of Eq. (123). As we see from the above, the current is neatly written in terms of U. We will need to use the EoM given in Eq. (A21), so it would be convenient to rewrite it using U as well: one can show that The existence of the automorphism given by Eq. (A11) implies that as well.
For the rest of this section, whenever we rewrite the current, we freely drop any terms involving the form of the EoM as in Eqs. (A23) and (A24), as well as any terms that are total derivatives, because the former lead to terms of higher orders in the derivative expansion, while the latter only contribute O(τ 2 ) effects in the soft expansion.
Similarly, we can use the following relations: to rewrite the current as which generates the vertices given by Eq. (126).

The current
Below we present the vertices V generated by the current.
Although we have a 3-pt vertex at O(1/Λ 2 ), the corresponding 3-pt amplitude still vanishes, so that it contributes nothing to the soft theorem in Eq. (B5). On the other hand, we can calculate the 5-pt amplitudes to be which agrees with Eq. (B13).