Ward Identity and Scattering Amplitudes for Nonlinear Sigma Models

We present a Ward identity for nonlinear sigma models using generalized nonlinear shift symmetries, without introducing current algebra or coset space. The Ward identity constrains correlation functions of the sigma model such that the Adler's zero is guaranteed for $S$-matrix elements, and gives rise to a subleading single soft theorem that is valid at the quantum level and to all orders in the Goldstone decay constant. For tree amplitudes, the Ward identity leads to a novel Berends-Giele recursion relation as well as an explicit form of the subleading single soft factor. Furthermore, interactions of the cubic biadjoint scalar theory associated with the single soft limit, which was previously discovered using the Cachazo-He-Yuan representation of tree amplitudes, can be seen to emerge from matrix elements of conserved currents corresponding to the generalized shift symmetry.


I. INTRODUCTION
Nonlinear sigma models (nlσm) [1] have wide-ranging applications in many branches of physics. It was realized early on that spontaneously broken symmetries play a central role in understanding the dynamics of nlσm. Such a realization was embodied in the current algebra approach [2], where currents corresponding to broken symmetry generators and their commutators allowed for computations of pion scattering amplitudes and resulted in the celebrated "Adler's zeros" in the single emission of soft pions [3]. Modern formulation of nlσm is based on the coset space construction by Callan, Coleman, Wess and Zumino (CCWZ) [4,5], where the Goldstone bosons parameterize the coset manifold G/H with G being the spontaneously broken symmetry in the UV and H the unbroken group in the IR.
Lately there has been a resurgence of efforts in understanding the infrared structure of quantum field theories, in particular in gravity and gauge theories [6]. The classic soft theorems [7] were re-derived using asymptotic symmetries and the related Ward identities, while the soft massless particles are interpreted as Goldstone bosons residing at the future null infinity [8][9][10][11].
On the other hand, our understanding of the Goldstone bosons in nlσm had stayed at the same level as in the 1960's, until Ref. [12] studied the double soft emission of Goldstone bosons in the context of scattering amplitudes, which sparked new efforts in this direction [13][14][15]. More recently Ref. [16] studied the subleading single soft limit of tree-level amplitudes in a variety of theories exhibiting Adler's zeros using the Cachazo-He-Yuan (CHY) representation of scattering equations [17][18][19]. They found in each case the subleading single soft factor can be interpreted as on-shell tree-amplitudes of a mysterious extended theory. Only the CHY representation of tree amplitudes in the extended theory is given, and little is known regarding how the extended theory emerges. For nlσm, the extended theory turns out to be a theory of cubic biadjoint scalars interacting with the Goldstone bosons.
In this work we aim to provide a common thread through the different perspectives on the infrared dynamics of Goldstone bosons. We first present a Ward identity governing the correlation functions of nlσm such that the Adler's zero is guaranteed for S-matrix elements. The Ward identity is derived using nonlinear shift symmetries [20,21], which makes transparent the infrared universality of the result, regardless of the underlying coset space. Using the identity we derive a single soft theorem, beyond the Adler's zero, that is valid both at the quantum level and to all orders in the Goldstone decay constant. For tree-level amplitudes, we obtain a novel set of Berends-Giele recursion relations, which leads to the subleading single-soft factor of flavor-ordered tree amplitudes. In particular, the derivation shed lights on the emergence of the extended cubic biadjoint scalar theory uncovered using the CHY approach in Ref. [16].

II. THE WARD IDENTITY
In Refs. [20,21] a new approach to constructing the effective Lagrangian for nlσm was proposed, without recourse to the current algebra or the coset construction. It is based on the simple observation that, for a spontaneously broken U (1) symmetry, the effective Lagrangian for the sole Goldstone boson π(x) can be constructed by imposing the shift symmetry: where ǫ is a constant. The constant shift symmetry enforces the Adler's zero condition. For a non-trivial unbroken group H, there are multiple Goldstone bosons π a (x) furnishing a linear representation of H and the constant shift symmetry is enlarged to respect both the Adler's zero condition and the linearly realized H symmetry. Choosing a basis such that generators of H, (T i ) ab , are purely imaginary and anti-symmetric, and adopting the bra-ket notation to define |T i π = T i |π , Eq. (1) can be generalized to [20,21]: where a k are numerical constants, |ǫ is a constant vector, and f is the Goldstone decay constant. By imposing the Adler's zero condition, an effective Lagrangian for the Goldstone bosons can be constructed, without specifying the broken group G in the UV, up to the overall normalization of f . The construction makes it clear that interactions of Goldstone bosons are universal in the IR and insensitive to the coset structure G/H. The highly nonlinear nature of the Goldstone interactions only serves two purposes: 1) fulfilling the Adler's zero condition and 2) linearly realizing the unbroken group H. The leading two-derivative Lagrangian is [20,21] Using the universality in Goldstone interactions, it is possible to derive the generalized nonlinear shift in Eq. (2) [22]: under which the nlσm Lagrangian is invariant.
It is now straightforward to derive the Ward identity corresponding Eq. (4) in path integral, by promoting the global transformation into a local one [23]: |ǫ → |ǫ(x) , which amounts to a change of variable in evaluating the path integral and leads to the Ward identity: Since we have not invoked any specific coset structure, Eq. (5) is universal. It is worth reiterating that we have only invoked the Adler's zero condition and the linearly realized unbroken symmetry H. This is in contrast with the usual vector and axial Ward identities considered in current algebra, which assumes the existence of broken symmetry generators as well as the associated current commutators.

III. A BERENDS-GIELE RELATION
The semi-on-shell amplitude is defined as J a1···an,a (p 1 , · · · , p n ) = 0|π a (0)|π a1 (p 1 ) · · · π an (p n ) . (7) Such objects were considered first by Berends and Giele in Ref. [24] as building blocks for computing S-matrix elements in Yang-Mills theories. In SU (N ) nlσm they were studied in Ref. [13] and a Berends-Giele type recursion relation was proposed using Feynman vertices from an effective Lagrangian. Eq. (7) can be obtained from a (n+1)-point correlation function via the Lehmann-Symanzik-Zimmermann (LSZ) reduction on n of the Goldstone fields [23]. We define and perform the LSZ reduction on the n Goldstone bosons by taking the on-shell limit p 2 i → 0, i = 1, · · · , n. We have also performed the Fourier transform with respect to q in the above, so that q = − n i=1 p i after the integration. In doing so, observe that the right-hand side (RHS) of Eq. (5) contains only (n − 1) single particle poles and, therefore, vanishes. The left-hand side (LHS) can be expanded in a power series in 1/f 2 using while the higher order terms are matrix elements of the form 0|Õ a k (q)|π a1 (p 1 ) · · · π an (p n ) , wherẽ The Ward identity now turns into q 2 J a1···an,a (p 1 , · · · , p n ) = ∞ k=1 (−4) k (2k + 1)! 0|Õ a k (q)|π a1 (p 1 ) · · · π an (p n ) , (11) which is valid at the quantum level. SinceÕ a k (q) is proportional to q µ , the Adler's zero is manifest when q µ → 0.
At the classical level, Eq. (11) can be turned into a recursion relation among tree-level semi-on-shell amplitudes. Let's define a (2k + 1)-point tree vertex fromÕ a k , where σ is a permutation of {1, 2, · · · , 2k + 1} and Then the Berends-Giele recursion relation is obtained from Eq. (11) by connecting the vertex in Eq. (12) with either an external leg or the off-shell leg of a sub-semion-shell amplitude, whose on-shell legs are a subset of {p 1 , · · · , p n }. In the end we arrive at Here d ℓ = {d ℓ,i } is a way to divide {1, 2, · · · , n} into 2k + 1 disjoint, non-ordered subsets. The jth element of d ℓ,i is denoted by d ℓ,i j and q d ℓ,i = j p d ℓ,i j .
Semi-on-shell amplitudes in nlσm are not invariant under field redefinitions and depend on the particular parameterization employed to write down the Lagrangian. It is then interesting to highlight the difference between Eq. (14) from that obtained using Feynman diagrams in Ref. [13]. The vertex in Eq. (14) arises from the operator insertion ofÕ a k (q), which carries momentum injection of q µ . In fact, as we will see, the cubic interaction of the extended biadjoint scalar is given by the matrix element of O a 1 (0), which is a three-point vertex. In addition, the explicit Adler's zero in the limit q µ → 0 in Eq. (12) greatly facilitates the calculation of the subleading single soft limit of the nlσm, which we turn to next.

IV. THE SUBLEADING SINGLE SOFT LIMIT
On-shell amplitudes are further obtained from the semi-on-shell amplitudes by where q = −(p 1 + · · · + p n ) ≡ p n+1 is the momentum of the (n + 1)-th leg. Using Eq. (11) it is simple to derive the single soft limit in p n+1 → τ p n+1 , τ → 0: This is the subleading single soft theorem in nlσm, valid at the quantum level. Notice there is no momentum injection at this order in τ and the operator behaves just like a "normal" Feynman vertex, which hints at interpreting the matrix element as scattering amplitudes.
The LHS of Eq. (21) can be turned into an on-shell amplitude by taking q µ = − n i=1 p µ i on-shell. Together with momentum conservation, the flavor-ordered vertex in Eq. (20) can be written as and Eq. (21) becomes M (1, 2, · · · , n + 1) = where p n+1 = q = − n i=1 p i . At this stage Eq. (23) is exact, having only taken the on-shell limit p 2 n+1 = 0. If we further take the soft limit, p n+1 → τ p n+1 , τ → 0, the RHS of Eq. (23) starts at linear order in τ , in accordance with the Adler's zero condition. Notice that at O(τ ), one can simply drop the τ dependence in the subamplitudes, by requiring n i=1 p i = 0. This is the next-to-leading order single soft factor of flavor-ordered tree amplitudes in nlσm.

V. THE CHY INTERPRETATION
In Ref. [16] the subleading single soft limit of flavorordered tree amplitudes in nlσm is studied using the CHY formulation of scattering equations [17][18][19]. The single soft limit is interpreted as relating the (n + 1)-point amplitudes in nlσm to the n-point amplitudes of a related, but different theory containing cubic interactions of biadjoint scalars. Specifically, at O(τ ), the proposal is where s ij = 2p i · p j . M (I n+1 ) is the (n + 1)-point flavorordered amplitude in nlσm with the ordering I n+1 = {1, 2, · · · , n + 1} and M nlσm⊕φ 3 (I n |1, n, i) denotes the npoint amplitudes of nlσm interacting with a cubic biadjoint scalar, where {1, n, i} is the flavor-ordering of the second adjoint index in the biadjoint scalar. Little is known about the nature of this "extended theory," and only the CHY representation of the flavor-ordered onshell amplitudes is given. Our results in the previous sections shed light on the interactions, in particular the Feynman vertices, of the extended theory. First of all, the emergence of a cubic scalar interaction is evident already in Eq. (16). Using the 4-point amplitude as an example and set p 4 = q as the soft momentum, the full tree amplitude from Eq. (16) is where we have used the on-shell condition, q 2 = 0, and momentum conservation, q = −(p 1 +p 2 +p 3 ). Setting the decay constant f = 1, and extracting the flavor-ordered single-soft factor using the CHY proposal in Eq. (24), we obtain the cubic interaction which agrees with the CHY representation of 3-point amplitude given in Ref. [16]. Tracing back the appearance of the cubic interaction we see it is rooted in the order 1/f 2 term in Eq. (5), which is a cubic operator.
To study Eq. (23) in the context of the CHY proposal, which only gives the flavor-ordered tree amplitudes but not Feynman rules, we make the following observations regarding the flavor-ordered Feynman rule of the biadjoint scalar φ: 1) no vertices exist with only one φ, 2) a flavor-ordered m-point vertex containing two φ's has the same flavor-ordered Feynman rule as in the nlσm, and 3) a (2k + 1)-point vertex involving three φ's has the following flavor-ordered Feynman rule V nlσm⊕φ 3 (1, 2, · · · , j, · · · , 2k + 1|1, 2k + 1, j) where p 1 , p j and p 2k+1 are the momenta of φ's. Similar to the 3-point vertex in Eq. (26), the (2k+1)-point vertex can be seen as emerging from the order 1/f 2k term in the Ward identity in Eq. (5). Using these Feynman rules, it is possible to show that the coefficients of s n+1,i in Eq. (23) are precisely the amplitudes in nlσm⊕φ 3 , in accordance with the CHY proposal in Eq. (24). In other words, these coefficients have consistent factorization and can be interpreted as scattering amplitudes [22].

VI. CONCLUSION
In this work we have explored consequences of nonlinear shift symmetries in nlσm and presented the associated Ward identity, which allowed us to study various aspects of scattering amplitudes in nlσm. In particular, we derived a next-to-leading order single soft theorem and studied the subleading single soft factor for flavorordered tree amplitudes, which provided a new perspective on the mysterious extended theory of cubic biadjoint scalars interacting with the Goldstone bosons.
There are many future directions. One example is whether the interpretation of an extended theory can be applied to the full scattering amplitudes of nlσm, instead of just the flavor-ordered amplitudes. Naively there is an obstacle in doing so, since the LHS of Eq. (24) carries one flavor index, while the biadjoint amplitude in the RHS carries two flavor indices. Another possibility is to extend the Ward identity to shift symmetries involving spacetime, and understand their soft theorems and the associated extended theories. Additionally, there is a new formulation of nlσm which makes the flavorkinematic duality transparent [25], in which the subleading soft theorems and the cubic biadjoint scalars can be accommodated. It would be interesting to understand the connection with the shift symmetry perspective.