Soft Bootstrap and Effective Field Theories

The soft bootstrap program aims to construct consistent effective field theories (EFT's) by recursively imposing the desired soft limit on tree-level scattering amplitudes through on-shell recursion relations. A prime example is the leading two-derivative operator in the EFT of $\text{SU} (N)\times \text{SU} (N)/\text{SU} (N)$ nonlinear sigma model (NLSM), where $ \mathcal{O} (p^2)$ amplitudes with an arbitrary multiplicity of external particles can be soft-bootstrapped. We extend the program to $ \mathcal{O} (p^4)$ operators and introduce the"soft blocks,"which are the seeds for soft bootstrap. The number of soft blocks coincides with the number of independent operators at a given order in the derivative expansion and the incalculable Wilson coefficient emerges naturally. We also uncover a new soft-constructible EFT involving the"multi-trace"operator at the leading two-derivative order, which is matched to $\text{SO} (N+1)/ \text{SO} (N)$ NLSM. In addition, we consider Wess-Zumino-Witten (WZW) terms, the existence of which, or the lack thereof, depends on the number of flavors in the EFT, after a novel application of Bose symmetry. Remarkably, we find agreements with group-theoretic considerations on the existence of WZW terms in $\text{SU} (N)$ NLSM for $N\ge 3$ and the absence of WZW terms in $\text{SO} (N)$ NLSM for $N\neq 5$.


I. INTRODUCTION
Soft bootstrap is a program to construct consistent EFT's by recursively imposing the desired soft limit on tree-level amplitudes. The first attempt dates back almost half-acentury ago in the context of pions in low-energy QCD, whose scattering amplitudes exhibit vanishing soft behavior known as the Adler's zero condition [1]. It was first shown by Susskind and Frye in Ref. [2] that, starting from the 4-point (pt) amplitude at O(p 2 ), both 6-pt and 8-pt amplitudes could be constructed by recursively imposing the Adler's zero. The relation of such an approach with the current algebra was clarified in Ref. [3].
At first glance it is surprising that tree amplitudes of pions could be constructed this way, as the Adler's zero condition makes no reference to the SU(2) L × SU(2) R chiral symmetry that is spontaneously broken; the only inputs are IR data. Operationally Susskind and Frye worked with flavor-stripped partial amplitudes, which is special to pions in the adjoint representation of SU(N) in that the color factor factorizes simultaneously with the kinematic factorization channel. One then computes the n-pt partial amplitude by connecting lower-pt amplitudes through a single internal propagator and summing over all factorization channels.
The resulting amplitude does not have the correct soft limit and an n-pt contact interaction is added by hand and fully constrained by the Adler's zero condition. Sample Feynman diagrams used in Ref. [2] to bootstrap the 6-pt amplitude are shown in Fig. 1. Going to higher multiplicities in n makes the procedure quite cumbersome, as higher-pt contact interactions have to be implemented manually, and little progress was made in the ensuing four decades.
More recently there are two new developments which shed new light on the soft bootstrap program. One is the realization that, for Nambu-Goldstone bosons (NGB's) arising from a global symmetry G spontaneously broken to a subgroup H, the effective Lagrangian depends only on the particular linear representation furnished by NGB's in the unbroken group H, and is independent of G, up to the normalization of the decay constant f [4,5]. More specifically one imposes a nonlinear shift symmetry in the IR and recursively constructs higher dimensional operators that are invariant under the shift symmetry. The Ward identity of the shift symmetry leads precisely to the Adler's zero condition on S-matrix elements [6], at the leading order in 1/f . Thus the nonlinear shift symmetry embodies the soft bootstrap program in the Lagrangian approach.
On a separate front, new progresses in the modern S-matrix program lead to on-shell recursion relations for EFT's exhibiting vanishing soft limit [7,8], which include NGB's and other more exotic scalar theories. The soft recursion relation allows one to compute the n-pt amplitudes directly using factorization channels involving lower-pt sub-amplitudes and sidesteps the need to introduce the n-pt contact interaction manually, which greatly streamlines the calculation. The on-shell soft bootstrap program initially focused on single scalar EFT's, and have since been expanded to supersymmetric theories whose scalar component corresponds to the scalar EFT's with vanishing soft limit [9]. Other related works on soft bootstrap can be found in Refs. [10][11][12][13][14][15][16][17][18][19][20][21].
In the context of NGB's, discussions in the on-shell approach so far concentrate on the leading two-derivative operator in the EFT, naturally. However, the essence of EFT lies in the existence of higher-derivative interactions which become more and more important toward the UV. The higher derivative operator each comes with an incalculable Wilson coefficient encoding the unknown UV physics. In addition, these operators also have more complicated flavor structures that are of multi-trace in nature. It is then interesting to expand the on-shell soft bootstrap program to higher derivative operators and study how these different aspects of EFT's arise from the IR, which is the aim of this work.
More broadly, studying higher derivative operators from the on-shell perspective could have far reaching implications on several other fascinating aspects of modern S-matrix program. One is the "double-copy" structure [22] that is prevalent among many quantum field theories, including scalar EFT's [23][24][25][26][27][28][29]. This structure is manifest in the Cachazo-He-Yuan (CHY) representation of scattering amplitudes for massless particles [30,31]. However, most of the studies so far are confined to the leading operator in the derivative expansion, except for Refs. [32,33] which considered higher derivative operators in gauge theories and gravity.
For the NLSM, work has only been done in the special cases when the higher order correc-tions satisfy certain properties, such as Bern-Carrasco-Johansson (BCJ) relations [9,24] or subleading double soft theorems [20]. Part of the purpose of this work is to initiate a study on the most general higher derivative operators in scalar EFT's in the on-shell approach.
An interesting aspect of higher derivative operators is that often they involve color/flavor factors that are of multi-trace in nature. Most quantum field theories studied by the "scattering amplitudes" community currently involve fields that carry no color/flavor charges or transforming under the adjoint representation of SU(N) color/flavor group. This is due to the U(1) decoupling relation for the adjoint of SU(N) theory [34], which allows the simulta- This work is organized as follows. In Section II we first consider soft-bootstrapping the leading two-derivative operator in EFT's with vanishing soft limit, with a focus on multiscalar EFT's and pointing out new subtleties involved. We introduce the notion of a "soft block" here, which serves as the seed of soft bootstrap, and present a new EFT with a flavor structure that is different from the commonly studied SU(N) adjoint representation.
In Section III we introduce soft blocks at the four-derivative order, which include both the parity-even and the parity-odd soft blocks, and study their soft-bootstrap. The parityodd soft block obviously maps to the Wess-Zumino-Witten term in the NLSM. Each soft block at this order comes with an undetermined free parameter, which is later shown to correspond to the Wilson coefficient in the EFT language. It is also in this section where we demonstrate the existence of two consistent EFT's in soft bootstrap. Then in Section IV we explicitly match these two EFT's to NLSM's based on the coset SU(N) × SU(N)/SU(N) and SO(N + 1)/SO(N), which is followed by the summary and outlook. We also provide two appendices on multi-trace flavor-ordered partial amplitudes and the IR construction of NLSM effective Lagrangians.

II. LEADING ORDER SOFT BOOTSTRAP
A. An Overview The modern approach to the soft bootstrap of EFT's was initiated in Refs. [7,8], which considered various kinds of one-parameter scalar effective theories. Much of the discussion there focused on single scalar EFTs. Since our main focus is NLSMs with multiple scalars, here we give an overview of soft bootstrap procedure adapted to multi-scalar EFTs.
We use the notation Φ = {φ 1 , φ 2 , · · · } to denote a generic set of scalars. By construction an EFT consists of an infinite number of operators organized with increasing powers of derivatives and fields. In general the EFT is a double expansion in the two parameters: where Λ and f are two mass scales which characterize the derivative expansion and the field expansion, respectively. Then the effective Lagrangian will have the schematic form where we have suppressed the Lorentz indices on the derivatives and Lorentz invariance implies an even number of derivatives. In addition [Φ] n denotes generic contractions of n φ i scalars. So at a given (2m, n) there could be many Wilson coefficients c 2m,n . The overall factor Λ 2 f 2 is dictated by a canonically normalized kinetic term for (∂φ) 2 , which also requires c 2,2 = 1.
A prime example of EFTs in the soft bootstrap program is the leading two-derivative operator in SU(N) × SU(N)/SU(N) NLSM, which describes a set of massless scalars transforming under the adjoint representation of SU(N). One major advantage of working with SU(N) NLSM is the existence of flavor-ordered partial amplitudes with a simple factorization property [6], much like the color-ordered partial amplitudes in the SU(N) Yang-Mills theory. The full amplitude at order O(p 2 ) can be written as M a 1 ···an (p 1 , · · · , p n ) ≡ σ∈S n−1 C a σ(1) ···a σ(n−1) an M(σ(1), · · · , σ(n − 1), n) , · · · a n−1 a n ∼ · · · · · · a 1 a k a k+1 a n where C a 1 a 2 ···an = tr (T a 1 T a 2 · · · T an ) , is the flavor factor, σ is a permutation of indices {1, 2, · · · , n − 1} and T a is the generator of SU(N) group. The full amplitude is permutation invariant among all external legs, while the flavor-ordered partial amplitude M(1, 2, · · · n) is invariant only under cyclic permutations of the external legs because of the cyclic property of the trace in Eq. (4).
At the two-derivative level, the flavor factor C a 1 a 2 ···an can always be written as a single trace operator involving T a . 1 Furthermore, the SU(N) generators satisfy the following completeness relation: a I tr (T a 1 · · · T a k T a I ) × tr (T a I T a k+1 · · · T an ) = tr (T a 1 T a 2 · · · T an ) − 1 N tr (T a 1 · · · T a k−1 ) × tr (T a k · · · T an ) .
In a tree diagram the disconnected 1/N term does not contribute due to the decoupling relations of the U(1) Goldstone boson in U(N) × U(N)/U(N) NLSM. (See, for example, Ref. [34].) This is an important property for the on-shell recursion relation, which expresses a higher-pt partial amplitude as the product of two or more lower-pt sub-partial amplitudes.
When dressing up each sub-partial amplitude with its own flavor factor, the recursed higherpt amplitude now has the following flavor factor: which is a single trace flavor factor that preserves the ordering of sub amplitudes. Note the summed-over index a I arises from the internal propagator iδ ab /p 2 , as shown in Fig. 2. This connection between soft recursion of partial amplitudes and proper factorization of flavor factors is very special for the adjoint representation of SU(N) group, and the main reason why so far most studies on NLSM utilizing partial amplitudes assume the field content to transform as the adjoint of SU(N). For example, generators in the adjoint of SO(N) group do not have as nice a completeness relation as in Eq. (5), and the corresponding partial amplitudes do not enjoy the simple factorization property. However, later we will see a new possibility to make compatible the partial amplitudes with factorization using multi-trace operators, which is matched to the SO(N + 1)/SO(N) coset, where the NGB's transform as the fundamental representation of SO(N).
Given the partial amplitudes, the soft bootstrap program then constructs higher-pt amplitudes from lower-pt amplitudes by using the soft recursion relation [7], which utilizes an all-leg shift for the external momenta of an n-pt amplitude M n , where z is a complex shift parameter and total momentum conservation requires, choosing all momenta to be incoming, The deformed amplitudeM n (z) with momentum variablesp i is still an on-shell amplitude, in the sense that all external momenta remain on-shell, and taking the soft limit of p i corresponds to setting z → 1/a i .
In D-dimensional spacetime Eq. when a solution exists, rescaling all a i 's simultaneously lead to a "degenerate" solution. In the end, the general solution for a i 's can be written as where {a (r) i }, r = 1, · · · , n − 5, are the non-trivial solutions which can be expressed in terms of kinematic invariants of external momenta, while A (r) and B are arbitrary constants reflecting the re-scaling degrees of freedom in both the non-trivial and trivial solutions.
It is rather intriguing that the solutions for a i 's have a "shift symmetry" and are defined "projectively." Since the Adler's zero condition requires the amplitude in NLSM to vanish linearly in the soft momentumM n (z) ∼p forp → 0 [1], one can define a soft factor, so that the Cauchy integral below always vanishes, dz zM This is because the integrand at large z vanishes like O(z 1−n ) and the residue at the infinity is zero. As a consequence of the Adler's zero condition, the integrand has no poles at z = 1/a i and the only poles come from z = 0 as well as the factorization channel I, the shifted internal momentum of the corresponding propagator beingP I (z) = P I + zQ I , where The residue theorem then relates the residue at z = 0, which is nothing but the n-pt amplitude M n , to the other residues atP 2 I (z ± I ) = 0: where M L and M R are the two lower-pt on-shell amplitudes associated with the factorization channel I. Therefore, starting with some "seed" amplitudes that does not factorize, one can use Eq. (13) to recursively construct on-shell amplitudes of all multiplicities in the theory.
We will refer to Eq. (13) as the soft recursion relation.
which has poles at z = 0 and z = 1/a i . The poles at z = 1/a i comes about because the individual factorization channel I does not satisfy Adler's zero condition in the soft limit; only after summing over all factorization channels is the Adler's condition satisfied and the 1/a i poles disappear. Then a second application of the residue theorem relates the residues of Eq. (14) at a = z ± I to the residues at z = 0, 1/a i : Res z=1/a iM (I) The first term on the right-hand side represents all Feynman diagrams contributing to M n that contain an internal propagator. The second term, on the other hand, must then be a local function of external momenta and relates directly to the n-pt contact operator in the effective Lagrangian.
The soft bootstrap program is predictive only when the higher-pt amplitudes constructed using the soft recursion relation are independent of the arbitrary coefficients A (r) and B in the general solution of a i 's in Eq. (9). Otherwise we would introduce more and more unknown parameters as we go to higher-pt amplitudes. Therefore, we define a consistent EFT in soft bootstrap to be when The amplitude M n obtained from the soft recursion relation is independent of the arbitrary constants A (r) and B for all n.
Otherwise the EFT one is trying to construct using soft bootstrap simply does not exist.

B. Introducing the Soft Blocks
At this point it is convenient to introduce the notion of a "soft block," • A soft block S (k) (p 1 , · · · , p n ) is a contact interaction carrying n scalars and k derivatives that satisfies the Adler's zero condition when all external legs are on-shell.
Because the soft blocks themselves satisfy the Adler's zero condition, they can be used as a seed amplitude in the recursion relation. As such, the soft block is an input to soft bootstrap.
For k ≤ 4, which we focus on in this work, the soft blocks exist only for n = 4 and n = 5.
It cannot exist for n = 3 because there is no non-trivial kinematic invariant built out of three on-shell real momenta satisfying total momentum conservation. Beyond n = 3, let's perform an all-leg-shift as in Eq. (7) on the external momenta. The "shifted block" is now a polynomial of degree k in z:Ŝ (k) =Ŝ (k) (z). However, in D = 4 there exists non-trivial We use the convention that a solid line represents an on-shell scalar particle, while a dashed line represents an off-shell scalar.
solutions for a i 's only when the number of external legs n ≥ 6. SinceŜ n (z) satisfies the Adler's zero condition by assumption, it must havê In other wordsŜ Next we identify soft blocks at O(p 2 ) that are invariant under cyclic permutations of all external legs, which we refer to as the single-trace soft block. Working with partial amplitudes, we start with the 4-pt flavor-ordered soft block S (2) (1, 2, 3, 4) such that 1. S (2) (1, 2, 3, 4) is quadratic in external momenta. (2) (1, 2, 3, 4) satisfies the Adler's zero condition.
At 4-pt level there are only two independent kinematic invariants s 12 and s 13 , where s ij ≡ (p i + p j ) 2 . Writing down the most general kinematic invariant and imposing the second and the third conditions lead to a unique soft block, up to total momentum conservation, where c 0 is a constant parameter. Using momentum conservation one could rewrite the righthand side as either s 24 , thereby exhibiting the cyclic property. This soft block is presented in Fig. 3 and corresponds to a two-derivative operator of the form in the effective Lagrangian in Eq. (2).
One could ask if there is a 5-pt soft block at the two-derivative level. In this case there where P i 1 i 2 ···in ≡ p i 1 + p i 2 + · · · + p in . This is a consistent amplitude because it does not depend on A (r) and B in the general solution for a i 's.
In soft-bootstrap the 6-pt amplitude is given by three factorized soft blocks shown in Fig. 4. Here we wish to make a distinction between "factorized soft blocks," which contribute to the right-hand side of soft recursion in Eq. (13), and the Feynman diagrams which may contain non-factorizable contact terms. More explicitly, we show in Fig. 1 the Feynman diagrams contributing to the 6-pt amplitude, which have three factorizable diagrams and one non-factorizable 6-pt contact term. Contributions from the three factorized soft blocks in Fig. 4 is equal to the four Feynman diagrams in Fig. 1.
The 6-pt contact interaction bootstrapped from the soft block is of the form whose presence makes the 6-pt amplitude conform to the Adler's zero condition. At the Lagrangian level, the structure and the coefficient of this operator is constrained by a shift symmetry in the IR construction of effective Lagrangians [4,5]. Once the 6-pt amplitude is soft-bootstrapped, one then proceeds to higher-pt amplitudes using in Eq. (13).
The preceding discussion on the 6-pt amplitude leads to the question: what operators in the effective Lagrangian in Eq. (2) can be soft-bootstrapped from S (2) (1, 2, 3, 4)?
To answer this question, we need to digress a little bit and recall some definitions to characterize the property of a Feynman diagram. Let's use b i and d i to represent the number of scalars and derivatives, respectively, carried by a particular operator O i in Eq. (2).
Generically a tree Feynman diagram with n external legs can be expressed as a rational function of degree d in external momenta. Then n and d are given as which comes about by counting the number of unconstrained momenta in a diagram: each internal propagator has a momentum integral and each insertion of vertex has a momentum delta function, and one delta function simply enforces total momentum conservation. Setting L = 0 to replace I B in Eqs. (21) and (22) we have If all operators in Eq. (2) are such that is a fixed non-negative rational number, then every Feynman diagram in the EFT will also have (d − 2)/(n − 2) = ρ, as can be seen from plugging ρ into Eqs. (24) and (25). In this sense operators carrying a definitive ρ form a closed set among themselves. The parameter ρ was first introduced in Refs. [7,8] to characterize a particular power counting order in derivative. 2 The operator corresponding to the soft block S (2) (1, 2, 3, 4) has n i = 4, d i = 2, and ρ = 0.
Thus all operators with ρ = 0 can be soft-bootstrapped from this particular soft block.
This can also be seen explicitly from Eq. (25): an arbitrary number of insertions of the two-derivative soft block will generate an amplitude with two powers of external momenta.
In addition, since we start with a soft block with n i = 4, Eq. (24) shows the number of external legs n must also be an even number. We conclude that the Wilson coefficients of It is worth commenting early on that a power counting in terms of ρ is of limited use for multi-scalar EFTs, especially when going beyond leading two-derivative order in NLSMs. In particular, O(p 4 ) operators in NLSMs involve an infinite number of operators carrying four derivatives but an arbitrary number of scalar fields, as we will see later. The reason a power counting based on ρ is useful for single scalar EFT's, at least for the leading interactions, is because there is a unique two-derivative operator that is the kinetic term. In fact, in single scalar EFT's all two derivative operators of the form, can be removed by a field re-definition, φ → φ + F (φ), for a suitably chosen F (φ). Equivalently, tree amplitudes of n ≥ 3 identical massless scalars must vanish at O(p 2 ). This can be seen easily because Bose symmetry requires the amplitude must be completely symmetric in interchange of any two external momenta, which vanishes due to total momentum conservation. However, when there is more than one In the previous subsection we constructed a soft block that is invariant under cyclic permutations of all external legs. One could ask if this assumption can be relaxed. Indeed here we consider a new soft block S (2) (1, 2|3, 4) that satisfies the first two requirements in Section II C and • S (2) (1, 2|3, 4) is invariant under separate cyclic permutations of (1, 2) and (3, 4).  The diagram on the right is the contact term.
We do not consider a soft block that is invariant under only cyclic permutations of three external legs such as S (2) (1, 2, 3|4), which would imply the soft block is not neutral under the flavor charge. We call S (2) (1, 2|3, 4) the "double trace" soft block, which is given by up to total momentum conservation. Diagrammatically we present the double trace soft block as in Fig. 5. Similar to the single trace case, we do not find any 5-pt soft blocks that are quadratic in external momenta.
Using S (2) (1, 2|3, 4) we can construct 6-pt amplitudes with different flavor-orderings. For example, using Eq. (15) we obtain the partial amplitude M (2) (1, 2|3, 4|5, 6) that is invariant under three separate (cyclic) permutations in (1, 2), (3,4) and (5,6). There are six factorized soft blocks shown in Fig. 6. It is important to draw a contrast with the three factorized blocks, in the case of single trace soft block, shown in Fig. 4. The additional factorized blocks, shown in the second row of Fig. 6, come about because we need to symmetrize with respect to the flavor indices (ij) that are connected through the internal propagator, so as to make the amplitude invariant under the cyclic permutation (ij) → (ji). In fact, if we only included the three factorized blocks in Fig. 4 for the double-trace soft blocks, the a i 's dependence would not cancel and the resulting amplitude is inconsistent. Only after summing over all six contributions in Fig. 6 did we arrive at a consistent amplitude, where the contact term M (2),c is given by Therefore, this particular 6-pt amplitude is which is manifestly invariant under the three separate cyclic permutations and satisfies the Adler's zero condition. It is interesting that requiring the recursed amplitude to be independent of a i forces a flavor structure that is of the "triple trace" nature.
Eq. (31) is a different amplitude from the 6-pt amplitude in Eq. (19), which is bootstrapped from the single trace soft block. The corresponding Feynman diagrams are shown in Fig. 7, which include the factorization channels as well as the 6-pt contact interaction.
In computing the 6-pt amplitude we do not have to explicitly plug in the solutions for a i because of the specialized recursion relation in Eq. (15). When going to 8-pt amplitude where the contact term is Res z=1/a i 1 zF 6 (z)ŝ 12 ŝ 46 If the "mixed" EFT exists, this expression needs to be independent of the constants A (r) and B when plugging in the general solution for a i 's in Eq. (9).
It is easy to see that the contact interaction However, it is not independent of B. This can be verified using a set of momenta numerically.
For example, we can choose the momenta p i to be, in arbitrary units, which satisfy p 2 i = 0 and i p i = 0. The general solution for a i , up to the overall scaling factor, can be written as Plugging the above into Eq. (35) we arrive at indicating that we cannot soft-bootstrap M (2) (1, 2|3, 4, 5, 6) using both soft blocks at O(p 2 ).
Therefore, the single-trace and double-trace soft blocks at O(p 2 ) cannot co-exist and there is no consistent EFT that follows.

III. HIGHER ORDERS IN DERIVATIVE EXPANSION
So far we have seen that soft bootstrap allows one to construct two-derivative operators in EFT that are to all orders in 1/f , with the only free parameter being c 0 or d 0 , which can be absorbed into the overall normalization of the scale f . In the Lagrangian approach, these operators resum to a single nonlinear operator invariant under the transformation of the nonlinear shift symmetry [4,5]. As is well-known, operators in the EFT of NLSM is organized in terms of an increasing powers of derivatives. At the leading two-derivative order, there is only one operator whose coefficient is fixed by the requirement of a canonically normalized scalar kinetic term. At higher orders in the derivative expansion, there exist several nonlinear operators in general, each with an incalculable Wilson coefficient encoding the unknown UV physics. Can the soft bootstrap program be extended to these higher derivative operators?
How do the unknown Wilson coefficients emerge in the soft bootstrap? A particularly interesting class of operators is the Wess-Zumino-Witten (WZW) term that captures the effect of anomaly in a NLSM [35,36]. Can the WZW term be soft-bootstrapped?
From the Lagrangian approach it seems the answer to above questions should be a definitive "yes." It is known that the Adler's zero condition corresponds to the Ward identity of the shift symmetry at the leading order in 1/f , which in turn is associated with the existence of degenerate vacua and the phenomenon of spontaneous symmetry breaking [19]. In the following we study soft bootstrap at higher orders in the derivative expansion.
4 , which has n i = d i = 4 in Eqs. (24) and (25). In terms of the derivative power counting parameter defined for single-scalar EFT's in Refs. [7][8][9], it has from single-scalar EFT's turns out to be incorrect. This is because there are two unknown vertices that would enter the 6-pt amplitude at the O(p 6 ): one has a multiplicity of 4 and the other has a multiplicity of 6, as shown in the second row of Fig. 9. Therefore, imposing the Adler's zero condition cannot determine the individual Wilson coefficients of these two O(p 6 ) vertices. In fact, the essence of the soft bootstrap program relies on the property that, at a given order in the derivatives and the multiplicity of external particles, only one unknown vertex would enter the amplitude, which is demonstrated in the first row of Fig. 9.
Consequently, the Adler's zero condition uniquely determines the coefficient of the unknown vertex. The program fails when two unknown vertices enter at the same time.
In fact, intuitions from the Lagrangian approach makes it clear that the nonlinear shift symmetry relates operators containing the same number of derivatives but an arbitrary number multiplicity in external fields [4,5].
To understand the situation more properly, one should go back to Eq.
can be soft-bootstrapped from S 4 , consistent with the expectation from nonlinear shift symmetry. In a general NLSM, operators with different ρ mix under soft bootstrap.
Using 4-pt and 5-pt soft blocks, we can start building 6-pt and higher-pt amplitudes using the soft recursion relation. Note thatM n (z) in the integrand of Eq. (11) is O(z 4 ) at large z, so that the integrand is at O(z 3−n ), and there is no pole at z → ∞ for n ≥ 6. Therefore, the on-shell recursion relation is still valid and can be used for soft bootstrap.
2 (1, 2, 3, 4) = Double-trace: S 2 (1, 2|3, 4) = where we have introduced four free parameters: c 1 , c 2 , d 1 and d 2 . The power counting of mass scales in these soft blocks is given by Eq. (2). Notice that these soft blocks not only satisfy the Adler's zero condition, but the soft degrees of freedom seem to be enhanced due to the increasing power of momenta. However, at this order the soft blocks themselves are not on-shell amplitudes, which still only vanish linearly in the soft momentum.
In Section II C we showed that there is no where M (2) (1, 2, 3, 4, 5, 6) is the single trace 6-pt amplitude at O(p 2 ) in Eq. (19), and The factorization channels, as well as the Feynman diagrams, of the above two kinds of which can be built using Eq. (13). We checked numerically that the amplitudes are consistent and independent of A (r) and B in the general solutions of a i . This indicates a consistent EFT can be built using the soft blocks At this order in derivative expansion, EFT 1 contains 4 free parameters {c 1 , c 2 , d 1 , d 2 } in Eqs. (41) and (42), with c 0 being absorbed into the normalization of f . We will see in Section IV that these four free parameters correspond precisely to the four Wilson coefficients in the SU(N) NLSM at O(p 4 ) order.
Next we consider the other case: and it has two free parameters {d 1 , d 2 } in Eq. (42). In Section IV we will match EFT 2 to the SO(N) NLSM, which has two Wilson coefficients at O(p 4 ).

C. 5-pt Soft Blocks: Wess-Zumino-Witten Terms
In this section we consider soft blocks with 5 external legs at O(p 4 ). We find one single trace soft block that is parity-odd, The expression is invariant under cyclic permutations upon total momentum conservation.

S (4)
− clearly corresponds to the Wess-Zumino-Witten (WZW) term [35,36], which accounts for the anomaly that may arise in a NLSM in D = 4. It is well-known that in the CCWZ construction the existence of WZW term, or the lack thereof, depends on the existence of a rank-5 totally anti-symmetric invariant tensor in the coset G/H [37,38]. Such information is clearly not available in soft bootstrap. Nevertheless we will see soon that the group-theoretic considerations based on G/H can be exactly reproduced in a remarkable way, after taking into account the Bose symmetry in the IR.
The WZW term has been considered previously in Refs. [8,9], however, only the leading 5-pt vertex in 1/f expansion was discussed. Here we are interested in soft-bootstrapping higher-pt amplitudes that are of O(p 4 ), using the WZW soft block. These amplitudes correspond to interactions that are higher orders in 1/f in the WZW term. What vertices can be soft-bootstrapped from S For example, there is only one unknown vertex, the 7-pt contact interaction, contained in the Feynman diagrams contributing to the 7-pt amplitude, as shown in Fig. 11. The Adler's zero condition then fixes the 7-pt vertex uniquely. These operators make up the WZW term to all orders in 1/f . Again, we need to discuss separately EFT 1 and EFT 2 . In EFT 1 we use S where the contact term is M (4),c (1, 2, 3, 4, 5|6, 7) Res z=1/a i 1 zF 7 (z)ŝ 67 ε(1234) .
It is easy to check numerically that the a i dependence does not cancel in the above when plugging in the general solution in Eq. (9). This indicates the absence of the WZW term in SO(N) NLSM, in general.
There is a subtlety in the preceding arguments, which involves the number of flavors N f and the Bose symmetry. If N f ≤ 5 in the EFT, two or more scalars in Eq. (49) are identical and Bose symmetry requires the amplitude must be symmetric in external momenta of identical scalars. As a result, the WZW soft block vanishes due to the anti-symmetric Levi-Civita tensor used in contracting the external momenta. Therefore, we arrive at the important observation: • S     M (1, 2, 3, 4, {5, 6, 7, 8, 9}). In the end the WZW amplitude in EFT 2 for N f = 5 has only one flavor structure that involves the cyclic permutation of the 5 distinct flavors.

IV. MATCHING TO LAGRANGIANS
Having constructed EFT 1 and EFT 2 in soft-bootstrap up to O(p 4 ), we match these two theories to effective Lagrangians of NLSM in this section. The general and top-down approach in the Lagrangian formulation for such effective interactions is given by Callan, Coleman, Wess and Zumino half-a-century ago [39,40]. The CCWZ construction requires knowledge of a spontaneously broken group G in the UV and an unbroken group H in the IR. The generators of G include the "unbroken generators" T i , which are associated with H, as well as the "broken generators" X a , which are associated with the coset G/H. The NGB's are then coordinates parameterizing the coset G/H. We will sometimes refer to CCWZ as the "the coset construction." At first sight it may seem rather improbable that soft bootstrap could (re)construct effective Lagrangians in the CCWZ approach, since one makes no reference to a spontaneously broken group G in the UV in soft bootstrap; all that is needed is the Adler's zero condition, an IR property of on-shell amplitudes. Indeed, the coset construction completely obscures the "infrared universality:" effective interactions of NGB's are dictated by their quantum numbers in the IR and independent of the broken group G in the UV.
Only recently was it realized that an IR construction of effective Lagrangians exists, without reference to the spontaneously broken symmetry G, which makes use of nonlinear "shift symmetries" acting on a set of massless scalars π a furnishing a linear representation of the (unbroken) group H [4,5]. It turns out that imposing the shift symmetry in the Lagrangian is equivalent to imposing Adler's zero condition on the on-shell amplitudes, which arises as a consequence of the Ward identity for the shift symmetry [41,42]. In this sense the IR construction can be viewed as the realization of soft bootstrap in the Lagrangian formulation [19]. Since the IR construction is more similar to the soft bootstrap program in philosophy, we will adopt the IR approach to consider effective Lagrangians corresponding to EFT 1 and EFT 2 .

A. The leading two-derivative Lagrangian
As a warm-up exercise to the eventual discussion of O(p 4 ) operators, as well as to set the notation, we briefly consider the leading two-derivative Lagrangian of NLSM. Consider a set of scalars π a transforming as a linear representation of an unbroken group H. Introducing the bra-ket notation (|π ) a = π a , we have where T i is the generator of H in the particular representation under consideration. Moreover, we will choose a basis such that T i is purely imaginary and anti-symmetric: We are interested in constructing an effective Lagrangian invariant under the following nonlinear shift symmetry [4,5], where (|ε ) a = ε a represents an infinitesimal constant "shift" in π a . Eq. (61) at the leading order is simply whose Ward identity leads to the Alder's zero condition [41,42]. Terms that are higher order in 1/f are dictated by the unbroken H-symmetry and the vanishing of n-pt tree amplitudes among identical massless scalars in Eq. (29).
The building block of effective Lagrangians consists of two objects: where In the above d a µ transforms covariantly under the shift symmetry in Eq. (61) while E i µ transforms in the adjoint representation of H like a "gauge field," where the specific form of h A (ε, π) does not concern us here. Then the leading two-derivative operator is unique: where the coefficient is fixed by canonical normalization of the scalar kinetic term.
A general discussion on four-derivative operators in the NLSM effective action is delegated to Appendix B. We note here that in the literature they have been enumerated in two contexts: chiral Lagrangian in low-energy QCD [43,44] and nonlinear Lagrangian for a composite Higgs boson [45][46][47][48]. In the former case π a 's furnish the adjoint representation of SU(N) group, while in the latter π a 's transform as the fundamental representation of SO(N) group. It turns out that these are precisely the two cases that are matched to EFT 1 and EFT 2 , respectively, which we will consider in the next two subsections.
We would like to finish this subsection with the power counting scheme based on the naive dimensional analysis (NDA) [49]. In the EFT of NLSM, each derivative ∂ µ , and as a result, d µ and ∇ µ = ∂ µ + iE i µ T i , is suppressed by an energy scale Λ; each field π a is suppressed by the coupling constant f . The Lagrangian with a canonically normalized kinetic term is given by Requiring that the change in the coupling of a particular operator due to loop-induced effects to be comparable to the natural size dictated by power counting in Eq. (69), one arrives at which is the cutoff of the effective Lagrangian.

B. Adjoint of SU(N )
In this subsection we consider a set of massless scalars π a 's which transform as the adjoint representation of SU(N) group. In the CCWZ construction this scenario could arise from the coset SU(N) × SU(N)/SU(N) and the O(p 2 ) tree amplitudes has been studied extensively from the on-shell perspective in Ref. [6]. As emphasized in Section II A, the full amplitudes Given that π a 's transform as the adjoint representation, we can write where T a is the generator of SU(N). We see that the leading two-derivative Lagrangian can be written as a single trace operator and the resulting partial amplitudes are symmetric in cyclic ordering of external particles. The two-derivative single trace soft block, S 4 (1234), is precisely the 4-pt vertex following from Eq. (71) [6] and the O(p 2 ) amplitudes bootstrapped from S Notice that there are two single-trace operators {O 1 , O 2 } and two double-trace operators In addition, there is the "parity-odd" WZW term, which can be expressed using the action The WZW term for SU(N) is also a single trace operator. Then the O(p 4 ) Lagrangian can be written as where C i , i = 1, 2, · · · , 5 are the unknown Wilson coefficients encoding the incalculable UV physics.
It is worth noting that, for SU(2) only two out of the four parity-even operators are independent. This is easily seen by using properties of Pauli matrices in the adjoint of SU(2).
For SU (3), three out of the four are independent. This can again be checked explicitly using the Gell-Mann matrices for SU(3). 6 Recall that in EFT 1 there exist five free parameters from the five O(p 4 ) soft blocks: c 1 , c 2 , d 1 , d 2 and c − . We can match the partial amplitudes from Eq. (77) with those from EFT 1 . This is achieved by calculating the 4-pt interactions in the Lagrangian: where we have adopted the shorthand notation Π ≡ π a T a . Thus the 4-pt vertices are generating the soft blocks Comparing with Eqs. (41) and (42), we see that The two single-trace soft blocks, c 1 and c 2 , could soft-bootstrap the amplitudes arising from the two single-trace operators O 1 and O 2 , and similarly for the two double-trace soft blocks and double-trace operators.
Similarly, by calculating the 5-pt vertex contributed by the WZW term we get so that The 7-pt local operator in the Lagrangian is which leads to the 7-pt vertex with cycl denoting the terms generated by cyclic permutation of momentum indices {1, 2, · · · , 7}. The 7-pt amplitude calculated using the Feynman rules completely agrees with Eq. (51). We have seen in Section III C that the corresponding soft block S The group generators satisfy the completeness relation In this case the IR construction of the effective Lagrangian simplifies considerably due to the property where T is defined in Eq. (61). Denote r ≡ π|π /(2f 2 ), we further have which allows one to simplify an arbitrary function F (T ): In this case the Goldstone covariant derivative also simplifies: and the leading two-derivative Lagrangian becomes The important observation following from Eq. (92) is that, because of the completeness relation in Eq. (87), the scalars in L (2) are pair-wise contracted by the Kronecker delta.
Because of the Bose symmetry, the amplitude must be symmetric in exchange of external momenta corresponding to pair-wise contracted scalars. This property agrees with that of the amplitudes soft-bootstrapped from the double-trace soft block at O(p 2 ) in Section II D.
At O(p 4 ), the number of independent operators is enumerated in Refs. [47,48] for the SO(5)/SO(4) coset, although we checked that the counting is valid for all N. 7 Again, using the completeness relation in Eq. (87) all flavor indices are contracted by Kronecker deltas: The number of independent operators matches the number of soft blocks at O(p 4 ) in EFT 2 , which has d 1 and d 2 as the free parameters. Furthermore, using O 1 and O 2 we calculate the 4-pt vertex to be V (4) (1, 2|3, 4) = 4 f 2 Λ 2 [2C 1 p 1 · p 2 p 3 · p 4 + C 2 (p 1 · p 3 p 2 · p 4 + p 1 · p 4 p 2 · p 3 )] , (100) which results in the soft block at O(p 4 ): Comparing with Eq. (42) we are able to identify As for the WZW term, it is shown in Section III C that EFT 2 does not have a n = 5 WZW soft block except for N f = 5, which corresponds to a fundamental representation in SO(5).
Indeed we show in Appendix B that there is no WZW term for the coset SO(N + 1)/SO(N), except for N = 5. For the coset SO(6)/SO(5), the WZW term can be expressed as 7 The non-existence of independent operator O 4 can be proved by using the fact that for the coset SO(N + 1)/SO(N ), all totally symmetric rank-4 tensors, with indices in the adjoint and restricted to ones associated with the "broken generators," can be expressed using Kronecker deltas.
Then the 5-pt operator in the Lagrangian is As the flavor factor is just ε abcde , the partial amplitude is given by with thus c − = 5C 5 . For the 7-pt amplitude, there are at least 3 external states of the same flavor. Suppose a 5 = a 6 = a 7 , the partial amplitude is defined as The 7-pt operator in the Lagrangian is from which we can calculate the 7-pt vertex: Using the 4-pt, 5-pt and 7-pt vertices, we have calculated the 7-pt partial amplitude, which exactly matches the result in Eq. (56) generated by the soft recursion.
We reach the conclusion that EFT 2 = SO(N) Fundamental NLSM.
As emphasized already, this is a new example where the partial amplitudes can be softbootstrapped in a simple manner. In particular, the WZW term in EFT 2 exists only for N = 5, in accordance with the expectation from group-theoretic arguments.

V. SUMMARY AND OUTLOOK
In this work we have considered soft bootstrapping four-derivative operators in a multiscalar EFT which satisfies the Adler's zero condition. We systematically introduced soft blocks, the seeds of soft bootstrap, at both the leading two-derivative order and the fourderivative order. We find 7 soft blocks in total, up to O(p 4 ), which are summarized in Table   I Remarkably, soft bootstrap is able to reproduce these results by considering the number of flavors involved and a novel application of Bose symmetry.
Our success of extending the soft bootstrap program to O(p 4 ) of NLSM strongly suggests that, by using the soft recursion relations, we should be able to construct the full EFT to all orders in the derivative expansion, at least for certain cosets. An advantage of such a method is that, at a given order in derivative expansion, it is remarkably easy to find the general set of independent operators: all we need to do is to enumerate all soft blocks that satisfy certain ordering properties as well as the Adler's zero condition, and make sure that they lead to consistent higher-pt amplitudes. Therefore, we are able to avoid applying the relations of nonlinear symmetries and equations of motion to reduce the number of independent operators in the Lagrangian, which become increasingly complicated when we go to higher orders. In this sense, our work is similar in spirit to recent attempts of classifying higher dimensional operators in the standard model EFT using an amplitude basis [52]. There also exist algorithms that enumerate independent operators in NLSM by utilizing the Hilbert series [53], and one should explore how they are related to the soft blocks.
Going There are many more future directions to consider. In particular, quantum field theories with matter content in the adjoint of SU(N) has been studied heavily in the scattering amplitudes community, because the color/flavor factor factorizes simply in each of the kinematic factorization channel, resulting in simple relations between the full and the partial amplitudes. The fundamental of SO(N) is a new example of quantum field theories enjoying such a nice property as in the SU(N) adjoint theory. It is possible that they may be related, e.g. by dimensional reduction [54]. At O(p 2 ), the flavor factor in SU(N) adjoint theory can be written as a single trace over group generators, resulting in two special properties • The partial amplitude is invariant under cyclic permutation of external particles.
• The factorization channel can only arise from adjacent momenta. Another interesting direction is related to the recent proposal to directly interpret treelevel amplitudes as canonical forms associated with the positive geometry in the space of kinematic invariants [55]. Geometric interpretations are given for a variety of theories, including pions transforming in the adjoint of SU(N) theory at the leading two-derivative order. It remains to be seen whether the O(p 4 ) amplitudes in SU(N) and/or the amplitudes in SO(N) theories can be incorporated in such a narrative. In particular, the scattering form proposed so far is projective. It was remarked earlier that the general solutions to a i 's, defined in the all-line shift in soft recursion relation, are also defined projectively (and enjoy a shift symmetry.) It is natural to wonder if these shift parameters can be given a meaning in the projective geometry in the space of kinematic invariants.
Last but not least, it is intriguing that a purely IR approach like the soft bootstrap could make statements on the existence of the WZW term, or the lack thereof, which relies on group-theoretic arguments previously. One could further ask whether it is possible to derive properties of the Lie group involved in the EFT's based simply on the notion of discrete ordering in soft bootstrap. For example, would it be possible to derive in EFT 1 that the number of flavors could only be N 2 − 1? If we only assume cyclic properties of the partial amplitudes, can one deduce that the flavor factors must satisfy a "Jacobi identity," thereby establishing its group nature?
We leave the study of these questions for future investigations.
The building blocks, d a µ and E i µ , are given in Eqs. (63) and (64). We assume the number of massless scalars is n and the number of generators in H is N g , so that the range of indices are {a, b, · · · } = {1, · · · , n} , {i, j, · · · } = {1, · · · , N g } , where {a, b, · · · } run in the linear representation R furnished by the massless scalars |π and {i, j, · · · } are indices in the adjoint of H. We have chosen a basis so that generators in the representation R is anit-symmetric and purely imaginary, Recall that the generators T i satisfy the Lie algebra where f ijk is the structure constant. The infrared data available to us in the low energies are therefore f ijk and (T i ) ab .
We will define two sets of (N g + n) × (N g + n) Hermitian matrices X and T where A a is an N g × n matrix, B i an N g × N g matrix and C i an n × n matrix: These matrices are defined entirely using IR data. However, it is possible to make connection with the coset construction by the identification using which one sees X a and T i are nothing but the "broken" and "unbroken" generators in the CCWZ construction.
Armed with the IR definition of X a and T i , one can now proceed to define the Cartan-Maurer one-form in the IR, Ω ≡ e iπ a X a /f , Under the nonlinear shift symmetry in Eq. (61), they transform covariantly and inhomogeneously as shown in Eqs. (66) and (67). Using the automorphism X a → −X a and T i → T i , we have Using these expressions we can work out the form of d a µ and E i µ explicitly, by calculating the derivative of the exponential map: where ad X Y ≡ [X, Y ], and 1 − e −ad X ad X = ∞ k=0 (−1) k (k + 1)! (ad X ) k .
Combining with Eqs. (B9) and (B10), we arrive at the expressions for d a µ and E i µ in Eqs. (63) and (64). Two important identities follow from Eqs. (B9) and (B10), where In the geometric construction of a symmetric coset the identities follow from the Maurer-Cartan equation [38].
The leading two-derivative Lagrangian is already presented in Eq. (68). Using d µ , ∇ µ d ν and E µ we write down the following 8 parity-even, O(p 4 ) operators, which implies O 5 and O 6 vanish up to O(∂ 4 ). So in the end we are left with four parity-even operators, O i , i = 1, · · · , 4, in general. However, as emphasized in Section IV, the number of independent operators could be further reduced, depending on the specific group structure, such as in SU (2), SU(3) and SO(N).
The operators considered so far are those that are invariant under the shift symmetry.
There is an operator that varies by a total derivative in the Lagrangian, which is the Wess-Zumino-Witten term [35,36]. To write down the Lagrangian density for the WZW term requires compactifying the spacetime to a 4-sphere M 4 and extending π a (x) →π a (x, s) such thatπ a (x, 1) = π a (x) andπ a (x, 0) = 0. One then defines a 5-ball B 5 with boundary M 4 and coordinates y α = {x µ , s}. The WZW action can be written as where d a α is the Goldstone covariant derivative in Eq. (63), suitably extended to B 5 . The totally anti-symmetric Levi-Civita tensor forces the rank-5 invariant tensor ω abcde to be totally anti-symmetric as well. Group-theoretically, the existence of WZW action now is related to the existence of a rank-5 totally anti-symmetric tensor in the particular representation R of H that is furnished by π a , which is given by the fifth de Rham cohomology group H 5 [37,38].
The fifth de Rham cohomology group of the symmetric space of simple Lie groups is wellknown. For π a 's furnishing the adjoint representation of SU(N) group, they can be thought of as coordinates parameterizing the coset space SU(N) × SU(N)/SU(N). For N ≥ 3, H 5 (SU(N), R) has a single generator which is precisely the integrand in Eq. (B24) [37,38].