Operator Counting and Soft Blocks in Chiral Perturbation Theory

Chiral perturbation theory (ChPT) is a low-energy effective field theory of QCD and also a nonlinear sigma model based on the symmetry breaking pattern ${\rm SU}(N_f)\times {\rm SU}(N_f)\to {\rm SU}(N_f)$. In the limit of massless $N_f$ quarks, we enumerate the independent operators without external sources in ChPT using an on-shell method, by counting and presenting the soft blocks at each order in the derivative expansion, up to ${\cal O}(p^{10})$. Given the massless on-shell condition and total momentum conservation, soft blocks are homogeneous polynomials of kinematic invariants exhibiting the Adler's zero when any external momentum becomes soft and vanishing. In addition, soft blocks are seeds for recursively generating all tree amplitudes of Nambu-Goldstone bosons without recourse to ChPT, and in one-to-one correspondence with the"low energy constants"which are the Wilson coefficients. Relations among operators, such as those arising from equations of motion, integration-by-parts, hermiticity, and symmetry structure, manifest themselves in the soft blocks in simple ways. We find agreements with the existing results up to NNNLO, and make a prediction at N$^4$LO.

Chiral perturbation theory describes the low-energy dynamics of QCD [1], in particular interactions of mesons and baryons below the energy scale Λ ∼ O(1 GeV), and is an integral component of our understanding of many nuclear processes. Conceptually ChPT is the earliest, and perhaps also the most elegant, example employing the modern technique of effective field theory (EFT) [2], which later found wide-ranging applications in diverse topics.
The success of ChPT lies in the key observation that, in most cases, symmetry alone is sufficient to capture the long wavelength dynamics of a physical system; short wavelength fluctuations are encoded in an infinite number of "coupling constants," or Wilson coefficients in the language of renormalization group evolution, which are given as inputs in the EFT.
In order to achieve any predictive power, a power counting scheme must be supplied to organize the relative importance of different coupling constants so that at low energies only a small number of them are needed to describe experimental processes relevant at a certain energy scale and a given precision.
In In addition to Λ, there is a second mass scale in ChPT, where J µ A (x) is the axial current for SU(N f ) A , p µ is the pion momentum and f π is the pion decay constant. This is the famous PCAC (Partially Conserved Axial Current) hypothesis in low-energy QCD [3], which states that the axial current would be exactly conserved if the pion were massless. At a fixed order n in the momentum expansion, there are actually an infinite number of operators carrying n derivatives and an arbitrary even power of pion fields (π/f π ) 2k . However this doesn't imply there are an infinite number of LEC's at a fixed derivative order, because operators carrying different powers of π/f π are often related to one another by the broken symmetry group in a nonlinear way. One can choose a parameterization for the nonlinearly realized broken symmetry and resum the corresponding 1/f π expansion into a compact analytic form, thereby arriving at an operator that is invariant under the complete SU(N f ) L × SU(N f ) R . Each invariant operator is multiplied by a LEC, which needs to be measured experimentally. In the end, the number of LEC's is finite at each order in the momentum expansion and independent of the nonlinear parameterization.
At the leading O(p 2 ), the lowest order term in the 1/f π expansion is the kinetic term of the pions, and all higher order terms in 1/f π are uniquely related to the pion kinetic terms by the broken symmetry, leading to one invariant operator carrying two derivatives. There is no LEC at this order because the coefficient of the O(p 2 ) operator is fixed by canonically normalizing the scalar kinetic term. One popular nonlinear parameterization is [2] L (2) = 1 2 ∂ µ π a ∂ µ π a (1 + π a π a /f 2 π ) 2 , a = 1, 2, 3 . ( Since the pions transform as the adjoint of SU(N f ) V , the invariance under the unbroken group is explicit. It is much less obvious that the above equation is also invariant under the broken SU(N f ) A . At O(p 4 ) there are four invariant operators in general and, therefore, four LEC's. At higher orders in the momentum expansion, it is customary to rescale each derivative ∂ µ → ∂ µ /Λ and each pion field π a → π a /f π , then the structure of ChPT reads where L (n) contains only operators carrying n derivatives. Moreover, demanding that quantum corrections to Eq. (3) cannot be larger than the natural size of each operator gives rise to the relation, This is the expectation from the so-called Naive Dimensional Analysis [4]. (approximate) symmetries of the SM; some influential works can be found in Refs. [5][6][7].
The effort intensified in recent years, mainly driven by the need to confront the SM with the ever improving experimental precision. This line of investigation has also led to interesting theoretical questions and new underlying structures of EFT's [8][9][10][11][12][13]. For ChPT, there are additional complications due to operator relations imposed by the nonlinearly realized broken symmetry, making the counting of operators more involved. In spite of the difficulty, the construction of the effective Lagrangian up to O(p 6 ) (NNLO) was completed a while ago [14][15][16] and O(p 8 ) (NNNLO) Lagrangian was obtained recently [17].
In this work we would like to take up a relatively modest task: counting the number of independent parity-even operators in ChPT, in the massless quark limit and external sources turned off, up to O(p 10 ) (N 4 LO) and provide an "on-shel" basis for the operators. This is less ambitious than building the complete N 4 LO effective Lagrangian, and our study could serve as a starting point toward that goal. Moreover our methodology is new, in the sense that we will employ the "soft bootstrap" approach developed in the modern S-matrix theory [18][19][20], which aims to (re)construct effective field theories from the peculiar soft limit of S-matrix elements. The approach relies on finding higher point (pt) tree amplitudes from lower pt amplitudes by imposing unitarity and soft limit, 2 which can be implemented easily via the soft recursion relation [18]. The seed amplitude for the recursion relation is called the soft block and in one-to-one correspondence with the invariant operator at each order in the momentum expansion [20]. In practice, the soft blocks are the lowest pt contact interaction from each invariant operator when all external legs are on-shell. Working with soft blocks has the advantage that the redundancies in operator relations manifest themselves in the on-shell method in simple ways.
This work is organized as follows. In Section II we provide a brief overview of recent progress in understanding the interactions of Nambu-Goldstone bosons from the soft limit of on-shell amplitudes, after which we discuss in Section III properties of soft blocks that are 2 The effort was actually initiated long time ago using the current algebra approach [21,22], which is quite cumbersome and difficult to systematize. relevant for our computation. A basis for the kinematic invariants we employ is also given there. Our main results are contained in Section IV, where soft blocks up to O(p 10 ) are presented. We also include an ancillary Mathematica notebook detailing the calculation. Then we summarize in Section V. Several long expressions, as well as an expository description of the Mathematica notebook, are relegated to the Appendices.

II. A BRIEF OVERVIEW OF IR FORMULATION
The construction of higher order effective operators in ChPT is usually based on the general formalism developed by Callan, Coleman, Wess, and Zumino (CCWZ) in Refs. [23,24], which is a top-down approach in that it requires specifying a broken group G in the ultraviolet (UV) and an unbroken group H in the infrared (IR). In the context of ChPT, the approach has been reviewed extensively; see for examples Refs. [25][26][27][28][29]. Here we give a brief overview of recent progress in describing interactions of Nambu-Goldstone bosons from the IR, without recourse to CCWZ, by specifying the soft limit of S-matrix elements.
The starting point of the IR formulation of ChPT is the Adler's zero condition [30], which states that the on-shell amplitudes of Nambu-Goldstone bosons must vanish when one external momentum p is taken to be soft: p → τ p and τ → 0. Although usually phrased as an on-shell statement and initially proven using the old-fashioned current algebra technique, the Adler's zero condition actually follows from the quantum Ward identity of a shift symmetry on the Nambu-Goldstone field π a in the Lagrangian formulation [31], where ε a is an infinitesimal constant shift. Terms neglected above arise from the non-trivial vacuum structure of the unbroken group H [32] and vanish trivially for unbroken U(1) (sub)group.
More specifically, let's consider a set of scalars {π a } transforming as a linear representation of an unbroken group H with generators T r . It is always possible to choose a basis such that T r is purely imaginary and anti-symmetric [33]: The shift symmetry enforcing the Adler's zero condition under the constraints of the H-symmetry is highly nonlinear [34,35], where f is the "decay constant" of the Nambu-Goldstone boson. The effective Lagrangian invariant under Eq. (7) can be constructed using the following two objects: where In the above d a µ transforms covariantly under the shift symmetry in Eq. (7), 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. The important observation here is that these two building blocks are constructed using only IR data, without reference to the broken group G, as long as the Closure condition is satisfied [34], which applies to many commonly encountered situations, such as the adjoint of SU(N ) group or the fundamental of SO(N ). Another important comment is the normalization of the decay constant f is undetermined in the IR formulation, in that the effective Lagrangian constructed this way is invariant under the rescaling of f → cf for arbitrary c. Moreover, a purely imaginary c corresponds to a non-compact coset, while a real c arises from a compact coset [34].
In ChPT, H = SU(N f ) and π a 's transform under the adjoint representation. We can then write d µ = d a µ T a and the leading two-derivative operator is unique: where the coefficient is fixed by canonical normalization of the scalar kinetic term. The leading order (LO) equation of motion is where At the next-to-leading order (NLO), there are four independent operators carrying four derivatives, and the NLO effective Lagrangian is In the above the coefficients L 4,i are the low-energy constants (LEC's) at NLO. In principle one could write several more operators at this order, but they are all equivalent to the above list upon integration-by-parts, equations of motion at LO, and the following two important identities, which can also be derived in the IR [20]. 3 The list of operators can be further reduced if we specialize to N f = 2, 3.
At NNLO, operators in ChPT for a general N f are first presented in Ref. [16] and later clarified in Ref. [17], while the N 3 LO Lagrangian was studied in Ref. [17]. In this work we shall focus on ChPT in the limit of massless N f quarks and without external sources.
The preceding IR construction in the Lagrangian approach has an on-shell counterpart, which is pioneered by Susskind and Frye in Ref. [21]. Starting from the 4-point (pt) amplitudes of pions, Susskind and Frye constructed the 6-pt pionic amplitudes, by introducing a 6-pt contact interaction so that the Adler's zero condition is satisfied; see Fig. 1. At the time up to 8-pt amplitudes were constructed this way, and the relation of this approach to the then popular current algebra was clarified in Ref. [22] The modern approach, which is developed in Refs. [18,37] and goes by the name of "soft bootstrap," relies on the powerful techniques of on-shell recursion relations for scalar 3 In the coset construction the identities follow from the Maurer-Cartan equation [36]. theories with vanishing single soft limit. While "normal" recursion relations for Yang-Mills theories and gravity rely on the amplitudes vanishing at infinity when deforming the real momentum into the complex plane, on-shell amplitudes for scalar theories do not have such nice properties. Instead, "soft" recursion relations make use of the vanishing single soft limit of the amplitudes to construct an integrand that vanishes at complex infinity. In essence, the recursion relation works by searching for an expression, constructed from lowerpt amplitudes, which possesses the correct single soft limit and factorization property of a higher-pt amplitudes.
Focusing on ChPT, where the Nambu-Goldstone bosons transform as the adjoint of the unbroken SU(N f ), it will be convenient to introduce flavor-ordered partial amplitudes [38], which are akin to the color-ordered partial amplitudes in the SU(N ) Yang-Mills theory. The full amplitude at the leading O(p 2 ) can be written as 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 f ) group. The full amplitude has permutation invariance among all external legs, while the partial amplitude M (1, 2, · · · n) is invariant only under cyclic permutations of the external legs due to Eq. (22). Beyond LO, the flavor factor C a 1 a 2 ···an may involve multitrace structure and the partial amplitudes will only have the corresponding partial cyclic invariance.
At a fixed order in the derivative expansion, the soft bootstrap program constructs higherpt amplitudes from lower-pt amplitudes through the soft recursion relation [18], which utilizes an all-leg shift for the external momenta of M n , where z is a complex shift parameter and total momentum conservation requires, where we have adopted the convention that all momenta are 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, non-trivial solutions for a i exist when n − 1 ≥ D + 1, and the number of distinct solutions is n − D − 1 [18].
The deformed amplitudesM n (z) does not vanish at the complex infinity z → ∞. However, given that the Nambu-Goldstone amplitude in ChPT must satisfy the Adler's zero condition,M n (z) ∼p forp → 0 [30], it is possible to define a soft factor [18], so that the Cauchy integral, dz zM contains poles only z = 0 and the factorization channel I, where the shifted momentum of an internal propagatorP I (z) becomes on-shell: Cauchy's theorem then relates the n-pt amplitude M n , which is the residue at z = 0, to the other residues atP 2 I (z ± I ) = 0: where M L and M R are the two lower-pt sub-amplitudes associated with the factorization channel I. As an example, the three factorization channels of 6-pt partial amplitudes are shown in Fig. 2, which is equivalent to the contribution of the four diagrams in Fig. 1.
Eq. (28) is the foundation of the soft bootstrap program, which determines higher-pt amplitudes from lower-pt amplitudes at a fixed order in the derivative expansion, by imposing the desired soft limit on the amplitudes. The nonlinear shift symmetry in Eq. (7) is the algebraic realization of soft bootstrap in the Lagrangian formulation, since the Adler's zero condition follows from the Ward identity corresponding to Eq. (7) [39].

III. POWER COUNTING AND SOFT BLOCKS
Since soft bootstrap constructs higher-pt amplitudes from lower-pt amplitudes, an input is needed to serve as the seed for soft bootstrap. These inputs are the soft blocks. 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. In other words, each soft block is a homogeneous polynomial of kinematic invariants {s ij ≡ 2p i · p j ; i, j = 1, · · · , n} of degree k/2, which satisfy the Adler's zero condition. Note that for scalar field theories k is always even.
For each derivative order k, the number of soft blocks is finite. First of all, no non-trivial soft blocks exist for n ≤ 3, because of total momentum conservation. For n > 3, let's perform the all-leg-shift in Eq. (23) and the soft block now becomes a polynomial of degree k in z:Ŝ (k) =Ŝ (k) (z). Taking the single soft limit in momentum p i is now equivalent to taking z → 1/a i . The Adler's zero condition then implieŝ That is 1/a i is a root ofŜ (k) (z). However, in D = 4 there exists non-trivial solutions for a i 's only when the number of external legs n ≥ 6 [18]. So we will distinguish two separate cases: i) k ≤ 4 and ii) k ≥ 6.
In case i),Ŝ (k) (z) is a polynomial in z of degree k ≤ 4, which has at most 4 distinct roots. Therefore soft blocks with n ≥ 6 cannot exist. For n = 4, 5, the a i 's are trivial and the argument fails. So soft blocks with 4 or 5 external legs could exist in principle and are indeed found in Ref. [20].
In case ii),Ŝ (k) (z) is a polynomial in z of degree k ≥ 6, so soft blocks with n ≤ k could be there. The goal of this study is to enumerate all soft blocks for k ≤ 10.
Furthermore, different operators related to each other under special relations such as those in Eq. (20) would result in identical S-matrix elements and, therefore, give rise to identical or dependent soft blocks. These properties greatly simplify the counting of independent operators in ChPT.
Last but not least, we would like to make a comment on the basis for kinematic invariants.
Since soft blocks with 2m-derivative are homogenous polynomials of degree m made out of kinematic invariants, and the number of kinematic invariants grew factorially with the number of external legs n, it is important to adopt an appropriate basis of independent kinematic invariant K n .
The Mandelstam kinematic invariant basis K n , for n-external momenta p i , i = 1, 2, 3, · · · , n, is defined by the symmetric invariants s ij ≡ (p i + p j ) 2 and antisymmetric invariants p 1 p 2 ···pn ≡ µ 1 µ 2 ···µn p µ 1 1 p µ 2 2 · · · p µn , where all the momenta are assumed to be incoming and µ 1 µ 2 ···µn is the antisymmetric tensor. The symmetric invariants are even under parity transformations and antisymmetric invariants are odd under parity transformations. In the case of n = 4 particles, the usual Mandelstam variables are defined as s ≡ s 12 = (p 1 + p 2 ) 2 , t ≡ s 13 = (p 1 + p 3 ) 2 , and u ≡ s 23 = (p 2 + p 3 ) 2 . In this work, we are interested in computing the even-parity soft-blocks and then compare these soft-blocks with the operators that have been obtained in Refs. [16,17,40]. Therefore, we impose parity invariance which eliminates the antisymmetric invariants involving the -tensor.
Among the s ij 's, on-shell condition and total momentum conservation together leave only n(n − 1)/2 − n = n(n − 3)/2 kinematic invariants. In addition, in d-dimensional spacetime, there can only be at most d linearly independent momenta. This implies the following n × n has a rank of at most d, leading to the condition that any (d + 1) × (d + 1) sub-Gram matrix has zero determinant. The Gram conditions are highly nonlinear relations among s ij . Following Refs. [16,17,40] we will work in d ≥ n − 1 spacetime dimensions and ignore the Gram conditions. It will be interesting to return to incorporating the Gram conditions in the future. In the end, the Mandelstam basis consists of choosing any n(n − 3)/2 invariants out of the set {s ij ; i < j, i, j = 1, · · · , n}.
In order to construct the n-pt soft blocks at O (p 2m ) in momenta, we start with the most general homogeneous polynomial of degree m in the kinematic invariants in K n , where the number of coefficients (number of monomials) in the polynomial p[s ij ] is The coefficients {c 1 , c 2 , · · · , c N } of the polynomial p[s ij ] in Eq. (40)    After the Adler's zero condition, we still need to impose the invariance under cyclic permutations and hermitian conjugation, only after then do we arrive at the number of independent n-pt soft blocks at O(p 2m ). In the following subsections, we explicitly construct the n-pt soft-blocks up to O(p 10 ) in a general spacetime dimension d. The 4-pt soft-blocks up to O(p 4 ) have already been studied in Ref. [20].

A. 4-pt Soft Blocks
In the subsection, we compute the 4-pt soft-blocks up to O(p 10 ). As an illustration, we explain in detail how to compute the 4-pt soft-block at O (p 6 ).
where d 0 and d 1 are another two LEC's.   Table I. Moreover, the Adler's zero condition is not automatic anymore and now gives nontrivial constraints on the coefficients c i in p[s ij ], which significantly reduces the number of monomials.
The kinematic invariants we adopt at 6-pt are: The Adler's zero condition corresponds to taking the soft limit for every external momentum.
For example, let's take p 1 → 0. This imposes constraints on four of the kinematic invariants in K 6 , thereby reducing the number of independent kinematic invariants down to five. It is obvious that the same is true for taking any other p i → 0. More explicitly, After Adler's zero condition we need to consider the possible trace structures at 6-pt.
There are four possibilities listed in Table III and we need to impose the corresponding cyclic invariance, exchange symmetry of the trace, as well as hermitian conjugation. In the end we found all 6-pt soft blocks up to O(p 10 ), whose numbers are summarized in Table III Table IV, while for 10-pt there are twelve possibilities listed in Table V.
We impose the cyclic invariance, trace exchange symmetry of the trace and the hermitian conjugation corresponding to the trace structure. The computation of the soft blocks is done in the accompanying Mathematica notebook. Here we list the number of soft blocks for 8-pt and 10-pt in Table IV and Table V    and O(p 8 ) are known in Refs. [14][15][16][17]. Our "on-shell" approach is independent from the traditional Lagrangian formulation utilizing the CCWZ method, and comparison of our results with the literature finds agreements after taking the appropriate limit. In particular, we make a prediction at O(p 10 ).
In addition to counting the soft blocks, we also presented the expressions for the soft  are free of ambiguities arising from operator relations typically present in the Lagrangian formulation.
One limitation of the current study is we are only considering ChPT in the massless quark limit and with external sources turned off, which is due to the fact that the soft bootstrap approach has not been extended to these cases. It would certainly be very interesting to study how this could be done in the future.
Although it is unlikely that results presented here could be of immediate phenomenological interest, it is perhaps worth recalling that ChPT has been playing a prominent role in our understanding of low-energy QCD in the past half-a-century, and also is the prototypical ex- where {c 0 , c 1 , · · · , c 14 } are the 15 coefficients remaining. This is the starting point of imposing cyclic invariance for various trace structures to compute the soft blocks.
where a 0 , a 1 , a 2 , a 3 , and a 4 are arbitrary constants.
where c 0 , c 1 , and c 2 are arbitrary constants.

Triple-Trace
The triple-trace soft blocks S (6) where d 0 , d 1 , and d 2 are arbitrary constants.