Amplitude/Operator Basis in Chiral Perturbation Theory

Effective field theories (EFTs) are powerful tools to understand salient behaviors of physical systems without detailed knowledge of ultraviolet (UV) physics, which is parameterized by some unknown constants. This remarkable approach, which has existed for half a century, is widely appreciated in several subfields of physics, from condensed matter to high energy and nuclear physics.

The philosophy of constructing an EFT is to use the relevant degrees of freedom to write down all effective operators consistent with symmetries of the physical system under consideration – what is not forbidden must be allowed – which generally results in an infinite number of operators. In order to be predictive, a power counting scheme must be supplied to estimate the relative importance of various effective operators so that only a small number of them contribute to the experimental process of interest.

Moreover, as measurements on a particular system become more and more precise, an increasing number of operators become relevant and need to be incorporated into the EFT. A prime example is the Chiral Perturbation Theory (ChPT) Weinberg:1968de ; Weinberg:1978kz describing interactions of pions with a single nucleon in a derivative expansion of $\partial/\Lambda$, where $\Lambda\sim O(1\ {\rm GeV})$. At higher orders in $\partial/\Lambda$, the construction of ChPT using the traditional Lagrangian approach becomes quite complicated, and the number of effective operators grows exponentially. In particular, there are subtleties associated with operator redundancies that are inherent in the Lagrangian formulation, rendering the enumeration of a complete and minimal set of operators a formidable task. After enduring efforts over a span of decades, the effective Lagrangian of ChPT is now known up to $O(\partial^{8})$ Gasser:1983yg ; Fearing:1994ga ; Bijnens:1999sh ; Bijnens:2018lez .

Another example, which has received much attention lately, is the Standard Model Effective Field Theory (SMEFT). Measurements at the Large Hadron Collider (LHC) push the validity of the Standard Model (SM) to unprecedented energy scales beyond the weak scale characterized by the Higgs vacuum expectation value $v=246$ GeV. As a consequence, any heavy, beyond-the-SM effects can be parameterized by irrelevant operators with increasing mass dimensions. It turned out there is only a single operator at the dim-5 order Weinberg:1979sa that is consistent with the symmetries of the SM. At the dim-6 order Buchmuller:1985jz ; Grzadkowski:2010es , the construction quickly becomes laborious, as in the case of the ChPT. Moreover, operator redundancies become constant sources of confusion and debate, especially when one attempts to go either to higher orders in the mass dimensions or beyond the approximate flavor symmetries of the SM.

The complexity of these EFTs at higher orders in the respective power counting schemes highlights the limitation of the traditional formulations and calls for new approaches to the problem. In the past few years, we have seen tremendous progress in applying new perspectives and methodology to constructing both ChPT and SMEFT. For the ChPT, which involves pseudo-Nambu-Goldstone bosons (NGB’s) and nonlinearly realized symmetries, a new infrared (IR) perspective based on the single soft theorem of NGB’s, i.e., the Adler’s zero Adler:1964um , was developed both in the Lagrangian formalism Low:2014nga ; Low:2014oga and in the on-shell formalism Cheung:2014dqa ; Cheung:2015ota . The IR perspective facilitated assembling an operator basis, in terms of the kinematic invariants $s_{ij}=2p_{i}\cdot p_{j}$, for independent operators up to $O(p^{10})$ in Ref. Low:2019ynd ; Dai:2020cpk .¹¹1 In addition to ChPT, the IR perspective also has important implications for extensions of the SM where the Higgs arises as a pseudo-Nambu-Goldstone bosonLiu:2018vel ; Liu:2018qtb . (If one is only interested in enumerating the number of operators, there is also the Hilbert series method in Ref. Graf:2020yxt .) In the SMEFT, advances were made in searching for an independent set of operator basis beyond the dim-6 order Henning:2015daa ; Li:2020gnx ; Li:2020xlh ; Li:2020xlh ; Li:2020zfq ; Graf:2020yxt ; Li:2020tsi ; Murphy:2020rsh . In particular, a standard operator basis called y-basis (y stands for Young tableau) was introduced in Refs. Li:2020gnx ; Li:2020xlh ; Li:2020tsi ; Li:2020zfq and a Mathematica package ABC4EFT was made public on HEPForge Li:2022tec .

There are, however, limitations to these new developments. For example, the studies in Refs. Bijnens:2018lez ; Dai:2020cpk were done without considering the kinematic constraint from the vanishing Gram determinant, which arises from the fact that in a spacetime dimension $D$, any $N>D$ momenta must be linearly dependent.²²2 More specifically, linear dependence of any $N>D$ vectors $X_{i}^{\mu}$ in $D$ dimensional spacetime implies vanishing of the Gram determinant $\det({X_{i}\cdot X_{j}})=0,\ \ i,j=1,\cdots,N$. These works further assumed a large number of flavor $N_{f}$, ignoring group-theoretical relations on the flavor factors which may exist at a small $N_{f}$. These issues will further reduce the number of independent operators in ChPT, where $D=4$ and $N_{f}=2,3$. Note that the vanishing Gram determinant is automatically fulfilled in on-shell spinor-helicity variables employed in the method presented in Refs. Li:2020gnx ; Li:2020xlh ; Li:2020tsi ; Li:2020zfq , which nonetheless cannot be applied directly to EFTs with nonlinearly realized symmetries such as the ChPT. In this work, we will fill in such a gap and present an algorithm, modified from Refs. Li:2020gnx ; Li:2020xlh ; Li:2020tsi ; Li:2020zfq , to systematically obtain an amplitude/operator basis for ChPT to arbitrary orders in the momentum expansion. Similar to Ref. Dai:2020cpk we work in the limit of massless quarks and focus on purely mesonic operators without external sources.

More specifically, in this work we construct an operator basis using on-shell variables with the following features:

• •

  Adler’s zero condition is satisfied.

• •

  Vanishing Gram determinants at $D=4$.

• •

  Parity-odd operators as well as the parity-even ones.

• •

  Reduction of the soft basis for finite $N_{f}$.

It is also shown that with the similar technique of y-basis reduction, the operator basis for ChPT with external sources of fermions or vectors can also be obtained Sun:2022ssa .

The paper will be organized as follows: In section 2, we present the general features of the Young Tableau basis (y-basis) of the Lorentz structure and show how it solves the subtleties in the literature and generates the soft blocks. Section 3 is dedicated to the flavor structures, where we explicitly show the redundancies of trace structures at small $N_{f}$. We combine the kinematics and flavor structures in section 4 by introducing the trace orbit, and show how the Gram determinant redundancy is taken into account in our algorithm. An explicit example of obtaining the $O(p^{6})$ 6-point soft basis is provided in section 5. Finally, we summarize in section 6. We provide some background on group theory and the $O(p^{8})$ 6-point soft basis in the appendices.

The starting point of our construction is the ”soft block” introduced in Ref. Low:2019ynd as the seed for the ”soft recursion” relation for Nambu-Goldstone bosons Cheung:2014dqa . Soft blocks are contact interactions that satisfy the Adler’s zero condition Adler:1964um when all external legs are on-shell. They are in one-to-one correspondence with the independent operators in momentum expansion in ChPT Low:2019ynd :

$$\begin{split}\text{Soft block }\mathcal{B}_{i}\quad\Leftrightarrow\quad\text{Invariant operators }\mathcal{O}_{i}\ .\end{split}$$

(1)

The Lorentz part of the soft block is a function of the on-shell momenta $p_{i}$ of the NGBs, which corresponds to the operator as

$$\begin{split}p_{i\mu_{1}}p_{i\mu_{2}}\cdots p_{i\mu_{n}}\quad\Leftrightarrow\quad\nabla_{(\mu_{1}}\nabla_{\mu_{2}}\cdots\nabla_{\mu_{n-1}}u_{\mu_{n})},\end{split}$$

(2)

where $u_{\mu}\equiv e^{-i\pi}\nabla_{\mu}e^{i\pi}$ is the field operator in the CCWZ formalism that generates an NGB $\langle\pi|u_{\mu}|0\rangle\sim p_{\mu}$ and is covariant under the leftover symmetry $H$. The Lorentz indices in the corresponding operator are totally symmetrized. Building blocks with indices not totally symmetrized can always be cast into the symmetrized one with extra pieces proportional to the so-called Equation of Motion (EOM) redundancy Li:2020zfq .

We will use “soft basis” to refer to an independent basis for the soft blocks, which can also be thought of as the leading contact on-shell interactions from each invariant operator in ChPT. Enumerating and constructing the soft blocks then gives an on-shell construction of ChPT via soft recursion relation. Ref. Dai:2020cpk listed the soft blocks for purely mesonic operators in the ChPT up to $O(p^{10})$ using momentum invariants $(p_{i}+p_{j})^{2}=2p_{i}\cdot p_{j}$, which did not take into account constraints from the Gram determinant. The Adler zero condition requires that the $N$-point soft blocks should satisfy the single soft limits of each NGB

$$\begin{split}\lim_{p_{i}\to 0}\mathcal{B}(p_{1},\dots,p_{i},\dots,p_{N})=0,\qquad\forall\ i=1,\dots,N.\end{split}$$

(3)

Hence we transform the question of obtaining an operator basis for ChPT into a problem of constructing local, contact amplitudes that satisfy eq. (3).

In general the amplitudes can be written as the product of a kinematic part $\mathcal{M}(p_{1},\dots,p_{N})$ and a flavor structure part $\mathcal{T}^{a_{1},\dots,a_{N}}$, where $a_{i}$ is the flavor index of the NGB under the unbroken global symmetry $H$. In ChPT, $a_{i}$ are adjoint indices of $SU(N_{f})$ group, and the invariant tensors $\mathcal{T}^{a_{1},\dots,a_{N}}$ can always be written as combinations of trace structures, which form a linear space with an independent basis $\{\mathcal{T}_{i}\}$. The kinematic part also forms a linear space with basis $\{\mathcal{M}_{i}\}$, all of which satisfy the Adler’s zero condition. Hence each soft block $\mathcal{B}_{i}$ can be expressed as the following linear combination:

$$\begin{split}\mathcal{B}_{i}=\sum_{j,k}\ C_{i}^{jk}\,\mathcal{M}_{j}(p_{1},\dots,p_{N})\ \mathcal{T}_{k}^{a_{1},\dots,a_{N}},\end{split}$$

(4)

where $C_{i}^{jk}$’s are some constants. The soft block must also satisfy the Bose symmetry and be invariant under permutations of external legs:

$$\begin{split}\sigma\circ\mathcal{B}_{i}\equiv\sum_{j,k}\ C_{i}^{jk}\,\mathcal{M}_{j}(p_{\sigma(1)},\dots,p_{\sigma(N)})\ \mathcal{T}_{k}^{a_{\sigma(1)},\dots,a_{\sigma(N)}}=\mathcal{B}_{i}\ ,\end{split}$$

(5)

where $\sigma\in S_{N}$ is an element of the permutation group.

In this section, we focus on obtaining a basis of kinematic factors $\mathcal{M}_{i}$. For purely NGB amplitudes, $\mathcal{M}_{i}$ are polynomials in the Lorentz invariant building blocks

$$\begin{split}s_{ij}\equiv g^{\mu\nu}p_{i\mu}p_{j\nu}\quad\text{and}\quad\epsilon(i,j,k,l)\equiv\epsilon^{\mu\nu\rho\sigma}p_{i\mu}p_{j\nu}p_{k\rho}p_{l\sigma}\,.\end{split}$$

(6)

However, polynomials in these building blocks are not all independent. The most important relations are massless on-shell condition $p_{i}^{2}=0$ and momentum conservation, which reads $\sum_{i}p_{i}=0$ in a convention where all momenta are incoming. They reduce, for instance, the set of independent kinematic invariants for 6-point (pt) amplitudes to the set

$$\begin{split}\{s_{24},s_{25},s_{26},s_{34},s_{35},s_{36},s_{45},s_{46},s_{56}\}\ ,\end{split}$$

(7)

which obviously is not a unique choice. In general, for $N$-pt amplitudes with $N\leq D+1$ the on-shell constraint and momentum conservation gives rise to $N$ linearly independent constraints:

$$\begin{split}\sum_{j=1,j\neq i}^{N}s_{ij}=0\ ,\qquad\forall\ \ i=1,\dots,N\ .\end{split}$$

(8)

These reduce the dimensionality of the kinematic space to ${N\choose 2}-N=N(N-3)/2$.

For $N>D+1$ there are additional relations, such as the CP even Gram determinant

$$\begin{split}\det(s_{ij})=0\ ,\qquad\forall\ \ i,j=1,2,\dots,D+1\,,\end{split}$$

(9)

and the CP odd combination

$$\begin{split}\sum_{\sigma\in S_{D+1}}{\rm sgn}(\sigma)\times s_{i,\sigma(j)}\epsilon(\underbrace{\sigma(k),\sigma(l),\dots}_{D})=0\end{split}$$

(10)

both due to the fact that $D+1$ momentum vectors in $D$ dimensional spacetime cannot be anti-symmetrized. In particular, the Gram determinant itself is a polynomial of degree $(D+1)$ in $s_{ij}$, which would reduce the number of independent polynomials of degree $D+1$ and more. Therefore the constraint from the Gram determinant becomes effective at the order of $p^{2(D+1)}$ or higher.³³3 In $D=4$ the Gram determinant only takes effect at the level of degree-5 polynomials in $s_{ij}$, or at $p^{10}$ order, as we will show explicitly later. Specializing to $D=4$ from now on, we will employ the spinor helicity variables $(\lambda,\tilde{\lambda})$ to construct the kinematic factors $\mathcal{M}_{j}(p_{\sigma(1)},\dots,p_{\sigma(N)})$. Recall that in $D=4$ a massless momentum $p_{\mu}$ satisfies the on-shell condition $p^{2}=p_{\mu}p^{\mu}={\rm det}(p\cdot\sigma)=0$, where $\sigma^{\mu}=(1,\vec{\sigma})$. In turn this implies $p\cdot\sigma$ has rank-1 and can be written as the outer product of two two-component spinors $(\lambda,\tilde{\lambda})$: $p_{\alpha\dot{\alpha}}\equiv(p\cdot\sigma)_{\alpha\dot{\alpha}}=\lambda_{\alpha}\tilde{\lambda}_{\dot{\alpha}}$.⁴⁴4 For real momentum $p\cdot\sigma$ is Hermitian and $\lambda^{\dagger}=\pm\tilde{\lambda}.$ The spinor helicity variables are intrinsically $D=4$ and take into account the constraint from the Gram determinant by default through the Schouten identities Elvang:2013cua :

$$\begin{split}&\langle ij\rangle\lambda_{k}+\langle jk\rangle\lambda_{i}+\langle ki\rangle\lambda_{j}=0,\quad\langle ij\rangle\equiv\lambda_{i}^{\alpha}\lambda_{j\alpha},\\ &[ij]\tilde{\lambda}_{k}+[jk]\tilde{\lambda}_{i}+[ki]\tilde{\lambda}_{j}=0,\quad[ij]\equiv\tilde{\lambda}_{i\dot{\alpha}}\tilde{\lambda}_{i}^{\dot{\alpha}}\ ,\end{split}$$

(11)

where the angle $\langle\cdot\rangle$ and square brackets $[\cdot]$ are defined as

$$\begin{split}&\langle ij\rangle\equiv\epsilon_{\alpha\beta}\lambda_{i}^{\alpha}\lambda_{j}^{\beta},\qquad[ij]\equiv\tilde{\epsilon}^{\dot{\alpha}\dot{\beta}}\tilde{\lambda}_{i}{}_{\dot{\alpha}}\tilde{\lambda}_{j}{}_{\dot{\beta}}\ .\end{split}$$

(12)

Moreover, using trace identities for Pauli matrices one can show

$$\begin{split}s_{ij}\equiv(p_{i}+p_{j})^{2}=2p_{i}\cdot p_{j}=[ij]\langle ji\rangle\ .\end{split}$$

(13)

Another advantage of adopting spinor variables is that external sources can be introduced in a consistent way Henning:2019enq ; Li:2020gnx ; Li:2020xlh . Last but not least, we have to impose the Adler’s zero condition.

In the ChPT, the NGB transforms under the adjoint representation of the unbroken flavor group $H=SU(N_{f})$. The operators must involve an invariant tensor with adjoint indices under $H$, called the flavor factor, which are usually written as traces over $T^{a}$, the $N_{f}\times N_{f}$ matrices satisfying the Lie algebra of $SU(N_{f})$ group. These traces are explicit $H$-invariant and easy to organize, but they are not always independent. In this section, we discuss the method of obtaining independent $H$-invariant tensors with rank given by the multiplicity of the NGB.

In general, the set of rank-$N$ independent invariant tensors can be obtained by taking the tensor product of $N$-copy of adjoint representations and focusing on the singlet representations in the decomposition. For example, for $SU(3)$ it is well-known that,

$$\begin{split}&\begin{array}[]{ccccccccccccc}\mathbf{8}&\times&\mathbf{8}&=&\mathbf{27}&+&\mathbf{10}&+&\mathbf{10}&+&\mathbf{8}\times 2&+&\mathbf{1}\end{array}\\ &\begin{array}[]{ccccccccc}\mathbf{8}&\times&(\mathbf{8}&\times&\mathbf{8})&=&\mathbf{1}\times 2&+&\dots\end{array}\end{split}$$

(34)

which implies there are 2 independent rank-3 invariant tensors in the adjoint of $SU(3)$, given by precisely the structure constants $f^{abc}$ and the totally symmetric $d^{abc}$. For general $SU(N_{f})$ the rank-$N$ invariant tensors are the collection of Clebsch-Gordan coefficients for the tensor product decomposition of the representations of the fields Li:2020gnx ; Li:2020xlh ; Li:2022tec , which can be computed repeatedly using the Littlewood-Richardson rule and is implemented in the package ABC4EFT. We list some results in the table 3 for reference.

When the rank of the invariant tensor becomes larger than the number of flavors, $N>N_{f}$, one subtlety arises because linear relations among the tensors due to the Cayley-Hamilton theorem starts to appear and reduce the number of independent flavor structures. The Cayley-Hamilton theorem relates the $N_{f}^{\rm th}$ power of an $N_{f}\times N_{f}$ matrix to a polynomial of degree $N_{f}-1$ in itself. For instance, for $N_{f}=3$ we have

$$\begin{split}A^{3}-{\mbox{tr}[A]}A^{2}+\frac{1}{2}\left({\mbox{tr}[A]}^{2}-{\mbox{tr}[A^{2}]}\right)A-(\det A)\,I_{3\times 3}=0\ ,\end{split}$$

(35)

where $A$ is an arbitrary square matrix. If ${\mbox{tr}[A]}=0$, which is the case for the $SU(N_{f})$ algebra, we can multiply $A$ to eq. (35) and take the trace to arrive at the relation:

$$\begin{split}{\mbox{tr}[A^{4}]}=\frac{1}{2}{\mbox{tr}[A^{2}]}^{2}\ .\end{split}$$

(36)

Now if we insert the linear combination $A=\sum_{a}\alpha_{a}\mathcal{T}^{a}$, where $\mathcal{T}^{a}$ are a basis for $SU(N_{f})$ generators and $\alpha_{a}$’s are arbitrary constants, to eq. (36), we arrive at many relations among the traces. For example, singling out the coefficient of $\alpha_{a}\alpha_{b}\alpha_{c}\alpha_{d}$, where $a,b,c,d$ are all distinct, we have

$$\begin{split}\sum_{\sigma\in S_{3}}{\mbox{tr}[{T}^{a}{T}^{\sigma(b)}{T}^{\sigma(c)}{T}^{\sigma(d)}]}={\mbox{tr}[{T}^{a}{T}^{b}]}{\mbox{tr}[{T}^{c}{T}^{d}]}+{\mbox{tr}[{T}^{a}{T}^{c}]}{\mbox{tr}[{T}^{d}{T}^{b}]}+{\mbox{tr}[{T}^{a}{T}^{d}]}{\mbox{tr}[{T}^{b}{T}^{c}]},\end{split}$$

(37)

which reduces the flavor structures starting at $N=4$ multiplicity for $N_{f}=3$.

For $N_{f}\geq N$ when the Cayley-Hamilton does not apply, the trace structures (modulo a residual symmetry such as the cyclic permutations of the traces) are independent. Yet the subtlety is particularly pronounced in ChPT, where $N_{f}=2,3$ and $N$ could easily exceed $N_{f}$. In order to find the independent set of invariant tensors for small $N_{f}$, we utilize the inner product between invariant tensors of the same rank introduced in Ref. Li:2022tec ,

$$\begin{split}(\mathcal{T}_{i},\mathcal{T}_{j})\equiv(\mathcal{T}_{i}^{\dagger})_{a_{1}\dots a_{N}}(\mathcal{T}_{j})^{a_{1}\dots a_{N}}\ .\end{split}$$

(38)

Notice the Hermitian conjugate inverts the traces, for example ${\mbox{tr}[ABCD]}^{\dagger}={\mbox{tr}[DCBA]}$, which renders the inner product positive-definite $(\mathcal{T},\mathcal{T})\geq 0$. From the inner products we define a metric tensor if $\{\mathcal{T}_{i}\}$ forms a basis: $\bar{g}_{ij}\equiv(\mathcal{T}_{i},\mathcal{T}_{j})$. When $\{\mathcal{T}_{i}\}$ is not linearly independent, the metric is degenerate $\det\bar{g}=0$. In this case a maximal independent subset $\mathfrak{T}$ can be obtained by demanding the corresponding minor metric $g_{ij}=\bar{g}_{ij},\ i,j\in\mathfrak{T}$, be non-singular $\det g\neq 0$.

To see how this works explicitly, consider the following 9 different trace structures for 4-pt amplitudes,

$$\begin{split}&\mathcal{T}_{i=1,\dots,6}={\mbox{tr}[{T}^{a}{T}^{\sigma_{i}(b)}{T}^{\sigma_{i}(c)}{T}^{\sigma_{i}(d)}]},\quad\sigma_{i}\in S_{3},\\ &\mathcal{T}_{7}={\mbox{tr}[{T}^{a}{T}^{b}]}{\mbox{tr}[{T}^{c}{T}^{d}]},\quad\mathcal{T}_{8}={\mbox{tr}[{T}^{a}{T}^{c}]}{\mbox{tr}[{T}^{d}{T}^{b}]},\quad\mathcal{T}_{9}={\mbox{tr}[{T}^{a}{T}^{d}]}{\mbox{tr}[{T}^{b}{T}^{c}]}.\end{split}$$

(39)

The inner product defined in eq. (38) can be computed by repeatedly using the orthogonality condition,

$$\begin{split}{T}^{a}_{ij}{T}^{a}_{kl}=\delta_{il}\delta_{kj}-\frac{1}{N_{f}}\delta_{ij}\delta_{kl}.\end{split}$$

(40)

For $N_{f}=3$, the resulting metric tensor is

$$\begin{split}\bar{g}_{ij}=\frac{16}{3}\times\left(\begin{array}[]{ccccccccc}19&-2&-2&-2&-2&4&8&-1&8\\ -2&19&-2&4&-2&-2&8&8&-1\\ -2&-2&19&-2&4&-2&-1&8&8\\ -2&4&-2&19&-2&-2&8&8&-1\\ -2&-2&4&-2&19&-2&-1&8&8\\ 4&-2&-2&-2&-2&19&8&-1&8\\ 8&8&-1&8&-1&8&24&3&3\\ -1&8&8&8&8&-1&3&24&3\\ 8&-1&8&-1&8&8&3&3&24\\ \end{array}\right)_{9\times 9}\,.\end{split}$$

(41)

It turned out $\bar{g}_{ij}$ in eq. (41) is a reduced rank, singular matrix, as expected; it has rank-8. We can find the redundancy relation by solving for the null space $\{\mathbf{n}^{j}\}$ of $\bar{g}$, which satisfies $\bar{g}_{ij}\mathbf{n}^{j}=0$. For eq. (41) we find

$$\begin{split}\bar{g}_{ij}\mathbf{n}^{j}=0\qquad\Rightarrow\quad\mathbf{n}^{i}T_{i}=T_{1}+T_{2}+T_{3}+T_{4}+T_{5}+T_{6}-T_{7}-T_{8}-T_{9}=0,\end{split}$$

(42)

which is precisely eq. (37), the redundancy relation obtained by using the Cayley-Hamilton theorem. Consequently we can remove any one of the 9 redundant tensors to arrive at an independent tensor basis for $N_{f}=3$, with the metric $g$ being the corresponding minor of $\bar{g}$. Entries marked red in Table 3 are the cases affected by the redundancy relations.