Scattering of Nambu–Goldstone Bosons

Abstract

Interactions of Nambu–Goldstone bosons are constrained by spontaneously broken symmetry just like their spectrum. This chapter is devoted to one specific manifestation of interactions: scattering. This is of particular importance in subatomic physics. The discussion of scattering of Nambu–Goldstone bosons is therefore restricted to relativistic, Lorentz-invariant systems. Some references to literature dealing with a general setting also applicable to condensed-matter physics are provided. The first half of the chapter revisits the Adler zero phenomenon. A necessary and sufficient condition for a scattering amplitude to vanish in the soft limit for a single particle is derived. There is a small class of effective theories whose scattering amplitudes are softer than the naive Adler zero would suggest. It is shown how such theories can be constructed by systematically imposing specific scaling of the scattering amplitudes in the soft limit. The chapter concludes with a brief introduction to on-shell recursion methods. These allow one to reconstruct all tree-level scattering amplitudes of a theory from a finite number of seed, low-point amplitudes by utilizing a priori knowledge about the infrared behavior of the amplitudes.

In our survey of spontaneous symmetry breaking (SSB), we have so far been mostly concerned with the ground state and excitation spectrum. Yet, the effective field theory (EFT) framework developed in Chap. 8 captures full nonlinear dependence of the action on the Nambu–Goldstone (NG) fields. It thus also includes all interactions among NG bosons and their interactions with other excitations, if present. Microscopic interactions in quantum systems may have many different macroscopic manifestations. However, there is an important class of observables that are amenable to both systematic computation and experiment: scattering amplitudes of free asymptotic states. I will therefore conclude the part of the book devoted to spontaneous breaking of internal symmetry with a short primer on scattering of NG bosons.

I will start in Sect. 10.1 by revisiting the Adler zero principle that scattering amplitudes of NG bosons tend to vanish in the long-wavelength limit. I will offer a general justification of this phenomenon, and underline the nature of exceptions to the rule. A modern approach to scattering of NG bosons that utilizes the geometry of the coset space of broken symmetry is introduced in Sect. 10.2. In Sect. 10.3, I demonstrate that there are exceptional EFTs whose long-wavelength scattering amplitudes are even softer that naively expected from the Adler zero principle. These theories set a benchmark for many modern developments in the rapidly expanding field of scattering amplitudes. One important line of development concerns the recursive reconstruction of all (tree-level) scattering amplitudes from a finite number of “seed” amplitudes. This is the subject of Sect. 10.4.

Scattering amplitudes as observables are arguably more important in fundamental (subatomic) physics than in other branches such as condensed-matter physics. I will therefore restrict the discussion to Lorentz-invariant systems. Here NG bosons are massless particles and the kinematics of their scattering is well-understood. Much of the material of this chapter can however be generalized to nonrelativistic EFTs with continuous spatial rotation symmetry; see [1, 2] for details. Let me stress that even amplitudes in relativistic systems have grown into a vast subject, of which this chapter only exposes a small corner. A reader interested in a broader introduction to the physics of scattering amplitudes is recommended to start with the lecture notes [3] or the book [4].

1 Adler Zero Revisited

One of the hallmarks of SSB is the existence of a local conserved current $J^\mu$ that couples to the NG boson state; cf. the proof of Goldstone’s theorem in Sect. 6.2.1. This can be used to extract nonperturbative information about scattering processes involving one or more NG bosons. We start with a generic process whose initial and final states $\left \lvert {\alpha }\right \rangle$ and $\left \lvert {\beta }\right \rangle$ may include any set of particles of arbitrary mass and spin, and inspect the matrix element $\left \langle {\beta }\middle \vert {J^\mu (x)}\middle \vert {\alpha }\right \rangle$. This receives two qualitatively different types of contributions, schematically shown in Fig. 10.1. We shall focus on the first contribution, which amounts to the creation or annihilation of a one-particle NG boson state by the current.

Fig. 10.1

Sketch of contributions to the matrix element $\left \langle \beta \vert J^\mu (x) \vert \alpha \right \rangle$. In the former, the current creates an intermediate one-particle NG state $\left \lvert \pi ( \boldsymbol p) \right \rangle$ (dashed line) that couples to $\left \lvert \alpha \right \rangle$ and $\left \lvert \beta \right \rangle$ through the amplitude ${\mathcal {A}_{\alpha \to \beta +\pi ( \boldsymbol p)}}$. In the latter, the current connects to $\left \lvert {\alpha } \right \rangle$, $\left \lvert {\beta } \right \rangle$ via a multiparticle intermediate state

The NG boson may be an elementary excitation of a field in the action. The contributions with an intermediate one-particle NG state then correspond to one-particle-reducible diagrams where cutting a single propagator disconnects the current from the $\alpha \to \beta$ process. However, the NG boson may also be composite. The NG state then reveals itself through a single-particle pole in the matrix element $\left \langle {\beta }\middle \vert {J^\mu (x)}\middle \vert {\alpha }\right \rangle$ at energy–momentum $p^\mu \equiv p^\mu _\alpha -p^\mu _\beta$. Contributions to the matrix element without such a single-particle pole are represented by the second diagram in Fig. 10.1.

We use the translation property of the current operator, $J^\mu (x)=\mathrm{e} ^{\mathrm{i} P\cdot x}J^\mu (0)\mathrm{e} ^{-\mathrm{i} P\cdot x}$, where $P^\mu$ is the operator of energy–momentum. This holds whenever the current does not depend explicitly on spacetime coordinates, which is the case for internal symmetry. According to the rules of polology (Sect. 19.2 of [5]), the residue of $\left \langle {\beta }\middle \vert {J^\mu (x)}\middle \vert {\alpha }\right \rangle$ at the one-particle NG pole is then $\mathrm{e} ^{-\mathrm{i} p\cdot x}\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle \mathcal {A}_{\alpha \to \beta +\pi (\boldsymbol p)}$. Here $\mathcal {A}_{\alpha \to \beta +\pi (\boldsymbol p)}$ is the invariant amplitude for the process where a NG boson of momentum $\boldsymbol p$ is added to the final state.¹With a Lorentz-invariant vacuum $\left \lvert {0}\right \rangle$, the matrix element $\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle$ transforms as a Lorentz vector, as one readily verifies using the properties of single-particle states (Sect. 2.5 of [6]). Lorentz invariance therefore dictates that $\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle =\mathrm{i} p^\mu _{\mathrm {on}}F$, where F is a constant and $p^\mu _{\mathrm {on}}\equiv (\left \lvert {\boldsymbol p}\right \rvert ,\boldsymbol p)$ is an on-shell energy–momentum with spatial part $\boldsymbol p$. The matrix element $\left \langle {\beta }\middle \vert {J^\mu (0)}\middle \vert {\alpha }\right \rangle$ can now be split into a pole part and a regular part as

$$\displaystyle \begin{aligned} \left\langle{\beta}\middle\vert{J^\mu(0)}\middle\vert{\alpha}\right\rangle =\left\langle{0}\middle\vert{J^\mu(0)}\middle\vert{\pi(\boldsymbol p)}\right\rangle _{\mathrm{off}}\frac 1{p^2}\mathcal{A}_{\alpha\to\beta+\pi(\boldsymbol p)}+R^\mu_{\beta\alpha}(p)\;, {} \end{aligned} $$

(10.1)

where . The as yet unspecified function collects all the nonpole contributions, and thus remains regular in the on-shell limit $p^2\to 0$.

Polology suggests that the numerator of the pole in (10.1) should be proportional to $\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle$. Replacing this with $\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle _{\mathrm {off}}$ amounts to trading $p_{\mathrm {on}}^\mu$ for $p^\mu$, and in turn to a redefinition of without changing the residue at the pole. The reason for doing so will become clear below when we work out an explicit example. Here let me just stress that $\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle _{\mathrm {off}}$ is a mere symbolic notation for $\mathrm{i} p^\mu F$. This object is not a well-defined matrix element but rather depends on the energy–momenta $p^\mu _\alpha$ and $p^\mu _\beta$.

We now combine (10.1) with current conservation expressed as $p_\mu \left \langle {\beta }\middle \vert {J^\mu (0)}\middle \vert {\alpha }\right \rangle =0$. This leads to a master identity which allows one to reconstruct the on-shell scattering amplitude from the remainder function ,

$$\displaystyle \begin{aligned} \mathcal{A}_{\alpha\to\beta+\pi(\boldsymbol p)}=\frac{\mathrm{i}}{F}p_\mu R^\mu_{\beta\alpha}(p)\;. {} \end{aligned} $$

(10.2)

Example 10.1

Recall the toy model (2.1), introduced in Chap. 2. To simplify the notation, I will only consider the limit of exact symmetry, $\epsilon \to 0$, and disregard the fermionic degrees of freedom. In the linear parameterization of the complex field, $\phi (x)=\mathrm{e} ^{\mathrm{i} \theta }[v+\chi (x)+\mathrm{i} \pi (x)]/\sqrt {2}$, the axial current in (2.3) becomes

$$\displaystyle \begin{aligned} J^\mu_{\mathrm{A}}=-2\mathrm{i}(\phi^*\partial^\mu\phi-\partial^\mu\phi^*\phi)=2v\partial^\mu\pi+2(\chi \partial^\mu\pi-\pi\partial^\mu\chi )\;, \end{aligned} $$

(10.3)

whence we read off $F=-2v$. Let us have a look at the process $\chi \pi \to \chi \pi$ in which a NG boson scatters off a massive (Higgs) particle. We set $\left \lvert {\alpha }\right \rangle =\left \lvert {\chi (\boldsymbol p_1)\pi (\boldsymbol p_2)}\right \rangle$ and $\left \lvert {\beta }\right \rangle =\left \lvert {\chi (\boldsymbol p_3)}\right \rangle$. The “missing” energy–momentum $p^\mu _4$ is supplied by the current. Using the Feynman rules (2.15), the remainder function collecting nonpole contributions to turns out to be

(10.4)

where the solid squares indicate an insertion of the axial current operator. Contracting this with $p^\mu _4$ gives

$$\displaystyle \begin{aligned} \mathrm{i} p_{4\mu}R^\mu_{\beta\alpha}(p_4)=-4\lambda v\biggl[1+m_\chi ^2\biggl(\frac 1s+\frac 3{t-m_\chi ^2}+\frac 1u\biggr)\biggr]\;. \end{aligned} $$

(10.5)

Using finally (10.2), this reproduces the previously calculated amplitude (2.20).

Note that when evaluating the remainder function , I ignored the contributions of Feynman diagrams where an external $\pi$-type leg terminates in the current operator. These arise from the part of the current linear in $\pi$, and are included in the pole part of (10.1). Had we used therein the on-shell matrix element $\left \langle {0}\middle \vert {J^\mu (0)}\middle \vert {\pi (\boldsymbol p)}\right \rangle$, we would have had to include in (10.4) the contributions of the mentioned diagrams with the pole canceled by the numerator factor . At the same time, the right-hand side of (10.2) would receive an additional factor of 2, owing to the fact that the on-shell limit of $p\cdot p_{\mathrm {on}}/p^2$ is $1/2$. The conclusion that the amplitude can be reconstructed from would however remain.

It is instructive to reproduce the same result also with the other parameterization considered in Chap. 2, $\phi (x)=\mathrm{e} ^{\mathrm{i} \theta }\exp [\mathrm{i} \pi (x)/v][v+\chi (x)]/\sqrt {2}$. In this case, the axial current reads

$$\displaystyle \begin{aligned} J^\mu_{\mathrm{A}}=2v\partial^\mu\pi+4\chi \partial^\mu\pi+\frac 2v\chi ^2\partial^\mu\pi\;, \end{aligned} $$

(10.6)

whereas the Feynman rules change to (2.28). The presence of the cubic term in the current adds another Feynman diagram to the expression for the remainder function,

(10.7)

This reproduces the same scattering amplitude, in the form shown in (2.30).

What is most interesting about (10.2) is the extra factor of $p_\mu$ on the right-hand side, arising from current conservation. This shows that

$$\displaystyle \begin{aligned} \lim_{\boldsymbol p\to\mathbf{0}}\mathcal{A}_{\alpha\to\beta+\pi(\boldsymbol p)}=0\;, {} \end{aligned} $$

(10.8)

provided the remainder function is nonsingular in this limit. This is the Adler zero. The same conclusion applies to processes with a NG boson inserted in the initial state. This distinction is trivial in Lorentz-invariant systems; a pedagogical proof of (10.8) for systems with mere spatial rotation invariance can be found in Sect. 3.1 of [1]. I will not reproduce the details here since they have no bearing on the result.

The function is by construction nonsingular in the on-shell limit $p^2\to 0$. The absence of singularities for $\boldsymbol p\to \mathbf {0}$ (or equivalently $p^\mu \to 0$) is an additional assumption. Finding a mechanism that leads to a violation of this assumption is required for understanding the origin of exceptions to the Adler zero principle.

1.1 Generalized Soft Theorem

Before we embark on a more detailed analysis of the conditions under which the Adler zero (10.8) is realized, let me introduce some terminology. Taking the momentum of a massless particle (a NG boson or, for instance, a gauge boson) to zero is generally referred to as the (single) soft limit. The statement of Adler zero is equivalent to the vanishing of the soft limit of scattering amplitudes of NG bosons. A universal statement about the soft limit of scattering amplitudes such as (10.8) is called a soft theorem. We will see that even in theories where the soft limit of is nonvanishing, one can still establish a generalized soft theorem. This relates the soft limit of $\mathcal {A}_{\alpha \to \beta +\pi (\boldsymbol p)}$ to the amplitude $\mathcal {A}_{\alpha \to \beta }$ with the NG boson removed.

We noticed in Chap. 2 that scattering the NG boson off the fermion within our toy model violates the Adler zero principle. Operationally, this could be understood as a consequence of the singularity of the fermion propagator in (2.21) in the soft limit for the NG boson. This singularity in turn arises from the cubic coupling between the NG boson and the fermion. The origin of the singularity is however purely kinematical and not specific to coupling of NG bosons to fermions; the same obstruction to the Adler zero may occur in purely bosonic theories. It is, in fact, commonplace in EFTs where all degrees of freedom are NG bosons, as constructed in Chap. 8. The Adler zero is therefore a much less robust consequence of SSB than, say, the very existence of gapless NG bosons.

Let us now examine the hypothesis that the violation of the Adler zero principle arises from the presence of cubic interaction vertices in the theory. To keep things simple, I will from now on restrict to relativistic EFTs of NG bosons, disregarding any non-NG degrees of freedom. The argument below closely follows [7].

Consider a set of NG fields $\pi ^a$ in a theory defined by the generic Lagrangian

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac 12\delta_{ab}\partial_\mu\pi^a\partial^\mu\pi^b+\frac 12\lambda_{abc}\partial_\mu\pi^a\partial^\mu\pi^b\pi^c+\mathcal{O}(\pi^4,\partial^4)\;. {} \end{aligned} $$

(10.9)

The coupling $\lambda _{abc}$ can be assumed to be symmetric in its $a,b$ indices. There are no terms without derivatives as guaranteed by the broken symmetry. Also, we can restrict the discussion to operators with two derivatives in line with the power counting laid out in Sect. 9.1.1. Finally, I assume that the kinetic term is diagonal in the NG flavor indices and properly normalized. This can always be ensured by a linear transformation of the NG fields. Other than that, however, I do not implicitly assume any particular parameterization of the fields.

The first thing to notice is that the cubic vertex in (10.9) alone cannot be the sole culprit in violating the Adler zero principle. Observable properties of scattering amplitudes must be unchanged by a field redefinition of the type

$$\displaystyle \begin{aligned} \pi^a\to\pi^{\prime a}(\pi)=\pi^a+\frac 12c^a_{bc}\pi^b\pi^c+\mathcal{O}(\pi^3)\;, {} \end{aligned} $$

(10.10)

where $c^a_{bc}$ is symmetric in $b,c$. In particular, choosing $c^a_{bc}=(1/2)\delta ^{ad}(\lambda _{dbc}+\lambda _{dcb}-\lambda _{bcd})$ eliminates the cubic interaction vertex altogether. In fact, even cubic operators with more than two derivatives can be removed analogously. This redundancy of cubic couplings of NG bosons is inherent to relativistic EFTs for SSB.

For the Lagrangian (10.9) to actually represent an EFT for NG bosons, it must preserve a symmetry that is spontaneously broken. We do not need to invoke the full machinery of nonlinear realizations at this stage. It is sufficient to assume the existence of a set of spontaneously broken symmetries, one for each NG field,

$$\displaystyle \begin{aligned} \updelta\pi^a\equiv \epsilon^b[F^a_b+G^a_{bc}\pi^c+\mathcal{O}(\pi^2)]\;. {} \end{aligned} $$

(10.11)

The independence of these transformations and the fact that they are all spontaneously broken amount to the condition that $F^a_b$ is an invertible square matrix. The assumed invariance of (10.9) then leads to the following set of Noether currents,

$$\displaystyle \begin{aligned} J^\mu_a=\delta_{bc}F^c_a\partial^\mu\pi^b+K_{abc}\partial^\mu\pi^b\pi^c+\mathcal{O}(\pi^3)\;, {} \end{aligned} $$

(10.12)

where $K_{abc}\equiv \delta _{bd}G^d_{ac}+F^d_a\lambda _{dbc}$.

Let us now evaluate, schematically, the amplitude $\mathcal {A}_{\alpha \to \beta +\pi ^a(\boldsymbol p)}$ for a process where $\left \lvert {\alpha }\right \rangle ,\left \lvert {\beta }\right \rangle$ are states with an arbitrary number of NG bosons. The soft NG boson state is added to the final state $\left \lvert {\beta }\right \rangle$. Then (10.1) is replaced with

(10.13)

From (10.12) we find that . Applying momentum conservation leads to . We expect the Adler zero property of $\mathcal {A}_{\alpha \to \beta +\pi ^b(\boldsymbol p)}$ to be violated if the limit $\boldsymbol p\to \mathbf {0}$ makes some of the propagators in the process on-shell. Barring fine tuning of the kinematics, this happens when the NG state is “attached” to an external leg of the $\alpha \to \beta$ process via a cubic interaction vertex. In terms of , such singular contributions map to insertions of the bilinear part of $J^\mu _a$ into one of the external legs of $\alpha \to \beta$. See Fig. 10.2 for a visualization and the index notation.

Fig. 10.2

Sketch of nonpole contributions to that are singular in the limit . These contributions arise from inserting the bilinear part of the Noether current (10.12) into an external leg of the $\alpha \to \beta$ process. The insertion is indicated by the solid square

Near the limit of vanishing $p^\mu$, inserting the current into an external leg with flavor b, carrying energy–momentum $p^\mu _b$, gives the singular contributions

$$\displaystyle \begin{aligned} \begin{aligned} R^\mu_{\beta\alpha a}(p)&\supset\sum_c(-\mathrm{i}\mathcal{A}^{b\to c}_{\alpha\to\beta})\frac{\mathrm{i}}{(p_b-p)^2}\big[-\mathrm{i} K_{abc}p_b^\mu+\mathrm{i} K_{acb}(p_b-p)^\mu\big]\\ &\approx\frac{\mathrm{i}}{2}\sum_c\mathcal{A}^{b\to c}_{\alpha\to\beta}\frac{p_b^\mu}{p\cdot p_b}(K_{abc}-K_{acb})\;. \end{aligned} \end{aligned} $$

(10.14)

The superscript $b\to c$ indicates a replacement of the flavor index on the leg where the insertion has been made. Upon summing over all possible insertions, we arrive at the final form of our generalized soft theorem,

$$\displaystyle \begin{aligned} \lim_{\boldsymbol p\to\mathbf{0}}\sum_bF^b_a\mathcal{A}_{\alpha\to\beta+\pi^b(\boldsymbol p)}=\frac 12\sum_{b\in\alpha\cup\beta}\sum_c(K_{abc}-K_{acb})\mathcal{A}^{b\to c}_{\alpha\to\beta}\;. {} \end{aligned} $$

(10.15)

Let me spell out some properties of this result explicitly. First, the soft theorem (10.15) relates the soft limit of an n-particle amplitude to a set of $(n-1)$-particle amplitudes, in which the soft NG leg is removed and the label on one other leg is changed. The only other ingredient needed is the set of algebraic coefficients $K_{abc}$, which is independent of the concrete scattering process considered. In this sense, the soft theorem (10.15) is universal.

Second, as is evident from (10.12), the coefficients $K_{abc}$ receive two types of contributions: one from the cubic coupling $\lambda _{abc}$, the other from the part $G^a_{bc}$ of the symmetry transformation (10.11), linear in NG fields. Importantly, the combination $K_{abc}-K_{acb}$, antisymmetric under the exchange $b\leftrightarrow c$, is manifestly invariant under the field redefinition (10.10). This ensures that the soft theorem (10.15) has a physical meaning independent of the choice of field parameterization.

1.2 Application to Coset Effective Theories

The above argument is fairly general and in principle also applies to scattering amplitudes in theories where some of the particles in $\left \lvert {\alpha }\right \rangle$ and $\left \lvert {\beta }\right \rangle$ are not NG bosons. The only essential constraint is that there are no cubic operators without derivatives in the Lagrangian. The sum in (10.15) is then to be restricted to c such that $\pi ^c$ has the same mass as $\pi ^b$. However, once the Lagrangian (10.9) does represent an EFT with only NG boson degrees of freedom, we can use the wealth of information accumulated in Chaps. 7 and 8.

Let us recall some of the basic relations that we are going to use here. First, the two-derivative part of the effective Lagrangian for the coset space $G/H$ reads

$$\displaystyle \begin{aligned} \mathcal{L}^{(2)}_{\mathrm{eff}}=\frac 12\kappa_{cd}\omega ^c_a(\pi)\omega ^d_b(\pi)\partial_\mu\pi^a\partial^\mu\pi^b\;, {} \end{aligned} $$

(10.16)

where the constant symmetric matrix $\kappa _{ab}$ satisfies the constraint $f^c_{\alpha a}\kappa _{cb}+f^c_{\alpha b}\kappa _{ac}=0$. The index $\alpha$ labels generators of the unbroken subgroup H. With the exponential parameterization of the coset space, $U(\pi )=\exp (\mathrm{i} \pi ^a Q_a)$, the Maurer–Cartan (MC) form can be computed explicitly as a power series in the NG fields,

$$\displaystyle \begin{aligned} \omega ^A_a(\pi)=\delta^A_a-\frac 12f^A_{ab}\pi^b+\mathcal{O}(\pi^2)\;, {} \end{aligned} $$

(10.17)

where the index A runs over all generators of G. Likewise, infinitesimal transformations induced on the coset space by G, $\updelta \pi =\epsilon ^A\xi ^a_A(\pi )$, are determined by a set of Killing vectors,

$$\displaystyle \begin{aligned} \xi^a_A(\pi)=\delta^a_A-\left(f^a_{Ab}-\frac 12\delta^e_Af^a_{eb}\right)\pi^b+\mathcal{O}(\pi^2)\;. {} \end{aligned} $$

(10.18)

To make the bilinear (kinetic) part of the Lagrangian (10.16) canonically normalized, one can set $\kappa _{ab}=\delta _{ab}$ by a suitable linear transformation of the NG fields. To preserve the form of the exponential parameterization and hence of (10.17) and (10.18), this has to be compensated by a linear transformation of the basis of broken generators $Q_a$. That in turn affects the values of the structure constants. Below, I will assume that such a choice of basis of broken generators has already been made.

With all these pieces at hand, it is easy to construct the Noether currents for the spontaneously broken generators of G,

$$\displaystyle \begin{aligned} \begin{aligned} J^\mu_a&=\frac{\partial{\mathcal{L}_{\mathrm{eff}}^{(2)}}}{\partial{(\partial_\mu\pi^b)}}\xi^b_a(\pi)\\ &=\delta_{ab}\partial^\mu\pi^b-\left(\delta_{bd}f^d_{ac}+\frac 12\delta_{ad}f^d_{bc}\right)\partial^\mu\pi^b\pi^c+\mathcal{O}(\pi^3)\;. \end{aligned} \end{aligned} $$

(10.19)

By matching this to (10.12), we can read off the algebraic coefficients needed in the soft theorem (10.15), namely $F^b_a=\delta ^b_a$ and

$$\displaystyle \begin{aligned} K_{abc}-K_{acb}=\delta_{cd}f^d_{ab}-\delta_{bd}f^d_{ac}-\delta_{ad}f^d_{bc}\;. {} \end{aligned} $$

(10.20)

Example 10.2

Let G be compact and semisimple. Using a faithful matrix representation of G, one can define the Cartan–Killing form on the Lie algebra $\mathfrak {g}$ as $\Delta _{AB}\equiv \operatorname {\mathrm {tr}}(Q_AQ_B)$. The cyclicity of trace, $\operatorname {\mathrm {tr}}([Q_A,Q_B]Q_C)= \operatorname {\mathrm {tr}}(Q_A[Q_B,Q_C])$, implies the identity

$$\displaystyle \begin{aligned} f^D_{CA}\Delta_{DB}+f^D_{CB}\Delta_{AD}=0\;. {} \end{aligned} $$

(10.21)

This encodes the invariance of the Cartan–Killing form under the adjoint action of G on $\mathfrak {g}$. At the same time, (10.21) ensures that the covariant structure constant $f_{ABC}\equiv \Delta _{AD}f^D_{BC}$ is fully antisymmetric in its three indices.

It is convenient to choose the basis of broken generators $Q_a$ so that the space $\mathfrak {g}/\mathfrak {h}$ is “orthogonal” to $\mathfrak {h}$, that is, $\Delta _{a\beta }=0$. The unbroken and broken indices can then be raised and lowered independently using $\Delta _{\alpha \beta }$ and $\Delta _{ab}$, respectively. The constraint $f^c_{\alpha a}\kappa _{cb}+f^c_{\alpha b}\kappa _{ac}=0$ can be rewritten as . Thus, the matrix commutes with the representation of H on the space $\mathfrak {g}/\mathfrak {h}$. Suppose in particular that this representation is irreducible, that is, all the NG modes of $G/H$ span a single irreducible multiplet of H. This is the case for a number of physically important coset spaces such as $\mathrm {SO}(n+1)/\mathrm {SO}(n)\simeq S^n$, $\mathrm {ISO}(n)/\mathrm {SO}(n)\simeq \mathbb {R}^n$, or $G_{\mathrm {L}}\times G_{\mathrm {R}}/G_{\mathrm {V}}$ with simple G. Then by Schur’s lemma, must be proportional to $\delta ^a_b$, and thus $\kappa _{ab}$ to $\Delta _{ab}$. The choice $\kappa _{ab}=\delta _{ab}$ here corresponds to the common normalization of symmetry generators such that $\operatorname {\mathrm {tr}}(Q_AQ_B)\propto \delta _{AB}$. Finally, (10.20) reduces to

$$\displaystyle \begin{aligned} K_{abc}-K_{acb}\propto f_{cab}-f_{bac}-f_{abc}=f_{abc}\;. \end{aligned} $$

(10.22)

In this special case, the right-hand side of (10.15) becomes up to an overall factor.

It is obvious from (10.15) and (10.20) that a sufficient condition for the presence of Adler zero is that the coset space $G/H$ is symmetric so that $f^a_{bc}=0$. In fact, for symmetric coset spaces, the reason for vanishing of the right-hand side of (10.15) is twofold. Namely, in the exponential parameterization, the automorphism $\mathcal {R}(U(\pi ))=U(\pi )^{-1}$ is equivalent to the inversion $\pi ^a\to -\pi ^a$. As a consequence, the components of the broken part of the MC form, $\omega ^a_b(\pi )$, are manifestly even functions of the NG fields. Hence, for symmetric coset spaces, the effective Lagrangian (10.16) can only give scattering amplitudes with an even number of particles. For theories with only even interaction vertices, the soft theorem (10.15) applied to an amplitude with an even number of particles yields Adler zero without any further requirements on the coefficients $K_{abc}$.

2 Geometric Framework for Scattering Amplitudes

In the analysis in Sect. 10.1.1, the freedom to redefine the field variables $\pi ^a$ was something of a nuisance. Indeed, we had to demonstrate explicitly that our soft theorem (10.15) is invariant under field reparameterization and thus physically meaningful. Here we will take control and make the freedom to redefine fields work for us. The discussion in this section loosely follows [8].

Suppose we are given a set of fields $\pi ^a$ taking values from some manifold $\mathcal {M}$. For the moment, we do not even have to assume that these are NG fields and $\mathcal {M}$ is a coset space, $\mathcal {M}\simeq G/H$. All we need is that the low-energy physics of the system is captured by the Lagrangian

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac 12g_{ab}(\pi)\partial_\mu\pi^a\partial^\mu\pi^b\;. {} \end{aligned} $$

(10.23)

In particular, this requires that there are no interactions without derivatives, and any higher-derivative operators can be treated as subleading. Under a point transformation, $\pi ^a\to \pi ^{\prime a}(\pi )$, the symmetric matrix function $g_{ab}(\pi )$ on $\mathcal {M}$ behaves as a rank-2 covariant tensor. Moreover, it must be positive-definite in some neighborhood of the origin, $\pi ^a=0$, to give a kinetic term with the correct signature. It can therefore be interpreted as a Riemannian metric on the target manifold $\mathcal {M}$.

Perturbative interactions among the fields $\pi ^a$ are generated by expanding the metric in powers of the fields. Individual interaction vertices are thus determined by derivatives of the metric at the origin. However, physical observables must at the same time be invariant under an arbitrary redefinition of the fields. We can use this fact and switch to the Riemannian (geodesic) normal coordinates on $\mathcal {M}$, in which (see Appendix A.6.4)

$$\displaystyle \begin{aligned} g_{ab}(\pi)=g_{ab}(0)-\frac 13\pi^c\pi^dR_{acbd}(0)-\frac 16\pi^c\pi^d\pi^e(\hat\nabla_eR)_{acbd}(0)+\mathcal{O}(\pi^4)\;. \end{aligned} $$

(10.24)

Moreover, we can always assume (and I will henceforth do so) that $g_{ab}(0)=\delta _{ab}$ so that the kinetic term in (10.23) is canonically normalized. This shows that any observable such as an on-shell scattering amplitude depends on $g_{ab}(\pi )$ solely through the Riemann curvature tensor and its covariant derivatives at the origin.

To see how the curvature tensor enters the amplitudes, let me spell out explicitly the simplest cases of the 4-particle and 5-particle amplitude,

(10.25)

Here I have introduced a compact notation for a scattering amplitude with particles of energy–momentum $p^\mu _i$ and flavor $a_i$. All the energy–momenta are (now and henceforth) oriented inwards. The variables generalize the standard Mandelstam variables to processes with an arbitrary number of particles. Finally, I have dropped the argument $(0)$ of the curvature tensor for notation simplicity.

The presentation of the amplitudes is highly ambiguous due to the redundancies among the Mandelstam variables and the components of the curvature tensor. To arrive at the expressions in (10.25), I have adopted the following conventions. First, the variables $s_{1i}$ with any $i\geq 2$ can be eliminated through $s_{1i}=-\sum _{j\geq 2}s_{ji}$, which follows from overall energy–momentum conservation for massless particles. Second, one of the remaining Mandelstam variables can be eliminated by means of . I use this to get rid of $s_{23}$. Similarly, all components $R_{a_ia_ja_ka_l}$ can be expressed solely in terms of those for which $i\leq \min (j,k,l)$ simultaneously with $k<l$ and $j\leq \max (k,l)$. This is guaranteed by the symmetries of the curvature tensor and the algebraic Bianchi identity (see Appendix A.6.3). Finally, the differential Bianchi identity (A.99) can be utilized to remove $\hat \nabla _{a_m}R_{a_ia_ja_ka_l}$ where $m<\min (k,l)$.

With the above remarks out of the way, let us have a look at the result. First, the 4-particle amplitude vanishes in the limit where any of the energy–momenta $p^\mu _i$ is taken to zero. This boils down to the very special kinematics of relativistic 4-particle scattering processes. The 5-particle amplitude is more interesting. Our expression is designed to make the soft limit for the last particle transparent,²

$$\displaystyle \begin{aligned} \notag \lim_{p_5\to0}\mathcal{A}_{a_1a_2a_3a_4a_5}(p_1,p_2,p_3,p_4,p_5)&=-s_{24}\hat\nabla_{a_5}R_{a_1a_2a_3a_4}-s_{34}\hat\nabla_{a_5}R_{a_1a_3a_2a_4}\\ &=\hat\nabla_{a_5}\mathcal{A}_{a_1a_2a_3a_4}(p_1,p_2,p_3,p_4)\;. \end{aligned} $$

(10.26)

This is a special case of a geometric soft theorem,

$$\displaystyle \begin{aligned} \lim_{p_n\to0}\mathcal{A}_{a_1\dotsb a_n}(p_1,\dotsc,p_n)=\hat\nabla_{a_n}\mathcal{A}_{a_1\dotsb a_{n-1}}(p_1,\dotsc,p_{n-1})\;, {} \end{aligned} $$

(10.27)

valid for any scalar theory that does not contain nonderivative interactions. There may be other, higher-order derivative couplings additional to (10.23). The covariant derivative of the amplitude in (10.27) is taken in the target space $\mathcal {M}$ with respect to the Levi-Civita (LC) connection defined by the metric $g_{ab}(\pi )$.

The reader is referred to [8] for a general nonperturbative proof of (10.27). Here we are mostly interested in EFTs for NG bosons, defined on a coset space of spontaneously broken symmetry. For this class of theories, I will derive below a dedicated geometric soft theorem using the more elementary approach of Sect. 10.1.

2.1 Geometric Soft Theorem for Nambu–Goldstone Bosons

Inspired by the above discussion, we shall now reinterpret the results of Sect. 10.1.2 in terms of the geometric properties of the coset space $G/H$. The Noether current for the spontaneously broken generator $Q_a$ can be written as

$$\displaystyle \begin{aligned} J^\mu_a=\frac{\partial{\mathcal{L}_{\mathrm{eff}}^{(2)}}}{\partial{(\partial_\mu\pi^b)}}\xi^b_a(\pi)=g_{bc}(\pi)\xi^b_a(\pi)\partial^\mu\pi^c\;. \end{aligned} $$

(10.28)

Expanding this in powers of the NG fields $\pi ^a$ gives

$$\displaystyle \begin{aligned} J^\mu_a=g_{bc}(0)\xi^b_a(0)\partial^\mu\pi^c+\partial_d(g_{bc}\xi^b_a)(0)\partial^\mu\pi^c\pi^d+\mathcal{O}(\pi^3)\;. {} \end{aligned} $$

(10.29)

I have already made the assumption that the kinetic term is canonically normalized, that is $g_{ab}(0)=\delta _{ab}$. Accordingly, the coupling of the current $J^\mu _a(x)$ to the one-particle state is defined by the matrix element . The bilinear part of the current (10.29) yields in turn the algebraic coefficient $K_{abc}=\partial _c(g_{db}\xi ^d_a)(0)=\partial _c(\boldsymbol \xi ^\flat _a)_b(0)$. We only need the part of this, antisymmetric under the exchange $b\leftrightarrow c$, which can be simplified to

$$\displaystyle \begin{aligned} K_{abc}-K_{acb}=(\hat\nabla_c\boldsymbol\xi^\flat_a)_b(0)-(\hat\nabla_b\boldsymbol\xi^\flat_a)_c(0)=2(\hat\nabla_c\boldsymbol\xi^\flat_a)_b(0) \end{aligned} $$

(10.30)

using the Killing equation (A.92). Putting all the pieces together, we infer from (10.15) a geometric soft theorem for the amplitude $\mathcal {A}_{a_1\dotsb a_n}(p_1,\dotsc ,p_n)$ for scattering of n NG bosons with flavors $a_i$ and energy–momenta $p^\mu _i$,

$$\displaystyle \begin{aligned} \begin{aligned} \lim_{p_n\to0}\xi^b_a(0)&\mathcal{A}_{a_1\dotsb a_{n-1}b}(p_1,\dotsc,p_n)\\ &=\sum_{i=1}^{n-1}(\hat\nabla^b\boldsymbol\xi^\flat_a)_{a_i}(0)\mathcal{A}_{a_1\dotsb a_{i-1}ba_{i+1}\dotsb a_{n-1}}(p_1,\dotsc,p_{n-1})\;. {} \end{aligned} \end{aligned} $$

(10.31)

In [8], the same result was obtained from (10.27) by invoking the G-invariance of the scattering amplitude in the schematic form

$$\displaystyle \begin{aligned} \mathcal{L}_{\boldsymbol\xi_a}\mathcal{A}_{a_1\dotsb a_{n-1}}=\xi^b_a\hat\nabla_b\mathcal{A}_{a_1\dotsb a_{n-1}}+\sum_{i=1}^{n-1}(\hat\nabla_{a_i}\boldsymbol\xi_a)^b\mathcal{A}_{a_1\dotsb a_{i-1}ba_{i+1}\dotsb a_{n-1}}=0\;. \end{aligned} $$

(10.32)

The formulation (10.31) of the soft theorem is manifestly covariant under field redefinitions that preserve the normalization of the kinetic term. One can take advantage of this and compute the algebraic coefficient in suitably chosen local coordinates on $G/H$. Specifically, in the exponential parameterization, we already have an expression for the Killing vector in (10.18). Likewise, the expression (10.17) for the MC form can be used to recover the metric $g_{ab}(\pi )$ and in turn the Christoffel symbols, needed to evaluate the covariant derivative of the Killing vector. Putting everything together, one finds

$$\displaystyle \begin{aligned} (\hat\nabla_c\boldsymbol\xi^\flat_a)_b(0)=\frac 12(\delta_{cd}f^d_{ab}-\delta_{bd}f^d_{ac}-\delta_{ad}f^d_{bc})\;, {} \end{aligned} $$

(10.33)

in accord with (10.20). This confirms that the soft theorems (10.15) and (10.31), formulated respectively in terms of the algebraic and geometric properties of the coset space, are equivalent.

2.2 Adler Zero or Not?

We have already observed that if the coset space $G/H$ is symmetric, the right-hand side of the soft theorem (10.15) or (10.31) automatically vanishes. Is the symmetry of $G/H$ also a necessary condition for the Adler zero? We cannot exclude the possibility that the soft limit of a specific amplitude in a given theory vanishes due to fine tuning of the effective couplings. However, in order that all amplitudes of the EFT vanish in the soft limit, the coefficients (10.33) must vanish for any choice of $a,b,c$. Using the shorthand notation $f_{abc}\equiv \delta _{ad}f^d_{bc}$, the vanishing of (10.33) is in turn equivalent to $f_{abc}=f_{bca}+f_{cab}$. Applying this twice gives

$$\displaystyle \begin{aligned} f_{abc}=f_{bca}+f_{cab}=(f_{cab}+f_{abc})+(f_{abc}+f_{bca})=2f_{abc}+f_{bca}+f_{cab}\;. \end{aligned} $$

(10.34)

This is only possible if $f_{abc}=0$. Thus, the set of all scattering amplitudes of the EFT satisfies the Adler zero principle if and only if the coset space $G/H$ is symmetric.

Incidentally, there is a neat geometric way to understand the one-way implication, ensuring Adler zero for symmetric coset spaces. Namely, the latter have the property that all covariant derivatives of the Riemann curvature tensor identically vanish (Theorem 10.19 in [9]). The Adler zero can then be seen as a direct consequence of (10.27). On coset spaces that are not symmetric, the nontrivial soft limit of scattering amplitudes is governed by the structure constants $f^a_{bc}$. According to Sect. 7.4.1, these are in a one-to-one correspondence with the (frame components of the) torsion 2-form of the canonical connection on $G/H$. Thus, the nonvanishing soft limit can be said to arise geometrically from the torsion of the coset space.

Example 10.3

One of the simplest symmetry groups leading to a nonvanishing soft limit of some scattering amplitudes is the Heisenberg group $\mathrm {H}_3$. This is a three-dimensional nilpotent Lie group. One can choose a basis of its Lie algebra $\mathfrak {h}_3$, $Q_a$ with $a=1,2$ and Q, so that the only nontrivial commutation relation among the generators is

$$\displaystyle \begin{aligned} [Q_a,Q_b]=\mathrm{i}\varepsilon_{ab}Q\;. {} \end{aligned} $$

(10.35)

Suppose the symmetry of a system under the Heisenberg group is completely broken. Denoting the NG fields associated with $Q_a$ and Q respectively as $\pi ^a$ and $\theta$, the coset space $\mathrm {H}_3/\{e\}\simeq \mathrm {H}_3$ can be parameterized by

$$\displaystyle \begin{aligned} U(\pi,\theta)=\exp\biggl(\frac{\mathrm{i}}{v}\pi^aQ_a\biggr)\exp\biggl(\frac{\mathrm{i}}{v}\theta Q\biggr)\;, {} \end{aligned} $$

(10.36)

where v is a positive dimensionful constant. The corresponding components of the MC form are $\omega ^a=(1/v)\mathrm{d} \pi ^a$ and $\omega ^\theta =(1/v)\mathrm{d} \theta +1/(2v^2)\varepsilon _{ab}\pi ^a\mathrm{d} \pi ^b$. The most general two-derivative effective Lagrangian would now be given by a generic rank-2 symmetric tensor built out of the MC form. For illustration, it is however sufficient to consider a particularly simple special case,

$$\displaystyle \begin{aligned} \mathcal{L}^{(2)}_{\mathrm{eff}}=\frac 12\delta_{ab}\partial_\mu\pi^a\partial^\mu\pi^b+\frac 12\left(\partial_\mu\theta+\frac 1{2v}\varepsilon_{ab}\pi^a\partial_\mu\pi^b\right)^2\;. {} \end{aligned} $$

(10.37)

This makes the calculation of various scattering amplitudes easy. Namely, the sole cubic interaction operator equals, up to a surface term, $-1/(2v)\theta \varepsilon _{ab}\pi ^a\Box \pi ^b$. As a consequence, any Feynman diagram where a $\theta$-propagator is attached to two external $\pi$-type legs will vanish on-shell.

Any on-shell 3-particle amplitude in a derivatively coupled relativistic theory of massless scalars automatically vanishes. Hence we need to consider a process with at least five particles to have a hope for a nonzero soft limit. Leaving out straightforward details, a simple example is

$$\displaystyle \begin{aligned} \mathcal{A}_{\theta1222}(p_1,p_2,p_3,p_4,p_5)=\frac{s_{12}}{2v^3}\;. {} \end{aligned} $$

(10.38)

In order to compare this to the prediction of our soft theorem (10.31), we have to keep in mind the factors of v in (10.36). These make the only nonzero structure constant effectively . Together with $\xi ^b_a(0)=\delta ^b_a$, (10.31) and (10.33) then predict

$$\displaystyle \begin{aligned} \begin{aligned} \lim_{p_5\to0}\mathcal{A}_{\theta1222}(p_1,\dotsc,p_5)=\frac 1{2v}[&\mathcal{A}_{1122}(p_1,\dotsc,p_4)\\ &-\mathcal{A}_{\theta\theta22}(p_1,\dotsc,p_4)]\;. \end{aligned} \end{aligned} $$

(10.39)

This is easily verified by calculating the two 4-particle amplitudes and finding that

$$\displaystyle \begin{aligned} \mathcal{A}_{1122}(p_1,p_2,p_3,p_4)=\frac{3s_{12}}{4v^2}\;,\qquad \mathcal{A}_{\theta\theta22}(p_1,p_2,p_3,p_4)=-\frac{s_{12}}{4v^2}\;. \end{aligned} $$

(10.40)

By permutation invariance, the same kind of result applies to the soft limit in which either or is taken to zero. On the other hand, the soft limit of $\mathcal {A}_{\theta 1222}(p_1,\dotsc ,p_5)$ for or manifestly vanishes, as is clear from (10.38).

The cubic interaction operator in (10.37) can be removed by the field redefinition with $c\equiv 1/(2v)$. This leads to the Lagrangian

$$\displaystyle \begin{aligned} \begin{aligned} \mathcal{L}^{(2)}_{\mathrm{eff}}={}&\frac 12(1+c^2\theta^2)\partial_\mu\boldsymbol\phi\cdot\partial^\mu\boldsymbol\phi+\frac 12(1-c^2\boldsymbol\phi^2)(\partial_\mu\theta)^2+\frac{c^2}4\partial_\mu(\theta^2)\partial^\mu(\boldsymbol\phi^2)\\ &+\frac{c^3}3\partial_\mu(\theta^3)\boldsymbol\phi\times\partial^\mu\boldsymbol\phi+\frac{c^2}2\big[(1+c^2\theta^2)\boldsymbol\phi\times\partial_\mu\boldsymbol\phi-c\boldsymbol\phi^2\partial_\mu\theta\big]^2\;, \end{aligned} {} \end{aligned} $$

(10.41)

where I used the notation $\boldsymbol \phi \equiv (\phi ^1,\phi ^2)$ and $\boldsymbol \phi \times \partial _\mu \boldsymbol \phi \equiv \varepsilon _{ab}\phi ^a\partial _\mu \phi ^b$ to avoid proliferation of indices. The first term on the second line of (10.41) is equivalent to $-(c^3/3)\theta ^3\boldsymbol \phi \times \Box \boldsymbol \phi$ by partial integration, and thus does not contribute to tree-level 5-particle amplitudes of the theory. All such amplitudes are therefore encoded in a single operator, $-c^3\partial ^\mu \theta \boldsymbol \phi ^2(\boldsymbol \phi \times \partial _\mu \boldsymbol \phi )$. This shows that $\mathcal {A}_{\theta 1222}$ and $\mathcal {A}_{\theta 2111}$ are the only nonzero 5-particle amplitudes, and makes it easy to reproduce (10.38).

2.3 Symmetric Coset Spaces

Expressing the scattering amplitudes in terms of the curvature tensor and its covariant derivatives is elegant and gives us useful insight. However, calculating the covariant derivatives of the curvature tensor from the metric $g_{ab}(\pi )$ in practice may be a tiresome task. The situation is much better for symmetric coset spaces where all the covariant derivatives of the curvature tensor vanish. Here we have a closed expression for the Riemannian metric in the normal coordinates (see Appendix A.6.4),

(10.42)

where . All we need to know are the constants . These can be extracted from our discussion of the geometry of coset spaces in Sect. 7.4. First, however, a word of caution is in place.

Throughout Appendix A, I dutifully distinguish components of tensors in a local frame from those in a local coordinate basis by using different types of indices. This has not been necessary in the main text of the book so far. Here, to avoid confusion, I will use lowercase indices $a,b,\dotsc$ to indicate a local coordinate system, and underlined indices for components in the frame defined by the MC form. Thus, for instance, a comparison of (10.16) and (10.23) shows that the G-invariant Riemannian metric on the coset space is . The components of the metric in the MC form basis are constant, as shown explicitly in Sect. 8.1.2.

According to Sect. 7.4.1, the same is true for the curvature 2-form of the LC connection on symmetric coset spaces,

(10.43)

Combining this with (10.42) gives upon a brief manipulation a practically more useful expression for the metric,

(10.44)

where . Here is a constant matrix that can easily be evaluated from the definition of the MC form, . Combining (10.23) with (10.44) determines the complete two-derivative effective Lagrangian on symmetric coset spaces in a closed form.

The structure constant $f^a_{\alpha c}$ in (10.44) defines the adjoint action of the unbroken subgroup H on the broken generators, and thus also the NG fields. For compact and semisimple groups G, the other type of structure constant in (10.44), $f^\alpha _{bd}$, can be related to the former using the Cartan–Killing form on $\mathfrak {g}$; cf. Example 10.2. This leads to the intriguing conclusion that the effective Lagrangian (10.16) is completely fixed by the set of low-energy constants $\kappa _{ab}$, spanning a symmetric invariant tensor of H, and certain linear representation of H. No specific information about the full symmetry group G is needed, as was first emphasized in [10].

Example 10.4

The requirement that G be compact and semisimple is essential. Contrast the coset spaces $\mathrm {SO}(n+1)/\mathrm {SO}(n)\simeq S^n$ and $\mathrm {ISO}(n)/\mathrm {SO}(n)\simeq \mathbb {R}^n$. Both of these are symmetric, share the same H, and in both the set of broken generators transforms as a vector of $\mathrm {SO}(n)$. Yet, the corresponding EFTs are different; $\mathrm {SO}(n+1)$ is compact but $\mathrm {ISO}(n)$ is not. In case of $\mathrm {SO}(n+1)/\mathrm {SO}(n)$, the low-energy EFT is most easily presented as a nonlinear sigma model for a linearly-transforming unit vector field, $\boldsymbol n\in S^n$. It implies nontrivial interactions among the n NG bosons. On the other hand, in case of $\mathrm {ISO}(n)/\mathrm {SO}(n)$, the leading-order EFT (10.16) reduces to a noninteracting theory, $\mathcal {L}_{\mathrm {eff}}=(1/2)\delta _{ab}\partial _\mu \pi ^a\partial ^\mu \pi ^b$. This follows from the fact that the broken generators of $\mathrm {ISO}(n)$ commute with each other so that $f^\alpha _{bc}=0$. The curvature tensor is trivial. After all, $\mathrm {ISO}(n)/\mathrm {SO}(n)$ is just the flat Euclidean space.

3 Beyond Adler Zero

Until now, we have been solely preoccupied with the question whether or not a given scattering amplitude vanishes in the soft limit for a chosen particle. This amounts to restricting the amplitude to a special hypersurface in the space of all allowed (on-shell and energy–momentum-conserving) kinematical configurations. There are however other interesting kinematical regimes that also probe the algebraic and geometric structure of the underlying EFT. One simple possibility is to take a consecutive soft limit for two particles. Here the geometric soft theorem (10.27) proves invaluable, giving immediately

$$\displaystyle \begin{aligned} \lim_{p_{n-1}\to0}\lim_{p_n\to0}\mathcal{A}_{a_1\dotsb a_n}(p_1,\dotsc,p_n)=\hat\nabla_{a_n}\hat\nabla_{a_{n-1}}\mathcal{A}_{a_1\dotsb a_{n-2}}(p_1,\dotsc,p_{n-2})\;. \end{aligned} $$

(10.45)

Of particular interest is the commutator of the two limits, probing the extent to which the limits interfere with each other,

(10.46)

where I used (A.79). Another possibility is to take a simultaneous soft limit for two or even more particles. This may lead to a kinematic singularity even in the absence of cubic interaction vertices. See [8, 11] for a precise formulation of such a simultaneous double-soft theorem and further details.

Here I will remain within the confines of the single soft limit, but look more closely at the asymptotic behavior of the scattering amplitude near the limit. I will promote the energy–momenta $p^\mu _i$ in an n-particle process to functions of a scaling parameter $z\in \mathbb {C}$ such that for all i and moreover . The other functions with $i=1,\dotsc ,n-1$ must be chosen so as to preserve overall energy–momentum conservation, , and remain on-shell, , for any $z\in \mathbb {C}$. Also, it is required that with $i=1,\dotsc ,n-1$ have a nonzero limit for $z\to 0$, that is, only the n-th particle becomes soft for small z.

For a generic kinematical configuration, the complexified tree-level amplitude will presumably be analytic in z in some neighborhood of the origin. It can thus be expanded in a series with a nonzero radius of convergence,

$$\displaystyle \begin{aligned} \hat{\mathcal{A}}_{a_1\dotsb a_n}(\hat p_1(z),\dotsc,\hat p_{n-1}(z),zp_n)=\sum_{k=0}^\infty z^kc^{(k)}_{a_1\dotsb a_n}\;. {} \end{aligned} $$

(10.47)

The soft theorems derived in Sects. 10.1 and 10.2 determine the leading term, , in terms of $(n-1)$-particle amplitudes with energy–momenta $\hat p^\mu _1(0),\dotsc ,\hat p^\mu _{n-1}(0)$. The asymptotic behavior of the amplitude at small z is dominated by the first term in (10.47) for which is nonzero. In the following, I will refer to the corresponding value of k as the soft scaling parameter and use the symbol $\sigma$;

$$\displaystyle \begin{aligned} \hat{\mathcal{A}}_{a_1\dotsb a_n}(\hat p_1(z),\dotsc,\hat p_{n-1}(z),zp_n)=z^\sigma c^{(\sigma)}_{a_1\dotsb a_n}+\mathcal{O}(z^{\sigma+1})\;. {} \end{aligned} $$

(10.48)

The notation used in (10.47) may suggest that the coefficients are independent of the detailed choice of functions $\hat p_i(z)$ beyond the basic requirements laid out above (10.47). I am not aware of any proof of this implicit assumption, or even a study addressing the issue. Still, it is plausible to assume that at least the leading term in (10.48) is independent of such arbitrary choices.

Clearly, $\sigma =0$ if the soft limit of the amplitude is nontrivial (nonvanishing). The Adler zero amounts to $\sigma \geq 1$. Barring accidental cancellations, we expect a generic EFT satisfying the Adler zero principle to have $\sigma =1$.

Example 10.5

Recall that for $n=4$, all the Mandelstam variables $s,t,u$ vanish in the soft limit for any of the four particles. This automatically guarantees Adler zero for the 4-particle amplitude in any derivatively coupled EFT of massless scalars. Moreover, in EFTs of a single species of massless scalar, permutation invariance allows only one linear function of Mandelstam variables, $s+t+u$, which is identically zero. Hence, in EFTs of a single NG boson, the 4-particle amplitude automatically satisfies $\sigma \geq 2$.

In case $\sigma \geq 2$ for all (tree-level) scattering amplitudes in a given EFT, we say that the soft limit of the amplitudes is enhanced. The rest of the present section is devoted to two immediate questions. Are there, in fact, any theories where scattering amplitudes feature an enhanced soft limit? If yes, is there any generic mechanism that makes the soft limit enhanced beyond the simple Adler zero?

3.1 Dirac–Born–Infeld Theory

The cleanest way to answer the first question is to find an explicit example. We will make an educated guess, loosely following [12]. Consider the class of EFTs of a single real NG field, $\pi$. We already know that $\sigma \geq 2$ for the 4-particle amplitude. In fact, the constraint $s+t+u=0$ leaves us with a single candidate, algebraically independent permutation-invariant 4-particle amplitude with $\sigma =2$, namely $\mathcal {A}(p_1,p_2,p_3,p_4)=(s^2+t^2+u^2)/(4\Lambda _D)$. Here $\Lambda _D$ is a nonzero constant of mass dimension D, the dimension of spacetime; the factor of 4 is a mere convention. This amplitude can be reproduced by a local quartic interaction operator, $-[(\partial _\mu \pi )^2]^2/(8\Lambda _D)$. This interaction in turn gives nontrivial amplitudes also for any even number of particles higher than four.

The problem is that already the next, $n=6$ amplitude fails to satisfy the desired scaling with $\sigma =2$. By dimensional analysis, the $n=6$ amplitude is a homogeneous function of degree 6 in energy–momenta. A detailed computation shows that the leading, term in its expansion (10.47) can be canceled by adding a new interaction operator carrying six derivatives, $[(\partial _\mu \pi )^2]^3/(16\Lambda _D)$. This pattern extends inductively to all higher orders. For any n, the $\sigma =2$ scaling of the (tree-level) $(2n)$-particle amplitude can be rescued by adding an interaction operator $[(\partial _\mu \pi )^2]^n$ and tuning its coefficient to a unique value. This ensures cancellation of the contribution to the $(2n)$-particle amplitude, generated by exchange diagrams with lower-point interaction vertices. Altogether, we end up with a unique theory modulo the choice of $\Lambda _D$; all the iteratively constructed interaction operators fold neatly into a closed-form Lagrangian,

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{DBI}}=\Lambda_D\sqrt{1+(\partial_\mu\pi)^2/\Lambda_D}\;. {} \end{aligned} $$

(10.49)

This is the so-called Dirac–Born–Infeld (DBI) theory.

A combination of (10.49) with the volume element, , admits a remarkable geometric interpretation. This is the induced volume measure on a D-dimensional hypersurface (“brane”), embedded in a flat $(D+1)$-dimensional spacetime and parameterized by the Minkowski coordinates $x^\mu$. The NG field $\pi (x)$ arises from spontaneous breaking of the symmetry under translations in the extra dimension by the brane. Properly normalized, the displacement of the brane along the extra dimension is $\pi (x)/\sqrt {\left \lvert {\Lambda _D}\right \rvert }$. The sign of $\Lambda _D$ determines the signature of the metric in the extra dimension. For positive $\Lambda _D$, the spacetime Lorentz group $\mathrm {SO}(d,1)$ is thus extended to $\mathrm {SO}(d,2)$, whereas for negative $\Lambda _D$ it is extended to $\mathrm {SO}(d+1,1)$. Up to an arbitrary choice of scale, we end up with two distinct DBI theories, differing by the geometry in the extra dimension.

The fact that the DBI Lagrangian (10.49) does not contain any fields without derivatives automatically guarantees the Adler zero. The realization of the $\sigma =2$ enhancement is however highly nontrivial. At first sight, it is not even obvious why the order-by-order cancellations required to make the soft limit enhanced should be possible at all. In Chap. 11, I will show that this is a consequence of the extended symmetry of the DBI theory. Let us therefore have a closer look at this symmetry. Under the translation in the extra dimension, the NG field shifts simply as $\pi (x)\to \pi '(x)=\pi (x)+\epsilon$. What is more interesting are the rotations connecting the D physical dimensions to the extra dimension. These can be parameterized by a Lorentz vector $\epsilon ^\mu$ and their infinitesimal form reads

$$\displaystyle \begin{aligned} \updelta\pi(x)=\sqrt{\left\lvert{\Lambda_D}\right\rvert }\epsilon_\mu x^\mu\;,\qquad \updelta x^\mu=-\frac{\operatorname{\mathrm{sgn}}\Lambda_D}{\sqrt{\left\lvert{\Lambda_D}\right\rvert }}\epsilon^\mu\pi(x)\;. {} \end{aligned} $$

(10.50)

Note that this is not a spacetime symmetry by the definition given in Sect. 4.1, since $\updelta x^\mu$ depends on the field $\pi (x)$. Rather, (10.50) is an example of a generalized local transformation. Its evolutionary form is

$$\displaystyle \begin{aligned} \pi'(x)-\pi(x)=\sqrt{\left\lvert{\Lambda_D}\right\rvert }\epsilon_\mu x^\mu+\frac{\operatorname{\mathrm{sgn}}\Lambda_D}{\sqrt{\left\lvert{\Lambda_D}\right\rvert }}\epsilon^\mu\pi(x)\partial_\mu\pi(x)\;, {} \end{aligned} $$

(10.51)

which depends on both $\pi (x)$ and its derivative. It is easy to verify that (10.49) is quasi-invariant under (10.51).

3.2 Galileon and Special Galileon Theory

Operationally, the form of the DBI theory is so strongly constrained because its Lagrangian only depends on the first derivatives of the NG field. One might hope to find more examples of theories with scattering amplitudes enhanced beyond the plain Adler zero by allowing for higher derivatives. Indeed, one can realize $\sigma =2$ trivially by setting

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{eff}}=\frac 12(\partial_\mu\pi)^2+\mathcal{L}_{\mathrm{int}}(\partial\partial\pi)\;, {} \end{aligned} $$

(10.52)

where $\mathcal {L}_{\mathrm {int}}$ is an arbitrary function of second derivatives of $\pi (x)$ whose Taylor expansion in $\pi$ starts at the fourth or higher order. This, while not very interesting per se, points in the right direction. Namely, the class of theories (10.52) is manifestly invariant under the so-called Galileon symmetry,

$$\displaystyle \begin{aligned} \updelta\pi(x)=\epsilon+\epsilon_\mu x^\mu\;. {} \end{aligned} $$

(10.53)

The interaction Lagrangian is strictly invariant, whereas the kinetic term is quasi-invariant. Mere invariance under the constant shift with scalar parameter $\epsilon$, together with the absence of cubic interaction vertices, already guarantees Adler zero. However, it is ultimately the full Galileon symmetry (10.53) that makes the soft limit of the scattering amplitudes enhanced; I will again return to this in Chap. 11. It is now natural to ask whether there might be any interactions that, similarly to the kinetic term, are merely quasi-invariant under (10.53). If present, such interactions would necessarily contain less than two derivatives per field, and thus realize a soft limit with $\sigma =2$ in a nontrivial manner. Moreover, by power counting, they would dominate over the interactions of type $\mathcal {L}_{\mathrm {int}}(\partial \partial \pi )$ in (10.52).

Finding all quasi-invariant Lagrangians requires an inspection of the relative Lie algebra cohomology of the spontaneously broken symmetry, as observed in a special case in Sect. 8.1. In case of the Galileon symmetry, a detailed analysis shows [13] that in D spacetime dimensions, there are $D+1$ algebraically independent quasi-invariant Galileon Lagrangians,

(10.54)

where $n=0,\dotsc ,D$. The quasi-invariance of all under the Galileon symmetry (10.53) is manifest. The first of these, , is a mere tadpole. This should be discarded in order for the EFT to be perturbatively well-defined. The next Galileon Lagrangian, , is just the kinetic term up to a rescaling and integration by parts. Each of the interaction Lagrangians with $n\geq 2$, or any linear combination thereof, realizes the $\sigma =2$ enhanced soft limit of scattering amplitudes. One might be concerned about the presence of the cubic operator, . This is however harmless. Namely, within the class of Lagrangians

$$\displaystyle \begin{aligned} \mathcal{L}_{\mathrm{Gal}}=\sum_{n=0}^Dc_n\mathcal{L}^{(n)}_{\mathrm{Gal}}\;, {} \end{aligned} $$

(10.55)

the couplings $c_n$ can be recalibrated by a one-parameter group of transformations, known as the Galileon duality [14, 15]. This amounts to a field redefinition, and thus does not affect the S-matrix of the theory. It can always be used to set $c_2$ to zero.

The 4-particle amplitude in the Galileon theory (10.55) is a homogeneous function of degree 6 in the particle energy–momenta. Permutation invariance together with the constraint $s+t+u=0$ implies that the amplitude equals $s^3+t^3+u^3$ up to overall normalization. Hence, in Galileon theories, the 4-particle amplitude is doubly enhanced, $\sigma =3$. Is it possible to adjust the values of $c_n$ in (10.55) so as to maintain $\sigma =3$ for all multiparticle amplitudes? The answer to this question is positive, and the resulting one-parameter family of theories is dubbed special Galileon.

The further enhancement of the soft limit of scattering amplitudes is associated with yet another, hidden symmetry [16]. Consider the infinitesimal transformation

$$\displaystyle \begin{aligned} \updelta_\epsilon\pi(x)\equiv\epsilon^{\mu\nu}\left[x_\mu x_\nu+\frac 1{\Lambda_{D+2}}\partial_\mu\pi(x)\partial_\nu\pi(x)\right]\;, {} \end{aligned} $$

(10.56)

where $\Lambda _{D+2}$ is a (nonzero) constant with mass dimension $D+2$ and $\epsilon ^{\mu \nu }$ is a traceless symmetric tensor of parameters. This is another nontrivial example of a generalized local transformation. To check whether (10.56) leaves the Lagrangian (10.55) quasi-invariant, it is sufficient to inspect the variation of the shift of the corresponding action, $\updelta _\epsilon S_{\mathrm {Gal}}$, with respect to $\pi$. A direct calculation gives

(10.57)

a reader wishing to verify this might want to take advantage of the auxiliary identity

(10.58)

The quasi-invariance of the Lagrangian (10.55) is now equivalent to the vanishing of (10.57), which leads to a recurrence relation for the couplings $c_n$, solved by

$$\displaystyle \begin{aligned} c_{2n}=\frac{(-1)^n}{\Lambda_{D+2}^n}\binom D{2n}\frac{c_0}{2n+1}\;,\qquad c_{2n+1}=\frac{(-1)^n}{\Lambda_{D+2}^n}\frac 1D\binom{D}{2n+1}\frac{c_1}{n+1}\;. \end{aligned} $$

(10.59)

What we have here are in principle two independent types of theories. However, the even set $c_{2n}$ inevitably includes the tadpole operator that we have discarded on physical grounds. We are thus left only with the odd terms $c_{2n+1}$, that is those Galileon operators (10.54) that are even in the NG field $\pi$. The parameter $c_1$ is fixed by demanding proper normalization of the kinetic term. The only genuine parameter of the special Galileon theory is therefore the scale $\Lambda _{D+2}$ that enters the symmetry transformation (10.56). Specifically in $D=4$ spacetime dimensions, we have $c_3=-c_1/(2\Lambda _6)$ as the sole effective coupling. Upon some integration by parts, the Lagrangian of the special Galileon theory can then be cast as

$$\displaystyle \begin{aligned} \mathcal{L}^{D=4}_{\mathrm{sGal}}\simeq\frac 12(\partial_\mu\pi)^2-\frac 1{12\Lambda_6}(\partial_\mu\pi)^2\big[(\Box\pi)^2-(\partial_\nu\partial_\lambda\pi)^2\big]\;. {} \end{aligned} $$

(10.60)

See [17] for a recent overview of the many intriguing properties of the special Galileon theory.

So far we have discovered three theories that realize in a nontrivial manner an enhanced soft limit of scattering amplitudes: DBI, Galileon and special Galileon. Remarkably, this is it as far as relativistic EFTs of a single NG boson are concerned [18]. Mapping the landscape of theories with $\sigma \geq 2$ and multiple NG bosons remains, as of writing this book, an open problem. Below, I outline a symmetry-based approach that makes the identification of candidate multiflavor EFTs in principle straightforward. In Sect. 10.4, I will then introduce an entirely different method, which utilizes solely known infrared properties of on-shell amplitudes and general quantum-field-theoretic principles. This approach is very efficient in proving that other nontrivial soft behavior than that observed in the above three theories is not possible.

3.3 Effective Theories with Enhanced Soft Limit from Symmetry

I have already hinted, so far without proof, that the enhanced soft limit of scattering amplitudes in the DBI and Galileon theories is a consequence of symmetry. The additional symmetry cannot be internal, for that would lead to mere linear constraints on scattering amplitudes or, even worse, additional NG bosons if spontaneously broken. Indeed, a glance at (10.50) and (10.53) shows that we are dealing with transformations with nontrivial dependence on spacetime coordinates, parameterized by a Lorentz vector. Guided by this observation, I will now show that the presence of such symmetries is severely constrained by group-theoretical consistency requirements. The discussion in this subsection is a special case of a framework developed in [19].

Consider an EFT for a single species of NG bosons. The symmetry of this EFT is generated by the operators of angular momentum $J_{\mu \nu }$ and energy–momentum $P_\mu$, and a spontaneously broken scalar generator Q. The commutation relations between $J_{\mu \nu }$ and $P_\mu$ are fixed by the Poincaré algebra. Furthermore, $[J_{\mu \nu },Q]=0$ expresses the fact that Q is a scalar, whereas $[P_\mu ,Q]=0$ is needed for Q to generate an internal symmetry. Let us now add a Lorentz-vector operator $K_\mu$, generating the tentative hidden symmetry responsible for enhancement of the soft limit of scattering amplitudes. Lorentz invariance restricts the commutation relations between $K_\mu$ and the other generators to³

$$\displaystyle \begin{aligned} \begin{alignedat}{3} [J_{\mu\nu},K_\lambda]&=\mathrm{i}(g_{\nu\lambda}K_\mu-g_{\mu\lambda}K_\nu)\;,&\qquad [P_\mu,K_\nu]&=\mathrm{i}(ag_{\mu\nu}Q+bJ_{\mu\nu})\;,\\ {} [K_\mu,K_\nu]&=\mathrm{i} cJ_{\mu\nu}\;,&\qquad [K_\mu,Q]&=\mathrm{i}(dP_\mu+eK_\mu)\;. \end{alignedat} {} \end{aligned} $$

(10.61)

The parameters $a,b,c,d,e$ cannot take arbitrary values, since to correspond to a well-defined Lie algebra, the commutators must satisfy a set of Jacobi identities. A straightforward calculation leads to the algebraically independent constraints $ae=0$, $b=0$ and $c=-ad$. Should $[P_\mu ,K_\nu ]$ be nonzero, which is necessary for the symmetry generated by $K_\mu$ to affect the momentum dependence of scattering amplitudes, a must be nonzero as well. Upon absorbing it into a redefinition of Q via $Q\to Q/a$, the three commutators in (10.61) not fixed by Lorentz invariance reduce to

$$\displaystyle \begin{aligned} [P_\mu,K_\nu]=\mathrm{i} g_{\mu\nu}Q\;,\qquad [K_\mu,K_\nu]=-\mathrm{i} vJ_{\mu\nu}\;,\qquad [K_\mu,Q]=\mathrm{i} vP_\mu\;, \end{aligned} $$

(10.62)

with the shorthand notation $v\equiv ad$.

The case of $v=0$ describes the Lie algebra of Galileon transformations (10.53). Here Q generates the constant shift with parameter $\epsilon$, whereas $K_\mu$ generates the linear shift with parameter $\epsilon _\mu$. It remains to address the case $v\neq 0$. Here the parameter can be nearly eliminated by rescaling both Q and $K_\mu$ by $\sqrt {\left \lvert {v}\right \rvert }$. What is left of v is just its sign, entering through $[K_\mu ,K_\nu ]=\mp \mathrm{i} J_{\mu \nu }$ and $[K_\mu ,Q]=\pm \mathrm{i} P_\mu$. Finally, merge $K_\mu$ with $J_{\mu \nu }$ and Q with $P_\mu$ by identifying $J_{\mu ,D+1}\equiv K_\mu$ and $P_{D+1}\equiv Q$ together with $g_{D+1,D+1}\equiv \pm 1$ and voilà, we find the $(D+1)$-dimensional Poincaré algebra. Its $v=+1$ version is based on the $\mathrm {SO}(d,2)$ group of spacetime rotations, whereas the $v=-1$ version is based on $\mathrm {SO}(d+1,1)$. These correspond precisely to the two mutations of the DBI theory.

We have successfully recovered the symmetries of the DBI and Galileon theories by a simple Lie-algebraic argument. The utility of the symmetry-based approach of course does not end here. It can be pursued further towards an explicit construction of effective actions, respecting the symmetry. This however requires a generalization of the techniques developed in Chaps. 7 and 8 to coordinate-dependent symmetries.

The classification of candidate symmetries leading to enhancement of the soft limit of scattering amplitudes, outlined here, can be extended in various directions. First, we may want to see how the special Galileon theory fits in. As suggested by (10.56), this requires adding another set of generators that span a traceless symmetric tensor representation of the Lorentz group. While the derivation of all the Lie-algebraic constrains is now more laborious, one eventually recovers (10.56) along with (10.60) as the only physically relevant solution [19]. A similar reasoning was used in [20] to show that in $D=4$ dimensions, there are no EFTs of a single NG boson that would realize $\sigma \geq 4$ enhancement of the soft limit in a nontrivial manner. Finally, the Lie algebra (10.61) can be extended to allow for multiple flavors of NG bosons. This turns out to be an efficient tool for carving out the landscape of potentially interesting EFTs [21]. The DBI and Galileon theories have a natural multiflavor generalization. However, the precise relation between the symmetry and the asymptotic behavior of scattering amplitudes in the soft limit that would allow us to unambiguously predict the value of $\sigma$, remains unknown.

4 Soft Recursion

No primer on scattering of NG bosons can be complete without at least briefly mentioning on-shell methods for scattering amplitudes, developed in the last decades. Given the context of this book, I will restrict the discussion to on-shell recursion relations for EFTs of NG bosons. These emerged as an adaptation of the framework, originally designed for the Yang–Mills theory by Britto, Cachazo, Feng and Witten. An interested reader will find an introduction to the latter in Chap. 3 of [4].

We return to the broad class of EFTs for multiple NG boson flavors. For notation simplicity, I will however suppress flavor and momentum labels and denote the on-shell tree-level amplitude for an n-particle scattering process simply as $\mathcal {A}_n$. Suppose now that we are able to complexify the energy–momenta in the process, $p^\mu _i\to \hat p^\mu _i(z)$, so that all the $\hat p^\mu _i(z)$ are linear functions of $z\in \mathbb {C}$, satisfying $\hat p^\mu _i(0)=p^\mu _i$. We of course still require that the $\hat p^\mu _i(z)$ add up to zero and remain on-shell, $[\hat p_i(z)]^2=0$, for any $z\in \mathbb {C}$. Then the scattering amplitude $\mathcal {A}_n$ is complexified to a meromorphic function $\hat {\mathcal {A}}_n(z)$ in the complex plane. Furthermore, for any holomorphic function $F_n(z)$ such that $F_n(0)=1$, the function $\hat {\mathcal {A}}_n(z)/[zF_n(z)]$ is also meromorphic and has a simple pole at $z=0$ with residue $\hat {\mathcal {A}}_n(0)=\mathcal {A}_n$. The residue theorem then implies that upon integration along an infinitesimal circle enclosing the origin,

$$\displaystyle \begin{aligned} \mathcal{A}_n=\frac 1{2\pi\mathrm{i}}\oint\mathrm{d} z\,\frac{\hat{\mathcal{A}}_n(z)}{zF_n(z)}\;. {} \end{aligned} $$

(10.63)

By the same residue theorem, the integral can also be expressed in terms of residues at the other poles in the complex plane, possibly including the residue at infinity. This is the central idea behind on-shell recursion techniques. To progress further, we need to be more specific about the complexification of the energy–momentum variables. We also need input on the asymptotic behavior of the amplitude at $z\to \infty$. Together, these will allow us to tailor the function $F_n(z)$ to the EFT at hand.

4.1 Complexified Kinematics

Our requirements that $\hat p^\mu _i(z)$ be linear in z and reduce to $p^\mu _i$ at $z=0$ are solved by

$$\displaystyle \begin{aligned} \hat p^\mu_i(z)=p^\mu_i+zq^\mu_i {} \end{aligned} $$

(10.64)

with arbitrary $q^\mu _i$. However, the on-shell condition $[\hat p_i(z)]^2=0$ is most easily satisfied if $q^\mu _i$ is parallel to $p^\mu _i$. I will parameterize such $q^\mu _i$ by a real constant $c_i$,⁴

$$\displaystyle \begin{aligned} \hat p^\mu_i(z)\equiv p^\mu_i(1-c_iz)\;. {} \end{aligned} $$

(10.65)

This is called the all-line soft shift of the energy–momentum variables. The last requirement we have to take care of is overall energy–momentum conservation, which leads to

$$\displaystyle \begin{aligned} \sum_{i=1}^nc_ip^\mu_i=0\;. {} \end{aligned} $$

(10.66)

I will always assume a generic kinematical configuration, disregarding accidental linear dependencies among the energy–momenta $p^\mu _i$. This is equivalent to the assumption that $p^\mu _i$ as a $D\times n$ matrix has the maximum possible rank consistent with energy–momentum conservation, that is $\min (D,n-1)$. By the rank–nullity theorem, the set of solutions to (10.66) then spans a vector space of dimension $n-\min (D,n-1)\geq 1$. The one-dimensional subspace that is always guaranteed to exist corresponds to $c_1=\dotsb =c_n$. However, it is desirable to have solutions for which all the $c_i$s are different. This will allow us to probe the single soft limit for the individual particles by tuning $z\to 1/c_i$. The existence of such solutions requires that $D<n-1$, or better $n\geq D+2$.

The all-line soft shift (10.65) cannot access the single soft limit of all scattering amplitudes, and the problem gets worse with increasing the spacetime dimension D. This motivated [18] to introduce alternative prescriptions, combining (10.64) and (10.65) for disjoint subsets of the n particles. To illustrate the idea, I will briefly outline the minimal modification of (10.65), dubbed the all-but-one-line soft shift. This applies (10.65) to the first $n-1$ particles but uses (10.64) for the last one,

$$\displaystyle \begin{aligned} \hat p^\mu_i(z)\equiv p^\mu_i(1-c_iz)\quad \text{for }i=1,\dotsc,n-1\;,\qquad \hat p^\mu_n(z)\equiv p^\mu_n+zq^\mu_n\;. {} \end{aligned} $$

(10.67)

Energy–momentum conservation and the on-shell condition now dictate that

$$\displaystyle \begin{aligned} q^\mu_n=\sum_{i=1}^{n-1}c_ip^\mu_i\;,\qquad q_n^2=p_n\cdot q_n=0\;. {} \end{aligned} $$

(10.68)

The first relation in (10.68) can be viewed as a definition of $q^\mu _n$. The second relation then constitutes two homogeneous constraints on $c_1,\dotsc ,c_{n-1}$, one linear and one quadratic. This defines an $(n-3)$-dimensional hypersurface in the $\mathbb {R}^{n-1}$ space of all $c_i$. The hypersurface always includes a one-dimensional linear subspace where $c_1=\dotsb =c_{n-1}$, as guaranteed by energy–momentum conservation. The existence of other, nontrivial solutions for $c_i$ requires that $n-3>1$, or $n\geq 5$, independently of the spacetime dimension D. We find that the all-but-one-line shift is applicable to a broader set of amplitudes than the all-line shift for any $D\geq 4$. The price to pay is that it only allows us to access the soft limit for $n-1$ particles out of n.

4.2 Recursion Relation for On-Shell Amplitudes

We are now ready to deal with the complexified amplitudes. Consider an EFT with soft scaling parameter $\sigma$. By definition, any complexified n-particle amplitude $\hat {\mathcal {A}}_n(z)$ in this theory has a zero of order $\sigma$ at $z=1/c_i$ for any of the particles to which (10.65) has been applied. It is then advantageous to set

$$\displaystyle \begin{aligned} F_{n,\sigma}(z)\equiv\prod_{i=1}^{n_{\mathrm s}}(1-c_iz)^\sigma\;, \end{aligned} $$

(10.69)

where $n_{\mathrm s}=n$ for the all-line shift and $n_{\mathrm s}=n-1$ for the all-but-one-line shift. This choice of $F_n(z)$ is optimized for maximum suppression of the integrand in (10.63) at large z without introducing any new poles in it. This suppression is sufficient to eliminate the pole at infinity provided

$$\displaystyle \begin{aligned} m<n_{\mathrm s}\sigma\;, {} \end{aligned} $$

(10.70)

where m is the degree of $\mathcal {A}_n$ as a function of energy–momenta.

Example 10.6

The constraint (10.70) is satisfied for a large class of EFTs. First, any EFT of the type (10.16), defined on a symmetric coset space, has $(m,\sigma )=(2,1)$ for any n. This follows from the relation $V=I+1$ between the numbers of interaction vertices V  and internal propagators I, valid for any connected tree-level Feynman diagram. Second, the Lagrangian of the DBI theory (10.49) contains one derivative on each field. As a result, the negative powers of energy–momentum due to propagators are exactly canceled by the adjacent interaction vertices. What is left is in effect one derivative per each external leg of the Feynman diagram, hence $(m,\sigma )=(n,2)$ for the DBI theory. Finally, the special Galileon theory in $D=4$ dimensions contains a single quartic interaction vertex with six derivatives. This gives $m=6V-2I$ and $n=4V-2I$, hence $(m,\sigma )=(2n-2,3)$.

With no pole at infinity and no new poles from $F_{n,\sigma }(z)$, all the $z\neq 0$ poles of the integrand in (10.63) come from the propagators inside $\hat {\mathcal {A}}_n(z)$. Consider a partition of all the particles in the scattering process into two disjoint subsets, I and $\tilde I$. Among all diagrams contributing to the tree-level amplitude $\hat {\mathcal {A}}_n(z)$, there is a subset of graphs where the particles in I and $\tilde I$ can be separated by cutting a single propagator. This subset of diagrams defines a factorization channel, which I will label simply with the letter I; see Fig. 10.3 for illustration. One can always assign the labels $I,\tilde I$ so that the n-th particle belongs to $\tilde I$. Then $c_i$ is well-defined for all particles in I even when the all-but-one-line shift is used. With the shorthand notation

$$\displaystyle \begin{aligned} P^\mu_I\equiv\sum_{i\in I}p^\mu_i\;,\qquad Q^\mu_I\equiv\sum_{i\in I}c_ip^\mu_i\;, \end{aligned} $$

(10.71)

the complex energy–momentum in the propagator separating I and $\tilde I$ is $P^\mu _I-zQ^\mu _I$. This leads to two poles at $z=z^\pm _I$ where

$$\displaystyle \begin{aligned} z^\pm_I=\frac 1{Q_I^2}\Bigl[P_I\cdot Q_I\pm\sqrt{(P_I\cdot Q_I)^2-P_I^2Q_I^2}\Bigr]\;. \end{aligned} $$

(10.72)

Near these poles, the complex amplitude $\hat {\mathcal {A}}_n(z)$ factorizes as

(10.73)

Here $\hat {\mathcal {A}}_I(z^\pm _I)$ is the complexified on-shell amplitude for scattering of particles in I, and likewise for $\hat {\mathcal {A}}_{\tilde I}(z^\pm _I)$. We arrive at the key result that the n-particle amplitude $\mathcal {A}_n$ can be reconstructed from on-shell amplitudes with lower numbers of particles, as long as it can be complexified using one of the prescriptions introduced in Sect. 10.4.1. Importantly, this soft recursion applies to the entire amplitudes. This observation lies at the heart of an approach to symbolic computation of scattering amplitudes that avoids the combinatorial explosion of ordinary perturbation theory. In practice, one only needs to calculate explicitly a small number of “seed” amplitudes. All higher-point amplitudes can then be reconstructed recursively.

Fig. 10.3

Example of a factorization channel in an 8-particle scattering process. The two sides of the factorization channel correspond respectively to $I=\{1,5,6,7\}$ and $\tilde I=\{2,3,4,8\}$. The gray disks represent the sum of all possible Feynman diagrams with the given external legs

Example 10.7

In $D=4$ spacetime dimensions, the all-line shift (10.65) can be used for any amplitude with $n\geq 6$. This leaves us with $\mathcal {A}_4$ and $\mathcal {A}_5$ as the seeds that need to be known a priori. For theories with just even interaction vertices, such as those from Example 10.6, one only needs to calculate the 4-particle amplitude explicitly. This is consistent with what we learned previously about the structure of EFTs on symmetric coset spaces. Namely, it follows from (10.42) that the quartic interaction vertex already fixes uniquely the entire effective Lagrangian. The same applies to the DBI theory.

Let us now see the recursion program through to the end. Upon summation over all factorization channels, (10.63) becomes [22]

$$\displaystyle \begin{aligned} \mathcal{A}_n=\sum_I\frac 1{P_I^2}\biggl[\frac{\hat{\mathcal{A}}_I(z^+_I)\hat{\mathcal{A}}_{\tilde I}(z^+_I)}{(1-z^+_I/z^-_I)F_{n,\sigma}(z^+_I)}+\frac{\hat{\mathcal{A}}_I(z^-_I)\hat{\mathcal{A}}_{\tilde I}(z^-_I)}{(1-z^-_I/z^+_I)F_{n,\sigma}(z^-_I)}\biggr]\;. {} \end{aligned} $$

(10.74)

The sum runs over all partitions of the particles, that is each pair $I,\tilde I$ is only counted once. This is our master equation, which allows recursive construction of tree-level scattering amplitudes by symbolic computation. However, the first step of the recursion can often be performed manually. Namely, if the functions $\hat {\mathcal {A}}_I(z),\hat {\mathcal {A}}_{\tilde I}(z)$ themselves do not contain any factorization poles, the summand in (10.74) can be interpreted in terms of the residues of the meromorphic function $\hat {\mathcal {A}}_I(z)\hat {\mathcal {A}}_{\tilde I}(z)/[zF_{n,\sigma }(z)(P_I-zQ_I)^2]$. This function does have also poles at $z=1/c_i$, since unlike $\hat {\mathcal {A}}_n(z)$ itself, $\hat {\mathcal {A}}_I(z)$ and $\hat {\mathcal {A}}_{\tilde I}(z)$ are not on-shell and thus do not necessarily vanish at these points. We can then use the residue theorem to rewrite the sum over poles at $z^\pm _I$ in terms of a sum over poles at $z=0$ and $z=1/c_i$,

$$\displaystyle \begin{aligned} \mathcal{A}_n=\sum_I\biggl[\frac{\hat{\mathcal{A}}_I(0)\hat{\mathcal{A}}_{\tilde I}(0)}{P_I^2}+\sum_{i=1}^{n_{\mathrm s}}\operatorname*{\mathrm{Res}}_{z=1/c_i}\frac{\hat{\mathcal{A}}_I(z)\hat{\mathcal{A}}_{\tilde I}(z)}{zF_{n,\sigma}(z)(P_I-zQ_I)^2}\biggr]\;. {} \end{aligned} $$

(10.75)

The first term takes into account diagrammatic contributions to $\mathcal {A}_n$ with an internal propagator. The second term must therefore match the contributions to $\mathcal {A}_n$ from contact n-point operators in the effective Lagrangian.

Example 10.8

The special Galileon theory (10.60) in $D=4$ dimensions has a single, quartic interaction vertex. When applied to the 6-particle amplitude, we therefore expect the second term in (10.75) to vanish. To check this, rewrite the Lagrangian as

(10.76)

The Feynman rule for the interaction vertex is $[-\mathrm{i} /(6\Lambda _6)]G(p_1,p_2,p_3)$, where $p^\mu _1,p^\mu _2,p^\mu _3$ are any three of the four four-momenta in the vertex and G denotes the corresponding Gram determinant. Since the latter is a homogeneous function of all its arguments of degree two, we have

$$\displaystyle \begin{aligned} \hat{\mathcal{A}}_I(z)\hat{\mathcal{A}}_{\tilde I}(z)=\frac 1{(6\Lambda_6)^2}G(\{p_i\}_{i\in I})G(\{p_j\}_{j\in\tilde I})\prod_{k=1}^6(1-c_kz)^2 {} \end{aligned} $$

(10.77)

for any partition of the six particles into two triples $I,\tilde I$. I have implicitly used the all-line shift. Note that the criterion (10.70) is satisfied even if we take $F_{n,\sigma }(z)$ with $\sigma =2$ instead of $\sigma =3$. The advantage of this choice is that the denominator term $F_{n,\sigma }(z)$ is then completely canceled by the last factor in (10.77). This indeed makes the second term in (10.75) disappear since the residues at $z=1/c_i$ trivially vanish.

The derivation of (10.74) does not apply to theories with a nonvanishing soft limit, where $\sigma =0$ and (10.70) cannot hold. Here we have two alternatives: either use $F_{n,0}(z)=1$ and deal with the residue at infinity, or use $F_{n,\sigma }(z)$ with $\sigma \geq 1$ and deal with the ensuing poles at $z=1/c_i$. The latter option is feasible, at least for two-derivative EFTs of the type (10.16), where sufficient suppression of the integrand in (10.63) is ensured by $F_{n,1}(z)$. The residues at the simple poles at $z=1/c_i$ can then be extracted from the geometric soft theorem (10.27) or its more explicit version (10.31). A minor modification of the steps leading to (10.74) now gives [8]

$$\displaystyle \begin{aligned} \mathcal{A}_n=\sum_I\frac 1{P_I^2}\sum_\pm\frac{\hat{\mathcal{A}}_I(z^\pm_I)\hat{\mathcal{A}}_{\tilde I}(z^\pm_I)}{(1-z^\pm_I/z^\mp_I)F_{n,1}(z^\pm_I)}+\sum_{i=1}^n\frac{\hat\nabla_{a_i}\hat{\mathcal{A}}_{n-1}(1/c_i)}{\prod_{j\neq i}(1-c_j/c_i)}\;. {} \end{aligned} $$

(10.78)

Here I used the shorthand notation $\hat {\mathcal {A}}_{n-1}$ for the amplitude where the i-th particle with flavor index $a_i$ has been removed. It is of course still possible to rewrite the first term in (10.78) as (10.75).

4.3 Soft Bootstrap

The modern scattering amplitude program strives to construct amplitudes, or even define what a quantum field theory is, without recourse to a Lagrangian or the symmetry thereof. It is therefore of great interest to understand what kind of soft behavior of scattering amplitudes is allowed on general grounds. Recursion relations such as (10.74) or its generalization (10.78) provide a versatile tool for constraining the landscape of possible EFTs.

In [18], soft recursion was used to derive a set of bounds on the soft scaling parameter $\sigma$. The most striking result is that in $D\geq 4$ spacetime dimensions, $\sigma =3$ is the maximum value that can be realized in a nontrivial manner. (Any $\sigma$ can be realized trivially by interactions with at least $\sigma$ derivatives per field.) Apart from this universal bound, the maximum achievable value of $\sigma$ in an EFT with a fixed set of interaction operators is also constrained by the number of derivatives per field. Consider a contact operator of the schematic type $(\partial \pi )^2\partial ^{m-2}\pi ^{n-2}$, and characterize the number of derivatives per field by the parameter $\varrho \equiv (m-2)/(n-2)$. We thus have $\varrho =1$ for the DBI theory and $\varrho =2$ for all the Galileon interactions (10.54). Then, the soft scaling parameter is bounded by $\sigma \leq \varrho +1$. This shows that the DBI and special Galileon theories are “exceptional” in the sense that they maximize $\sigma$ in their respective classes of theories with fixed $\varrho$. Similar bounds can be derived for translationally and rotationally invariant nonrelativistic EFTs [1].

The values of $\sigma$ and $\varrho$ alone do not necessarily specify a unique EFT. Here one can gain further insight by a procedure known as the soft bootstrap, see for instance [23, 24]. In $D\leq 4$ dimensions, the only seed amplitudes needed to apply soft recursion in combination with the all-line shift are $\mathcal {A}_4$ and $\mathcal {A}_5$. These are contact amplitudes without any exchange contributions. As such, they are given by polynomials of degree m in the particle energy–momenta, or $m/2$ in the Mandelstam variables. These polynomials are restricted by energy–momentum conservation and permutation invariance, and their complete classification is usually straightforward. With a list of candidate seed amplitudes at hand, one then proceeds to construct higher-point amplitudes via (10.74) or (10.78). The result of the recursion must be independent of the choice of parameters $c_i$ satisfying (10.66) or (10.68). Failure to pass this $c_i$-independence test indicates that the seed amplitudes do not correspond to any well-defined local field theory.

Soft bootstrap cannot, strictly speaking, be used to prove that a theory based on a fixed set of seed amplitudes exists. That would require checking the $c_i$-independence of recursively constructed amplitudes to all orders of the recursion. In this aspect, soft bootstrap is nicely complemented by more conventional EFT approaches based on symmetry. The latter are efficient in isolating a set of candidate EFTs with desired particle composition and soft behavior, as illustrated in Sect. 10.3.3. What soft bootstrap can do is show that other theories with given soft behavior than those known a priori do not exist.