j^{1}f}^{L}}{\partial{\bf x}^{J}}d{\bf x}^{I}\otimes d{\bf x}^{J}\,,$ (113)

where we employ a compact notation in which

$\\{{\bf x}^{I}\\}=\\{x^{\mu},u^{i},x_{\mu}^{i}\\}$ (114)

runs over all the 1-jet bundle coordinates, and here ${\bf j^{1}f}^{I}={\bf
x}^{I}\circ j^{1}f$. When we pull back the transformed metric along the
original section $j^{1}\phi$, we get

$\displaystyle(j^{1}\phi)^{\ast}(j^{1}f)^{\ast}(g^{(1)})=g^{(1)}_{KL}({\bf
x}\circ j^{1}f\circ j^{1}\phi)$ $\displaystyle\frac{\partial({\bf x}^{K}\circ
j^{1}f\circ j^{1}\phi)}{\partial{\bf x}^{I}}\frac{\partial({\bf x}^{L}\circ
j^{1}f\circ j^{1}\phi)}{\partial{\bf x}^{J}}$ $\displaystyle\times d({\bf
x}^{I}\circ j^{1}\phi)\otimes d({\bf x}^{J}\circ j^{1}\phi)\,.$

This can be expanded out in the various pieces, analogous to the expansion
(5.3), using Eqs. (108) and (88) to evaluate objects like ${\bf x}\circ
j^{1}f\circ j^{1}\phi$, but the result would be very cumbersome and so we do
not write it explicitly.

####  5.5.4 Example: free particle quantum mechanics

To see a more explicit expression, it is helpful to lose this full generality
and consider a specific example. We consider the quantum mechanics of a single
real scalar field, described by a $1$d sigma model on $M=\mathbb{R}$ (we will
treat this example in full generality in §6.1), for which we view a field as a
section $\phi$ of a bundle
$(E\cong\mathbb{R}_{t}\times\mathbb{R}_{u},\mathbb{R}_{t},\pi:(t,u)\mapsto
t)$. The 1-jet bundle is just 3-dimensional, on which we specify an initial
fibred coordinate system $(t,u,u_{t})$. We start from the simplest viable
Lagrangian, just ${\cal L}=\frac{1}{2}\dot{\phi}^{2}$ that describes a free
particle. As mentioned at the end of §6.1, one valid choice of metric
is252525One cannot take $g^{(1)}=dtdt+dudu$ on the 1-jet bundle, because that
is not invertible and so does not define a metric.

$g^{(1)}=u\,dt\,du_{t}+2\,du\,du\,.$ (115)

To recap, given a section $\phi:t\mapsto(t,\phi(t))$ which we prolong, this
gives the Lagrangian

${\cal
L}[\phi,g^{(1)}]=\frac{1}{2}\langle\eta^{-1},(j^{1}\phi)^{\ast}g^{(1)}\rangle=\frac{1}{2}\phi\ddot{\phi}+\dot{\phi}^{2}\sim\frac{1}{2}\dot{\phi}^{2}$
(116)

after using IBPs.

Now, consider the base-point preserving bundle morphism

$f:(t,u)\mapsto(t,u^{2})\,,$ (117)

which, similar to the field theory example considered in §2.3, implements a
non-derivative field redefinition sending our initial section $\phi$ to a
section $\psi$ whose components satisfy $\psi=\phi^{2}$. Applying Eq. (108),
the prolongation of this morphism to the 1-jet bundle is

$j^{1}f:(t,u,u_{t})\mapsto(t,u^{2},2uu_{t})\,,$ (118)

as expected from the chain rule. Now apply this morphism to the metric
$g^{(1)}$, using (113), to obtain

$g^{(1)}\mapsto(j^{1}f)^{\ast}g^{(1)}=2u^{2}u_{t}dt\,du+2u^{3}dt\,du_{t}+8u^{2}du\,du\,.$
(119)

Finally, pulling this back and forming the transformed Lagrangian gives

${\cal
L}[\phi,(j^{1}f)^{\ast}g^{(1)}]=\frac{1}{2}\langle\eta^{-1},(j^{1}\phi)^{\ast}(j^{1}f)^{\ast}g^{(1)}\rangle=\phi^{2}\dot{\phi}^{2}+\phi^{3}\ddot{\phi}+4\phi^{2}\dot{\phi}^{2}\buildrel\text{IBPs}\over{\sim}2\phi^{2}\dot{\phi}^{2}\,,$
(120)

after doing IBPs. We see that the formulae for prolonging the bundle morphism
to the 1-jet, where the original morphism corresponds to a particular non-
derivative field redefinition, gives the consistent transformation of the
Lagrangian when computed using a 1-jet metric. We expect the same applies for
any non-derivative field redefinition, which we would write as
$\phi\mapsto\psi=\phi^{2}$ at the level of coordinates implemented via the
general formula (108), and for general Lagrangians (with up to 4-derivatives).

#### Derivative field redefinitions

We emphasize that, mirroring our discussion in §3.3 of 2-derivative
Lagrangians using the field space bundle picture, one can always implement
more general derivative field redefinitions just by pulling back the metric
$g^{(1)}$ along a different section $j^{1}\psi$, as described in §5.4; but, in
this more general situation, one loses the equivalent description in terms of
a prolongated morphism on the jet bundle.

We can exhibit this by considering an explicit derivative field redefinition,
still in the context of this QM example for simplicity. Consider a change of
section $\phi\to\psi$ which, in components, is defined by
$\phi\to\psi=\phi-\epsilon\ddot{\phi}$, analogous to the field theory example
(66). Starting from the same metric (66) describing the free particle (when
pulled back along $j^{1}\phi$), but pulling back along the different section
$\psi$, we get

$\displaystyle{\cal L}[\phi,g^{(1)}]\mapsto{\cal L}[\psi,g^{(1)}]$
$\displaystyle=\frac{1}{2}\langle\eta^{-1},(j^{1}\psi)^{\ast}g^{(1)}\rangle$
(121)
$\displaystyle=\frac{1}{2}\langle\partial_{t}\otimes\partial_{t},dt\otimes
dt\rangle\left((u\circ j^{1}\psi)\partial_{t}(u_{t}\circ
j^{1}\psi)+2\partial_{t}(u\circ j^{1}\psi)\partial_{t}(u\circ
j^{1}\psi)\right)$
$\displaystyle=\frac{1}{2}(\phi-a\ddot{\phi})\partial_{t}(\dot{\phi}-a\dddot{\phi})+\partial_{t}(\phi-a\ddot{\phi})^{2}$
$\displaystyle=\frac{1}{2}\left(\phi\ddot{\phi}-a\ddot{\phi}^{2}-a\phi\ddddot{\phi}+a^{2}\ddot{\phi}\ddddot{\phi}\right)+\left(\dot{\phi}-a\dddot{\phi}\right)^{2}\buildrel\text{IBPs}\over{\sim}\frac{1}{2}\left(\dot{\phi}-a\dddot{\phi}\right)^{2}\,.$

As for the example considered above in §3.3.4, the formulae for implementing
this field redefinition are essentially trivial. Changing section corresponds
to ‘subbing in’ ${\bf x}\circ\psi$ everywhere in place of ${\bf x}\circ\phi$,
which are precisely the manipulations one would do to ‘change fields’ at the
level of the Lagrangian.

##  6 Examples

In this Section we consider concrete examples of scalar field theories, to
better acquaint the reader with the jet bundle formalism described in §4, and
the notion of geometry thereon (§5). The examples start simple and increase in
complexity.

###  6.1 Quantum mechanics on the line

We begin with an example that could not be much simpler, namely the quantum
mechanics of a point particle moving on the real line $M\cong\mathbb{R}$,
which is described by a $(0+1)$-dimensional sigma model on $\mathbb{R}$. The
1-jet bundle is here only 3-dimensional, meaning it is feasible to write down
metrics explicitly and thereby get a feel for the general formalism.

It is worth remarking that, in $0+1$ dimensions, the scalar field $\phi$ has
negative (classical) mass dimension, and so operators with more insertions of
the field $\phi$ become more and more relevant (in contrast to the story we
are used to in $3+1$ dimensions). Therefore, the ‘EFT expansion’ we consider
in this Section should be regarded as a formal construction, in which we
expand in both number of fields and number of derivatives, not to be
interpreted as an expansion in increasingly irrelevant operators.

The sigma model field is simply a real function
$\phi:\mathbb{R}\to\mathbb{R}:\,t\mapsto\phi(t)$. For our symmetries, we take
a pair of $O(1)\cong\mathbb{Z}_{2}$ factors that act as inversions on both the
source and target lines:

$\displaystyle T$ $\displaystyle:t\mapsto-t,$ $\displaystyle\text{(time
reversal)}\,,$ (122) $\displaystyle P$ $\displaystyle:\phi\mapsto-\phi,$
$\displaystyle\text{(parity)}\,.$ (123)

This defines a simple toy quantum field theory to which we can apply the jet
bundle formalism, which nonetheless is structurally similar to the Higgs EFTs;
the $O(1)_{t}\times O(1)_{u}$ symmetry, like the $O(3,1)_{\Sigma}\times
O(4)_{M}$ of the HEFT or SMEFT, allows only terms with even numbers of
derivatives, and even numbers of fields.

The first step is to pass from the formulation in terms of maps to field space
$M\cong\mathbb{R}$, to the formulation in which the scalar field is a section
of a bundle $E$. For simplicity, let us take the field space bundle here to be
(globally) a product manifold,
$E=\mathbb{R}_{t}\times\mathbb{R}_{u}\to\mathbb{R}_{t}$, on which our field

$\phi:t\mapsto(t,\phi(t))$ (124)

is a section. The 1-jet bundle $J^{1}E$ can then be modelled as the real
3-manifold

$J^{1}E\cong T^{\ast}\mathbb{R}_{t}\times\mathbb{R}_{u}\,,$ (125)

with local fibred coordinate system

$y^{I}=\\{t,u,u_{t}\\}\,.$ (126)

Along the section $j^{1}\phi$ of $J^{1}E\to\Sigma$, these coordinates evaluate
to

$\displaystyle t\circ j^{1}\phi$ $\displaystyle=t,$ (127) $\displaystyle
u\circ j^{1}\phi$ $\displaystyle=\phi(t),$ (128) $\displaystyle u_{t}\circ
j^{1}\phi$ $\displaystyle=\dot{\phi}(t)\,.$ (129)

Following our general formalism, an invariant metric on this 1-jet bundle can
be used to define a general 4-derivative Lagrangian for this system.

Before writing down the 1-jet metric, it is helpful to know where we want to
end up: using the usual procedure, let us first write down the most general
Lagrangian for this toy EFT with up to 4-derivatives in a non-redundant basis,
consistent with our pair of $O(1)$ symmetries. At the 2-derivative level,
there are two independent operators $\phi\ddot{\phi}$ and $\dot{\phi}^{2}$,
one of which can be removed (we remove $\phi\ddot{\phi}$) by an IBP relation.
At the 4-derivative level, there are 5 operators and 3 IBP relations. We
choose to remove the operators $\ddddot{\phi}$ and $\dddot{\phi}\dot{\phi}$,
which cannot in fact be written using the 1-jet bundle (they contain too many
derivatives acting on a single field), plus the operator
$\ddot{\phi}\dot{\phi}\dot{\phi}$, to leave a non-redundant basis spanned by
the 4-derivative operators $\dot{\phi}^{4}$ and $\ddot{\phi}^{2}$. In general
all these operators (including the 0-derivative constant piece) are multiplied
by arbitrary even functions of $\phi$.

Now let us see how to recover this general Lagrangian using the 1-jet bundle
described above. We write down the most general metric

$g=g_{IJ}dy^{I}\otimes dy^{J}$ (130)

on $J^{1}E$ that is consistent with our $O(1)_{t}\times O(1)_{u}$ global
symmetry. We truncate each metric component to include all contributions that
will pull back to Lagrangian terms with up to and including four time
derivatives. We have:

$\displaystyle g_{u_{t}u_{t}}$ $\displaystyle=A(u),$ (131) $\displaystyle
g_{uu_{t}}$ $\displaystyle=u_{t}uB(u),$ (132) $\displaystyle g_{uu}$
$\displaystyle=C(u)+u_{t}^{2}D(u),$ (133) $\displaystyle g_{tu_{t}}$
$\displaystyle=uE(u)+uu_{t}^{2}F(u),$ (134) $\displaystyle g_{tu}$
$\displaystyle=u_{t}G(u)+u_{t}^{3}H(u),$ (135) $\displaystyle g_{tt}$
$\displaystyle=-2V(u)+u_{t}^{2}J(u)+u_{t}^{4}K(u),$ (136)

where $A(u)$, … $V(u)$ are even functions of $u$, as dictated by our symmetry.
Written as a symmetric matrix of components:

$[g_{IJ}]=\begin{pmatrix}-2V+u_{t}^{2}J+u_{t}^{4}K&\quad
u_{t}G+u_{t}^{3}H&\quad uE+uu_{t}^{2}F\\\ \cdot&C+u_{t}^{2}D&uu_{t}B\\\
\cdot&\cdot&A\end{pmatrix}\,,$ (137)

in our local fibred coordinates (126).

To form a Lagrangian, we pull back this metric along the section $j^{1}\phi$,
and contract with the inverse metric on the 1d spacetime:

${\cal
L}[\phi,g]=\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle\,.$
(138)

Here, the inverse metric is just

$\eta^{-1}=\partial_{t}\otimes\partial_{t}.$

The resulting object is a scalar function on the worldline, i.e. a function of
$t$, that is our Lagrangian. We get

$\displaystyle{\cal
L}[\phi,g]=\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle=$
$\displaystyle\frac{1}{2}g_{tt}+g_{tu}\dot{\phi}+g_{tu_{t}}\ddot{\phi}+\frac{1}{2}g_{uu}\dot{\phi}^{2}+g_{uu_{t}}\dot{\phi}\ddot{\phi}+\frac{1}{2}g_{u_{t}u_{t}}\ddot{\phi}^{2}$
(139) $\displaystyle=$
$\displaystyle-V(\phi)+\frac{1}{2}\dot{\phi}^{2}(C+2G+J)+\ddot{\phi}\phi E$
(140)
$\displaystyle+\frac{1}{2}\dot{\phi}^{4}(D+2H+K)+\ddot{\phi}\dot{\phi}^{2}\phi(B+F)+\frac{1}{2}\ddot{\phi}^{2}A\,.$

where $V,A,B\dots K$ are implicitly understood to be even functions of the
field: $V(\phi),A(\phi)$, etc. As anticipated, we do not obtain the operators
$\ddddot{\phi}$ and $\dddot{\phi}\dot{\phi}$ with more than two derivatives on
a single field, but this is not an issue because those operators can be
removed from the Lagrangian using IBPs.

We do, however, obtain 2 (3) operator structures with 2 (4) derivatives, only
1 (2) of which are independent, as discussed. The extra operators can be
removed via IBP:

$\displaystyle\ddot{\phi}\phi E$
$\displaystyle=-\dot{\phi}^{2}E-\dot{\phi}\phi\,\partial_{t}E=-\dot{\phi}^{2}E-\dot{\phi}\phi\,2\phi\dot{\phi}\frac{dE}{d(\phi^{2})}=-\dot{\phi}^{2}(E+2\phi^{2}E^{\prime})\,,$
(141) $\displaystyle\ddot{\phi}\dot{\phi}^{2}\phi(B+F)$
$\displaystyle=-\frac{1}{3}\dot{\phi}^{4}\left(B+F+2\phi^{2}(B^{\prime}+F^{\prime})\right)\,,$
(142)

where $E^{\prime}=dE/d(\phi^{2})$ and analogously for $B^{\prime},F^{\prime}$.
With this notation, $B^{\prime},E^{\prime},F^{\prime}$ are even functions of
$\phi$. The Lagrangian (139) can now be written in a non-redundant basis:

$\displaystyle{\cal L}[\phi,g]=$
$\displaystyle\text{IBP}\circ\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$
(143) $\displaystyle=$
$\displaystyle-V+\frac{1}{2}\dot{\phi}^{2}(C+2G+J-2E-4\phi^{2}E^{\prime})$
(144)
$\displaystyle+\dot{\phi}^{4}\left(\frac{1}{2}(D+K)+H-\frac{1}{3}(B+F)-\frac{2}{3}\phi^{2}(B^{\prime}+F^{\prime})\right)+\frac{1}{2}\ddot{\phi}^{2}A\,.$

In fact we know a little more; the 2-derivative term must be non-vanishing,
and its leading order piece is fixed by canonical normalisation of $\phi$. We
require

$C(0)+2G(0)+J(0)-2E(0)=1\,.$ (145)

This defines a map from metric $g$ to ${\cal L}[\phi,g]$, the Lagrangian
functional (of $\phi$), in a particular basis with no residual IBP
redundancies. The map is, importantly, surjective, meaning that the most
general Lagrangian with up to 4-derivatives can be captured by pulling back a
metric on the 1-jet bundle.

The map $g\mapsto{\cal L}[\phi,g]$ is not, however, injective. This is a
general feature that will recur in every example: there are redundancies in
the map from 1-jet metrics to Lagrangians, in the sense that many different
metrics map to the same Lagrangian. A natural next step is to try to define
equivalence classes of metrics that map to the same Lagrangian (up to a total
derivative), in order to fix up a map ${\cal L}[\phi,g]$ that is both
surjective and injective. Generally, two metrics $g$ and $\tilde{g}$ are
equivalent iff

$\eta^{-1}(j^{1}\phi)^{\ast}(g-\tilde{g})=\partial_{\mu}K^{\mu}\,.$ (146)

However, it is not so straightforward to make a consistent choice of
representative for each such equivalence class, as is made apparent even in
this toy example where we can explicitly ‘read off’ the redundancies. For
example, one might try to define representative metrics by setting the
functions $G$, $J$, $E$ to zero (capturing the 2-derivative term via $C$), and
setting $H$, $K$, $B$, and $F$ to zero also (capturing the $\dot{\phi}^{4}$
term via $D$). But this can lead to pathologies when trying to describe
certain (physically reasonable) Lagrangians, such as the free theory ${\cal
L}=\frac{1}{2}\dot{\phi}^{2}$; with the ‘representative’ above, the ‘metric’
would have components $[g_{IJ}]=\text{diag}(0,1,0)$, which is singular and so
not a metric at all. A valid metric that does map to this Lagrangian,
accounting for IBPs in the manner described above, would be
$[g_{IJ}]=\text{diag}(u/2,2,u/2)$.

###  6.2 Real scalar in 4d

We now consider the case of a single real scalar field in Minkowski spacetime.
We consider the scalar field $\phi$ to be a section of the bundle
$\pi:E=\mathbb{R}^{1,3}\times\mathbb{R}_{u}\to\mathbb{R}^{1,3}$:

$\phi:x^{\mu}\mapsto\left(x^{\mu},\phi(x^{\mu})\right)\,.$ (147)

We require Poincaré invariance, but do not impose any internal symmetries on
the field. The 1-jet bundle $J^{1}E$ is a real 9-manifold

$J^{1}E\cong T^{*}\mathbb{R}^{1,3}\times\mathbb{R}_{u}\,,$ (148)

with local fibred coordinates

$y^{I}=\\{x^{\mu},u,u_{\mu}\\}\,.$ (149)

Along the section $j^{1}\phi$ of $J^{1}E\to\Sigma$, they evaluate to

$\displaystyle x^{\mu}\circ j^{1}\phi$ $\displaystyle=x^{\mu},$ (150)
$\displaystyle u\circ j^{1}\phi$ $\displaystyle=\phi(x^{\mu}),$ (151)
$\displaystyle u_{\mu}\circ j^{1}\phi$
$\displaystyle=\partial_{\mu}\phi(x^{\mu})\,.$ (152)

Following the same steps as in the previous example, we first write down the
Lagrangian we wish to obtain.

In this case, we are after a complete Green’s basis for a scalar EFT, i.e. a
set of operators that is non-redundant under IBP (but can in general contain
redundancies under Equations-of-Motion), with up to 4 derivatives and
arbitrarily large number of field insertions. We also require no more than 2
derivatives acting on each field and that boxes
$\square=\partial_{\mu}\partial^{\mu}$ are absent. The former condition is
necessary, because it is not possible to obtain such structures from a 1-jet
bundle metric. The complete absence of operators with boxes is not strictly
required by the metric structure, but turns out to be a very convenient
choice: the Lagrangian obtained pulling back the most general 1-jet bundle
metric contains _all_ box-less operators with up to 4 derivatives, but only a
subset of those with boxes, see §B. Restricting to box-less bases guarantees
that the scalar Lagrangian will be manifestly matched by pulling back the
1-jet bundle metric, without further manipulations and without introducing
cumbersome evaluations of which operators with boxes can or cannot be
obtained. The fact that all operators with boxes can always be removed from a
Green’s basis is not obvious a priori, but can be proven explicitly for any
scalar theory at any derivative order, see §B.

In the case of a single real scalar with interactions with up to 4
derivatives, the conditions listed above identify unambiguously:262626With
$N\geq 2$ field insertions, there are 2 operator structures with 2 derivatives
and 1 IBP. With $N=2$ (3) fields, one can write 4 (6) structures with 4
derivatives, related by 3 (4) IBP. With $N\geq 4$ fields, there are 7
structures with 4 derivatives and 4 IBP. The case $N=1$ is never relevant as
any such operator is automatically a total derivative. The number of
independent operators for each field multiplicity is automatically reflected
in the Lagrangian (153).

$\displaystyle\mathcal{L}=$
$\displaystyle\,\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\mathcal{F}_{0}(\phi)-\mathcal{V}(\phi)$
(153)
$\displaystyle+\frac{1}{\Lambda^{2}}(\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\partial^{\nu}\phi)\,\mathcal{F}_{1}(\phi)+\frac{1}{\Lambda^{3}}(\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\phi\partial^{\nu}\phi)\,\mathcal{F}_{2}(\phi)+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}\,\mathcal{F}_{3}(\phi)\,.$

where $\mathcal{F}_{0}(0)=1$ to have a canonical kinetic term and in general
$\mathcal{F}_{i}(0)=c_{i}$ for $i=1,2,3$, accounting for arbitrary Wilson
coefficients.272727We will see in §8 that the Wilson coefficient
$\mathcal{F}_{1}(0)$ is not completely arbitrary, but must satisfy a
positivity bound (293). In particular, its sign is negative Brivio:2014 . In
the context of our jet bundle geometry, this translates to a condition on the
signature of the metric.  The operator bases in Eqs. (153) and (193) below
were cross-checked with BasisGen Criado:2019ugp .

The Lagrangian (153) can be recovered from a 1-jet bundle metric which is
formally similar to the one in the 1d example. The metric components,
truncated such that only terms with up to 4 derivatives are generated when
pulling back to the Lagrangian, are

$\displaystyle g_{uu}^{\mu\nu}$ $\displaystyle=\eta^{\mu\nu}A(u)\,,$ (154)
$\displaystyle g_{uu}^{\mu}$
$\displaystyle=\frac{u^{\mu}}{\Lambda^{2}}B(u)\,,$ (155) $\displaystyle
g_{uu}$ $\displaystyle=C(u)+\frac{u_{\rho}u^{\rho}}{\Lambda^{4}}D(u)\,,$ (156)
$\displaystyle g_{\mu u}^{\nu}$
$\displaystyle=\delta^{\nu}_{\mu}E(u)+\frac{u^{\nu}u_{\mu}}{\Lambda^{4}}F_{1}(u)+\delta^{\nu}_{\mu}\frac{u_{\rho}u^{\rho}}{\Lambda^{4}}F_{2}(u)\,,$
(157) $\displaystyle g_{\mu u}$
$\displaystyle=\frac{u_{\mu}}{\Lambda^{2}}G(u)+\frac{u_{\mu}\,u_{\rho}u^{\rho}}{\Lambda^{6}}H(u)\,,$
(158) $\displaystyle g_{\mu\nu}$
$\displaystyle=-\frac{\eta_{\mu\nu}}{2}V(u)+\left(\frac{u_{\mu}u_{\nu}}{\Lambda^{4}}+\frac{\eta_{\mu\nu}}{4}\frac{u_{\rho}u^{\rho}}{\Lambda^{4}}\right)\frac{J(u)}{2}+\left(\frac{u_{\mu}u_{\nu}}{\Lambda^{4}}+\frac{\eta_{\mu\nu}}{4}\frac{u_{\rho}u^{\rho}}{\Lambda^{4}}\right)\frac{u_{\sigma}u^{\sigma}}{\Lambda^{4}}\frac{K(u)}{2}\,,$
(159)

where the coordinates $u^{\mu}$ are defined by raising indices with the
spacetime metric $u^{\mu}=\eta^{\mu\nu}u_{\nu}$ and we have restored the
powers of $\Lambda$ to make the dimensionalities manifest. The functions
$V,A\dots K_{i}$ are defined to be dimensionless, and therefore functions of
$(u/\Lambda)$. Moving from 1d to 4d Minkowski spacetime, we find a larger
number of independent tensor structures: the terms proportional $F_{1}$ and
$F_{2}$ were degenerate in the QM example, but become distinguishable in this
case. In Eq. (159) the functions $J$ and $K$ multiply a combination of two
Lorentz structures: they were grouped together because they yield the same
result upon contracting with $\eta^{\mu\nu}$:

$\eta^{\mu\nu}u_{\mu}u_{\nu}=u_{\mu}u^{\mu}=\frac{\eta^{\mu\nu}\eta_{\mu\nu}}{4}u_{\rho}u^{\rho}\,.$
(160)

Pulling back along the section $j^{1}\phi$ and contracting with the inverse
metric:

$\displaystyle\mathcal{L}[\phi,g]$
$\displaystyle=\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$
$\displaystyle\text{where}\quad\eta^{-1}$
$\displaystyle=\eta^{\rho\sigma}\partial_{\rho}\otimes\partial_{\sigma}\,,$
(161)

we get

$\displaystyle\mathcal{L}[\phi,g]=$
$\displaystyle\,\frac{\Lambda^{4}}{2}\eta^{\mu\nu}g_{\mu\nu}+\Lambda^{2}g_{\mu
u}\partial^{\mu}\phi+\Lambda g_{\mu
u}^{\nu}\partial^{\mu}\partial_{\nu}\phi+\frac{1}{2}g_{uu}\partial_{\mu}\phi\partial^{\mu}\phi$
(162)
$\displaystyle+\frac{g_{uu}^{\mu}}{\Lambda}\partial^{\nu}\phi\partial_{\mu}\partial_{\nu}\phi+\frac{1}{2}\frac{g_{uu}^{\mu\nu}}{\Lambda^{2}}\partial_{\rho}\partial_{\mu}\phi\partial^{\rho}\partial_{\nu}\phi$
$\displaystyle=$
$\displaystyle\,-\Lambda^{4}V+\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi)\left(C+2G+J\right)+\Lambda\square\phi
E$ (163)
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\partial^{\nu}\phi)}{\Lambda^{2}}\frac{A}{2}+\frac{\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\phi\partial^{\nu}\phi}{\Lambda^{3}}\left(B+F_{1}\right)+\frac{(\square\phi)(\partial_{\mu}\phi\partial^{\mu}\phi)}{\Lambda^{3}}F_{2}$
$\displaystyle+\frac{(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}}{\Lambda^{4}}\frac{D+2H+K}{2}\,,$

where $V,A\dots K$ are understood to be functions of $\phi/\Lambda$.

The terms proportional to $E$ and $F_{2}$ contain boxes. Both arise from
pulling back the $g_{\mu u}^{\nu}$ metric element, which, at the 1-jet level,
is the only one that allows the contraction of two derivatives acting on the
same field. As is evident in Eq. (162), box contractions can’t be implemented
in any of the other terms. Following the reasoning above and the argument
proved in §B, all operators with boxes obtained from the 1-jet bundle are
redundant with the box-less ones, and can be removed using IBPs. Let us do
this explicitly: if $E$ is a constant, the corresponding term is a total
derivative and therefore can be removed. If $E$ is at least linear in $\phi$,
then it can be recast as:

$\square\phi
E=-\partial_{\mu}\phi\partial^{\mu}E=-\frac{\partial_{\mu}\phi\partial^{\mu}\phi}{\Lambda}\frac{dE}{d(\phi/\Lambda)}\,.$
(164)

The term proportional to $F_{2}$ can always be removed via IBP, even if
$F_{2}$ is a constant:

$(\square\phi)\partial_{\mu}\phi\partial^{\mu}\phi
F_{2}=-2\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\phi\partial^{\nu}\phi
F_{2}-\frac{\partial_{\mu}\phi\partial^{\mu}\phi\,\partial_{\nu}\phi\partial^{\nu}\phi}{\Lambda}\frac{dF_{2}}{d(\phi/\Lambda)}$
(165)

No further manipulation is required in order to obtain a non-redundant
Lagrangian, that manifestly matches Eq. (153):

$\displaystyle\mathcal{L}[\phi,g]$ $\displaystyle=\,{\rm
IBP}\circ\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$
(166)
$\displaystyle=\,-\Lambda^{4}V+\frac{1}{2}(\partial_{\mu}\phi\partial^{\mu}\phi)\left(C+2G+J-2E^{\prime}\right)$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\partial^{\nu}\phi)}{\Lambda^{2}}\frac{A}{2}+\frac{\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\phi\partial^{\nu}\phi}{\Lambda^{3}}\left(B+F_{1}-2F_{2}\right)$
$\displaystyle+\frac{(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}}{\Lambda^{4}}\frac{D+2H+K-2F_{2}^{\prime}}{2}\,,$

where now $E^{\prime}=dE/d(\phi/\Lambda)$ and analogously for $F_{2}$. A
canonical kinetic term requires

$C(0)+2G(0)+J(0)-2E^{\prime}(0)=1\,.$ (167)

The results obtained for a real scalar in Minkowski spacetime share several
features with the case of quantum mechanics on the line: the map from metric
to Lagrangian is surjective, and a degeneracy is observed between the same
groups of functions, namely $(B,F_{i})$, $(C,G,J,E)$, $(D,H,K)$. The presence
of $E^{\prime}$ alone in this example, rather than both $E$ and $E^{\prime}$
as in the 1d case, is simply due to the different internal symmetry assumed.

Finally, we comment on the conditions for $g^{(1)}$ with elements (154)-(159)
to be invertible, which is a required to have a valid metric. The necessary
condition to have $\det[g^{(1)}]\neq 0$ is that at least one of the following
9 polynomials is non-vanishing:282828 Note that, in general, this requirement
does not preclude the possibility of coordinate singularities, i.e. conditions
in which the metric is singular only at specific values of the coordinates
$(u,u_{\mu})$, which should be examined case by case.

$\displaystyle C(2E^{2}+AV)\,,$ (168) $\displaystyle D(8F_{2}^{2}-AK)\,,$
(169) $\displaystyle B(4F_{1}G-BJ)-2CF_{1}^{2}\,,$ (170) $\displaystyle
B(4F_{1}H-BK)-2DF_{1}^{2}\,,$ (171) $\displaystyle
4D(2E^{2}+AV)+C(16EF_{2}-AJ)\,,$ (172) $\displaystyle
C(8F_{2}^{2}-AK)+D(16EF_{2}-AJ)\,,$ (173) $\displaystyle 4E(BG-
CF_{1})+A(CJ-2G^{2})+B^{2}V\,,$ (174) $\displaystyle 16E(BH-
DF_{1})+16F_{2}(BG-CF_{1})+4A(DJ+CK-4GH)-B^{2}J\,,$ (175) $\displaystyle
16F_{2}(BH-DF_{1})+4A(DK-2H^{2})-B^{2}K\,.$ (176)

In the generic case where 4-derivative operators with non-zero Wilson
coefficients are present in the Lagrangian, the conditions (168)–(176) leave
large freedom to choose a possible metric that pulls back to Eq. (153). A
minimal example is:

$\displaystyle g^{\mu\nu}_{uu}$
$\displaystyle=2\eta^{\mu\nu}\mathcal{F}_{1}(u)\,,$ $\displaystyle
g_{uu}^{\mu}$ $\displaystyle=\frac{u^{\mu}}{\Lambda^{2}}\mathcal{F}_{2}(u)\,,$
$\displaystyle g_{uu}$
$\displaystyle=\mathcal{F}_{0}(u)+2\frac{u_{\mu}u^{\mu}}{\Lambda^{4}}\mathcal{F}_{3}(u)\,,$
(177) $\displaystyle g_{\mu u}^{\nu}$ $\displaystyle=0\,,$ $\displaystyle
g_{\mu u}$ $\displaystyle=0\,,$ $\displaystyle g_{\mu\nu}$
$\displaystyle=-\frac{1}{2}\eta_{\mu\nu}\frac{\mathcal{V}(u)}{\Lambda^{4}}\,.$

If we restrict the Lagrangian to the 2- and 0-derivative terms, setting to
zero all functions except $V,C,E$, then the conditions for invertibility
collapse into $2CE^{2}\neq 0$, which implies that both $C$ and $E$ must be
non-vanishing. This indicates that, even though it produces a physically
redundant operator, the $E$ term is necessary for the jet-bundle metric to be
non-singular for any value of the 4-derivative Wilson coefficients,
consistently with what observed in §§5.5.4 and 6.1. Restricting further to
0-derivatives only is incompatible with both physics (as the kinetic term is
removed) and jet-bundle geometry, as the metric would be inevitably singular.

#### Equation-of-Motion redundancies

The operator basis considered up to this point is non-redundant under IBP,
i.e. it is a Green’s basis for operators with up to 4 derivatives and
arbitrary number of fields. It is interesting to note that the 1-jet bundle
geometry automatically accounts for all IBP among operators with up to 4
derivatives, except the residual $E$ and $F_{2}$ function
redundancies.292929The fact that no redundant box-less operators are produced
by the jet bundle metric is due to the fact that, in this particular case,
there are exactly as many operators with boxes as independent IBP. Therefore
the set of _all_ box-less operators with up to 4 derivatives forms a complete
and non-redundant Green’s basis.

When constructing EFTs, we are typically interested in removing Equation-of-
Motion (EOM) redundancies as well. EOM relations express invariance properties
of the $S$-matrix. Because this information is not contained in the geometric
formulation, they need to be imposed as additional conditions on top of the
Lagrangian in Eq. (153).

Formally, the removal of EOM redundancies in an EFT is performed by applying
_derivative_ field redefinitions on the Lagrangian. Therefore, based on the
discussion in §5, two jet bundle metrics pulling back to EOM-equivalent
Lagrangians are in general not related by a morphism. In practice, we do not
expect to find an operation on the jet bundle metric that is mathematically
equivalent to basis reduction via EOM. Nevertheless, it is instructive to
consider the EOM-reduced Lagrangian for the toy example under study, and ask
what classes of jet bundle metrics would pull back to it. We will come back to
the inadequacy of jet bundle morphisms (even prolonged ones) at the end of
this subsection.

To this end, we take a step back and review the construction of the operator
basis. This operation is necessary because, being (153) “box-less”, EOM
redundancies are not manifest. At the same time, we hope to give a useful
illustration of the application of derivative field redefinitions in an EFT.

At the 2-derivatives level and with $N$ fields, there are two allowed
operators structures:

$\displaystyle O^{(2)}_{1}$ $\displaystyle=(\square\phi)\phi^{N-1}\,,$
$\displaystyle O^{(2)}_{2}$
$\displaystyle=(\partial_{\mu}\phi\partial^{\mu}\phi)\phi^{N-2}\,,$ (178)

which are related by the IBP

$O^{(2)}_{1}+(N-2)O^{(2)}_{2}=0\,.$ (179)

To obtain (153), we have used the IBP to remove $O_{1}^{(1)}$ for each value
of $N\geq 2$. This is always the most convenient choice for $N=2$, since in
this case $O_{2}^{(2)}$ is the kinetic term. However, for $N\geq 3$, we can
choose to remove $O_{2}^{(2)}$ via IBP, and then use the EOM to trade
$O_{1}^{(2)}$ for 0-derivative interactions, effectively obtaining a basis
without any 2-derivative operator, besides the kinetic term. In practice, the
EOM removal implies that we start with a Lagrangian

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\mathcal{V}(\phi)+\sum_{N=3}^{\infty}\frac{c_{1,N}^{(2)}}{\Lambda^{N-2}}(\square\phi)\phi^{N-1}+\dots\,,$
(180)

where the scalar potential is as in Eq (96) and the dots stand for other
operators that are present in general. Then we apply the derivative
redefinition

$\phi\mapsto\phi-\sum_{k=0}^{\infty}\frac{\epsilon_{k}}{\Lambda^{k+2}}(\square\phi)\phi^{k}\,,$
(181)

and truncate again at 2-derivatives:

$\displaystyle\mathcal{L}$
$\displaystyle\to\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\mathcal{V}(\phi)-\frac{d\mathcal{V}(\phi)}{d\phi}\sum_{k}\,\frac{\epsilon_{k}}{\Lambda^{k+2}}(\square\phi)\phi^{k}+\sum_{N=3}^{\infty}\frac{c_{1,N}^{(2)}}{\Lambda^{N-2}}(\square\phi)\phi^{N-1}+\dots$
(182)
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\mathcal{V}(\phi)+(\square\phi)\left[-\sum_{n,k}\frac{\epsilon_{k}nV_{n}}{\Lambda^{n+k-2}}\phi^{n+k-1}+\sum_{N=3}^{\infty}\frac{c_{1,N}^{(2)}}{\Lambda^{N-2}}\phi^{N-1}\right]+\dots$
(183)
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\mathcal{V}(\phi)+\sum_{N=3}^{\infty}\left[c_{1,N}^{(2)}-\sum_{n=1}^{N}\epsilon_{N-n}nV_{n}\right]\frac{\phi^{N-1}}{\Lambda^{N-2}}(\square\phi)+\dots$
(184)

Choosing the constants $\epsilon_{k}$ such that
$\sum_{n=1}^{N}\epsilon_{N-n}nV_{n}=c_{1,N}^{(2)}$ removes the infinite tower
of $O_{1}^{(2)}$ operators with $N\geq 3$.303030This is an infinite system of
equations, each labeled by $N$ and depending on $N$ free parameters
$\epsilon_{0}\dots\epsilon_{N-1}$. It always admits a solution that can be
found recursively.

At the level of 4 derivatives and with $N$ fields, we can write the 7
structures

$\displaystyle O_{1}^{(4)}$ $\displaystyle=(\square\square\phi)\phi^{N-1}\,,$
$\displaystyle O_{5}^{(4)}$
$\displaystyle=(\square\phi)(\partial_{\mu}\phi\partial^{\mu}\phi)\phi^{N-3}\,,$
(185) $\displaystyle O_{2}^{(4)}$
$\displaystyle=(\partial_{\mu}\square\phi)(\partial^{\mu}\phi)\phi^{N-2}\,,$
$\displaystyle O_{6}^{(4)}$
$\displaystyle=(\partial_{\mu}\partial_{\nu}\phi\,\partial^{\mu}\phi\partial^{\nu}\phi)\phi^{N-3}\,,$
(186) $\displaystyle O_{3}^{(4)}$
$\displaystyle=(\square\phi)(\square\phi)\phi^{N-2}\,,$ $\displaystyle
O_{7}^{(4)}$
$\displaystyle=(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}\phi^{N-4}\,,$ (187)
$\displaystyle O_{4}^{(4)}$
$\displaystyle=(\partial_{\mu}\partial_{\nu}\phi)(\partial^{\mu}\partial^{\nu}\phi)\phi^{N-2}\,,$
(188)

which are related by the 4 IBPs

$\displaystyle O_{1}^{(4)}+(N-1)O_{2}^{(4)}$ $\displaystyle=0\,,$ (189)
$\displaystyle O_{2}^{(4)}+O_{3}^{(4)}+(N-2)O_{5}^{(4)}$ $\displaystyle=0\,,$
(190) $\displaystyle O_{2}^{(4)}+O_{4}^{(4)}+(N-2)O_{6}^{(4)}$
$\displaystyle=0\,,$ (191) $\displaystyle
2O_{6}^{(4)}+O_{5}^{(4)}+(N-3)O_{7}^{(4)}$ $\displaystyle=0\,.$ (192)

To obtain Eq (153), we used the four IBP to remove the four operators with
boxes. On the other hand, the more widely used procedure to construct minimal
operator bases is to first use IBP to remove as many operators _without_ boxes
as possible, and then employ EOMs to reduce box operators to lower derivative
structures, by using field redefinitions similar to Eq (181). With $N=2,3$
fields, this procedure reveals that all 4-derivative operators are redundant
with lower-derivative ones. However, for $N\geq 4$, the operator $O_{7}^{(4)}$
cannot be removed, neither with IBP nor with EOMs.

In summary, a IBP+EOM reduced Lagrangian with up to 4 derivatives is

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\mathcal{V}(\phi)+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}\mathcal{F}_{3}(\phi)\,,$
(193)

which is simply a subset of the Green’s basis in Eq (153), as expected. This
indicates that the EOM-reduced Lagrangian can be obtained by pulling back a
correspondingly reduced metric $g$ on the 1-jet bundle. Specifically, Eq.
(193) can be obtained requiring

$\displaystyle C(\phi)+2G(\phi)+J(\phi)-2E^{\prime}(\phi)$
$\displaystyle=1\,,$ (194) $\displaystyle V(\phi)$
$\displaystyle=\mathcal{V}(\phi)/\Lambda^{4}$ (195) $\displaystyle A(\phi)$
$\displaystyle=0\,,$ (196) $\displaystyle B(\phi)+F_{1}(\phi)-2F_{2}(\phi)$
$\displaystyle=0\,,$ (197)
$\displaystyle\frac{1}{2}\left(D(\phi)+K(\phi)\right)+H(\phi)-F_{2}^{\prime}(\phi)$
$\displaystyle=\mathcal{F}_{3}(\phi)\,.$ (198)

This still leaves much freedom to identify a suitable metric, that respects
the conditions (168)-(176) as well. A minimal example is

$\displaystyle g^{\mu\nu}_{uu}$ $\displaystyle=g^{\mu}_{uu}=g_{\mu u}=0\,,$
$\displaystyle g_{uu}$
$\displaystyle=2\frac{u_{\rho}u^{\rho}}{\Lambda^{4}}\mathcal{F}_{3}(u)\,,$
(199) $\displaystyle g_{\mu\nu}$
$\displaystyle=-\frac{\eta_{\mu\nu}}{2}\frac{\mathcal{V}(u)}{\Lambda^{4}}\,,$
$\displaystyle g^{\nu}_{\mu u}$
$\displaystyle=-\frac{\delta^{\nu}_{\mu}}{2}\frac{u}{\Lambda}\,.$ (200)

#### Can prolonged bundle morphisms capture ‘EOM field redefinitions’?

Derivative field redefinitions of the form

$\phi\mapsto\psi\quad\text{such that}\quad
u\circ\psi=\psi\left(u\circ\phi,\partial_{\mu}\partial^{\mu}(u\circ\phi)\right)\,,$
(201)

namely in which the new section $\psi$ is a function of $\phi$ and $\Box\phi$,
have played an important role in our discussion of this 4d real scalar
example. Before our introduction of jet bundles, in §3.3 we discussed to what
extent a morphism on the field space bundle $(E,\Sigma,\pi)$, which recall can
capture any non-derivative field redefinition, can capture the effects of
derivative field redefinitions such as this (which we can understand
generically to be changes of section). There are indeed fundamental reasons
why such morphisms cannot induce derivative maps on sections (see lemma 3.1).

Nonetheless, for the particular example of a 4d real scalar with canonical
kinetic term and a $\phi^{4}$ potential, we saw that a bundle morphism of the
form $(x^{\mu},u)\mapsto(x^{\mu},u+\epsilon\lambda(u))$ can replicate the
shift of the Lagrangian under a derivative change of section,
$\phi\mapsto\psi=\phi-\epsilon\Box\phi$, at least when the Lagrangian is
truncated to remain at 2-derivative order, and keeping only operators with up
to 4-fields. One might ask whether similar ‘accidents’ can happen when we pass
to 4-derivative Lagrangians using our 1-jet bundle formalism. In other words,
can a morphism on the 1-jet bundle replicate, at the Lagrangian level, the
effects of derivative field redefinitions that depend on $\Box\phi$? We here
suggest that the answer to this question is no. (More precisely, we expect
that such an equivalence will only occur for very special examples, truncated
in a particular way.)

To see this, let us consider the effects of a prolongated bundle morphism, as
defined in §5.5, on the 1-jet bundle metrics constructed for the 4d real
scalar in this Subsection. We specialise to the case of a base-point
preserving ‘perturbative morphism’ on the bundle $E$ that is consistent with
Poincaré symmetry, considered in §3.3.4, of the form

$f_{E}:(x^{\mu},u)\mapsto(x^{\mu},f(u))=(x^{\mu},u+\epsilon\lambda(u))\,.$
(202)

Using Eq. (108), the prolongation of this bundle morphism to the 1-jet bundle
is

$j^{1}f_{E}:(x^{\mu},u,u_{\mu})\mapsto\left(x^{\mu},f(u),u_{\mu}\frac{\partial
f}{\partial u}\right)=\left(x^{\mu},u+\epsilon\lambda,u_{\mu}+\epsilon
u_{\mu}\lambda^{\prime}\right)\,,$ (203)

where $\lambda^{\prime}:=d\lambda/du$ etc. The coordinate basis 1-forms
transform as

$\displaystyle
j^{1}f_{E}:(dx^{\mu},du,du_{\mu})\mapsto\left(dx^{\mu},du(1+\epsilon\lambda^{\prime}),du_{\mu}(1+\epsilon\lambda^{\prime})+\epsilon
u_{\mu}\lambda^{\prime\prime}du\right)\,,$ (204)

from which we can compute the transformation law for the 1-jet bundle metric.

Let us start with the 1-jet metric written in the form of Eq. (177)

$\displaystyle g^{(1)}=$
$\displaystyle-\frac{\eta^{\mu\nu}}{2}\frac{\mathcal{V}(u)}{\Lambda^{4}}dx^{\mu}dx^{\nu}+\left(\mathcal{F}_{0}(u)+\frac{2u_{\mu}u^{\mu}}{\Lambda^{4}}\mathcal{F}_{3}(u)\right)dudu$
(205)
$\displaystyle+\mathcal{F}_{2}(u)\frac{u^{\mu}}{\Lambda^{3}}dudu_{\mu}+\frac{2\eta^{\mu\nu}}{\Lambda^{2}}\mathcal{F}_{1}(u)du_{\mu}du_{\nu}\,,$

which pulls back to give the Lagrangian in Eq. (153). Doing our prolongated
jet bundle morphism and keeping only terms that are linear in the perturbation
$\epsilon$, we compute the transformed metric to be

$\displaystyle j^{1}f_{E}^{\ast}g^{(1)}$
$\displaystyle=-\frac{1}{2}\eta^{\mu\nu}\mathcal{V}(u+\epsilon\lambda)dx^{\mu}dx^{\nu}+\mathcal{F}_{0}(u+\epsilon\lambda)(1+2\epsilon\lambda^{\prime})dudu$
(206)
$\displaystyle+\frac{u_{\mu}u^{\mu}}{\Lambda^{4}}\left[2\mathcal{F}_{3}(u+\epsilon\lambda)(1+4\epsilon\lambda^{\prime})+\epsilon\mathcal{F}_{2}(u+\epsilon\lambda)\lambda^{\prime\prime}\right]dudu$
$\displaystyle+\frac{u^{\mu}}{\Lambda^{2}}\left[\mathcal{F}_{2}(u+\epsilon\lambda)(1+3\epsilon\lambda^{\prime})+4\epsilon\mathcal{F}_{1}(u+\epsilon\lambda)\lambda^{\prime\prime}\right]dudu_{\mu}$
$\displaystyle+2\eta^{\mu\nu}\mathcal{F}_{1}(u+\epsilon\lambda)(1+2\epsilon\lambda^{\prime})du_{\mu}du_{\nu}+O(\epsilon^{2})\,.$

Expanding also each ‘structure function’ $\mathcal{V}$,
$\mathcal{F}_{0,1,2,3}$ about $u$, we can capture the effects of this
transformation in terms of the following shifts at leading order in $\epsilon$
(the terms neglected are $O(\epsilon^{2})$):

$\displaystyle\mathcal{V}$
$\displaystyle\mapsto\mathcal{V}+\epsilon\lambda\mathcal{V}^{\prime}\,,$ (207)
$\displaystyle\mathcal{F}_{0}$
$\displaystyle\mapsto\mathcal{F}_{0}+\epsilon(2\lambda^{\prime}\mathcal{F}_{0}+\lambda\mathcal{F}_{0}^{\prime})\,,$
(208) $\displaystyle\mathcal{F}_{1}$
$\displaystyle\mapsto\mathcal{F}_{1}+\epsilon(2\lambda^{\prime}\mathcal{F}_{1}+\lambda\mathcal{F}_{1}^{\prime})\,,$
(209) $\displaystyle\mathcal{F}_{2}$
$\displaystyle\mapsto\mathcal{F}_{2}+\epsilon(3\lambda^{\prime}\mathcal{F}_{2}+4\lambda^{\prime\prime}\mathcal{F}_{1}+\lambda\mathcal{F}_{2}^{\prime})\,,$
(210) $\displaystyle\mathcal{F}_{3}$
$\displaystyle\mapsto\mathcal{F}_{3}+\epsilon(4\lambda^{\prime}\mathcal{F}_{3}+\frac{1}{2}\lambda^{\prime\prime}\mathcal{F}_{2}+\lambda\mathcal{F}_{3}^{\prime})\,.$
(211)

If we then pullback this metric to $\Sigma$ along the original (prolongated)
section $j^{1}\phi$, we obtain the transformed Lagrangian

$\displaystyle{\cal L}[\phi,(j^{1}f_{E})^{\ast}g^{(1)}]=$
$\displaystyle\,\,\frac{1}{2}\left(\mathcal{F}_{0}+\epsilon(2\lambda^{\prime}\mathcal{F}_{0}+\lambda\mathcal{F}_{0}^{\prime})\right)(\partial\phi)^{2}-(\mathcal{V}+\epsilon\lambda\mathcal{V}^{\prime})$
(212)
$\displaystyle+\frac{1}{\Lambda^{2}}\left(\mathcal{F}_{1}+\epsilon(2\lambda^{\prime}\mathcal{F}_{1}+\lambda\mathcal{F}_{1}^{\prime})\right)\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\partial^{\nu}\phi$
$\displaystyle+\frac{1}{\Lambda^{3}}\left(\mathcal{F}_{2}+\epsilon(3\lambda^{\prime}\mathcal{F}_{2}+4\lambda^{\prime\prime}\mathcal{F}_{1}+\lambda\mathcal{F}_{2}^{\prime})\right)\partial_{\mu}\partial_{\nu}\phi\partial^{\mu}\phi\partial^{\nu}\phi$
$\displaystyle+\frac{1}{\Lambda^{4}}\left(\mathcal{F}_{3}+\epsilon(4\lambda^{\prime}\mathcal{F}_{3}+\frac{1}{2}\lambda^{\prime\prime}\mathcal{F}_{2}+\lambda\mathcal{F}_{3}^{\prime})\right)(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}\,\,.$

Thus, upon doing the prolongated bundle morphism to transform the metric, we
see that the transformation of the Lagrangian has a particularly constrained
structure; operators with a certain number $D$ of derivatives (here 0, 2, or 4
due to our symmetry assumptions) mix only into operators with the same value
of $D$.

In contrast, if we were to perform a derivative change of section, we would
see operators of lower-derivative-order mix into operators of higher-
derivative-order, as in the $\phi^{4}$ example considered in §3.3.4: for
example, it is straightforward to compute that $\mathcal{V}$ ‘mixes into’ the
2-derivative structure function $\mathcal{F}_{0}$ and all three 4-derivative
structure functions $\mathcal{F}_{1,2,3}$; the 2-derivative term
$\mathcal{F}_{0}$ likewise mixes into all of $\mathcal{F}_{1,2,3}$. For the
2-derivative toy example considered in §3.3.4 we exploited the fact that there
was only a single such operator mixing, namely of the potential into the
2-derivative term, that could be cancelled by a bundle morphism that shifts
the kinetic term (explicitly, this fixed $\lambda$ in terms of
$\partial_{u}V$) provided the potential were high enough order that we could
neglect its own variation. This was a very special situation which enabled the
derivative change of section to be matched by a bundle morphism. In this more
general 4-derivative example, we must still fix $\lambda$ by matching the
shift of the 2-derivative term (even if that is zero, for example in the case
of vanishing potential), but then we cannot hope to simultaneously match the
shift in any 4-derivative term that is present. So one cannot even contrive a
theory with 2- and 4-derivative terms for which the
$\phi\to\psi(\phi,\Box\phi)$ change of section can be described via an
(appropriately prolongated) morphism of the jet bundle. Nonetheless, we
reiterate that the derivative field redefinition can be implemented as a
prolonged change of section, as discussed in §5.4.

###  6.3 Two real scalars in 4d, without internal symmetries

We generalize the previous example by introducing two scalar fields
$\phi^{1},\phi^{2}$, without imposing any internal symmetries relating them.
The main goals in considering this example are twofold: (i) to cover a case in
which non-trivial field indices appear in the metric, and (ii) to check that
the 1-jet bundle gives a complete Green’s basis for a theory with multiple
fields but without internal symmetries.

The formalism is identical to the previous case, except the 1-jet bundle is
now a real 14-manifold with local fibred coordinates

$j^{I}=\\{x^{\mu},u^{i},u^{i}_{\mu}\\},\quad i=1,2\,,$ (213)

that, along the section $j^{1}\phi$, evaluate to

$\displaystyle x^{\mu}\circ j^{1}\phi$ $\displaystyle=x^{\mu}\,$ (214)
$\displaystyle u^{i}\circ j^{1}\phi$ $\displaystyle=\phi^{i}(x^{\mu})\,$ (215)
$\displaystyle u^{i}_{\mu}\circ j^{1}\phi$
$\displaystyle=\partial_{\mu}\phi^{i}(x^{\mu})\,$ (216)

As in the previous cases, we start by writing the EFT Lagrangian containing a
complete and IBP-non-redundant basis with up to 4 derivatives, requesting that
no more than 2 derivatives act on each field and that there are no boxes. This
identifies the Green’s basis:

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j})\mathcal{F}^{0}_{ij}-\mathcal{V}(\phi_{1},\phi_{2})$
(217)
$\displaystyle+\frac{1}{\Lambda^{2}}(\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\mu}\partial^{\nu}\phi^{j})\mathcal{F}^{1}_{ij}+\frac{1}{\Lambda^{3}}(\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\mu}\phi^{j}\partial^{\nu}\phi^{k})\mathcal{F}^{2}_{ijk}+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j})(\partial_{\nu}\phi^{k}\partial^{\nu}\phi^{l})\mathcal{F}^{3}_{ijkl}\,.$

where, in general, the scalar potential $\mathcal{V}$ and the $\mathcal{F}$
functions have the form

$\displaystyle\mathcal{F}$
$\displaystyle=f_{0}+f_{1}\frac{\phi^{1}}{\Lambda}+f_{2}\frac{\phi^{2}}{\Lambda}+f_{11}\frac{(\phi^{1})^{2}}{\Lambda^{2}}+\dots=\sum_{a,b=0}^{\infty}f_{\footnotesize\underbrace{1\dots
1}_{a}\underbrace{2\dots
2}_{b}}\frac{(\phi^{1})^{a}(\phi^{2})^{b}}{\Lambda^{a+b}}$ (218)

Each such function contains $(n+1)$ independent index assignments for every
number of fields $n=a+b$. Therefore, accounting for the symmetry in $(ij)$,
with $N$ field insertions in the Lagrangian, the 2-derivative operator has
$3(n+1)=3(N-1)$ independent index assignments. For the 4-derivative operators
there are a total of $3(5N-11)$ independent assignments for $N\geq 4$. This
number is reduced to 12 for $N=3$ and to 3 for $N=2$, due to the fact that
some operator structures are unavailable in these cases. These numbers, as
well as the EOM-reduced counting below, have been cross-checked with BasisGen.
Canonical kinetic terms require $f^{0}_{11}=f^{0}_{22}=1,f^{0}_{12}=0$.

We now turn to the jet bundle metric from which we would like to recover
(217). The generic metric components, truncated as in the previous examples,
are

$\displaystyle g_{ij}^{\mu\nu}$ $\displaystyle=\eta^{\mu\nu}A_{ij}(u)\,,$
(219) $\displaystyle g_{ij}^{\mu}$
$\displaystyle=\frac{u^{k\mu}}{\Lambda^{2}}B_{ijk}(u)\,,$ (220) $\displaystyle
g_{ij}$
$\displaystyle=C_{ij}(u)+\frac{u^{k}_{\rho}u^{l\rho}}{\Lambda^{4}}D_{ijkl}(u)\,,$
(221) $\displaystyle g_{\mu j}^{\nu}$
$\displaystyle=\delta^{\nu}_{\mu}E_{j}(u)+\frac{u^{k\nu}u^{l}_{\mu}}{\Lambda^{4}}F_{1,jkl}(u)+\delta^{\nu}_{\mu}\frac{u^{k}_{\rho}u^{l\rho}}{\Lambda^{4}}F_{2,jkl}(u)\,,$
(222) $\displaystyle g_{\mu j}$
$\displaystyle=\frac{u^{k}_{\mu}}{\Lambda^{2}}G_{jk}(u)+\frac{u^{k}_{\mu}\,u^{l}_{\rho}u^{m\rho}}{\Lambda^{6}}H_{jklm}(u)\,,$
(223) $\displaystyle g_{\mu\nu}$
$\displaystyle=-\frac{\eta_{\mu\nu}}{2}V(u)+\left(\frac{u^{k}_{\mu}u^{l}_{\nu}}{\Lambda^{4}}+\frac{\eta_{\mu\nu}}{4}\frac{u^{k}_{\rho}u^{l\rho}}{\Lambda^{4}}\right)\frac{J_{kl}(u)}{2}+\left(\frac{u^{k}_{\mu}u^{l}_{\nu}}{\Lambda^{4}}+\frac{\eta_{\mu\nu}}{4}\frac{u^{k}_{\rho}u^{l\rho}}{\Lambda^{4}}\right)\frac{u^{m}_{\rho}u^{n\rho}}{\Lambda^{4}}\frac{K_{klmn}(u)}{2}\,,$
(224)

where $u^{i\mu}=\eta^{\mu\nu}u^{i}_{\nu}$ and all the functions $V,A\dots K$
are dimensionless functions of $(u^{i}/\Lambda)$ . We also assume that the
metric is fully symmetric: $g^{\mu\nu}_{ij},g^{\mu\nu},g^{\nu}_{\mu j}$ are
symmetric in $\mu\nu$, $g^{\mu\nu}_{ij},g_{ij}$ are symmetric in $ij$, and
$g_{\nu i}^{(\mu)}=g_{\mu j}^{(\nu)},g_{ij}^{\mu}=g_{ij}^{\nu}$ upon
relabeling the free indices. All repeated indices are understood to be summed
over. Because we are not assuming internal symmetries, they simply represent
labels on the functions.

Pulling back along the section $j^{1}\phi$ and contracting with the inverse
metric

$\displaystyle{\cal L}[\phi,g]$
$\displaystyle=\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$
$\displaystyle\text{where}\quad\eta^{-1}$
$\displaystyle=\eta^{\rho\sigma}\partial_{\rho}\otimes\partial_{\sigma}\,,$
(225)

we get

$\displaystyle\mathcal{L}[\phi,g]=$
$\displaystyle\,\frac{\Lambda^{4}}{2}\eta^{\mu\nu}g_{\mu\nu}+\Lambda^{2}g_{\mu
j}\partial^{\mu}\phi^{j}+\Lambda g_{\mu
j}^{\nu}\partial^{\mu}\partial_{\nu}\phi^{j}+\frac{1}{2}g_{ij}\partial^{\mu}\phi^{i}\partial^{\mu}\phi^{j}$
(226)
$\displaystyle+\frac{1}{2}\frac{g_{ij}^{\mu}}{\Lambda}\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\nu}\phi^{j}+\frac{1}{2}\frac{g_{ij}^{\nu}}{\Lambda}\partial^{\mu}\phi^{i}\partial_{\mu}\partial_{\nu}\phi^{j}+\frac{1}{2}\frac{g_{ij}^{\mu\nu}}{\Lambda^{2}}\partial_{\rho}\partial_{\mu}\phi^{i}\partial^{\rho}\partial_{\nu}\phi^{j}$
$\displaystyle=$
$\displaystyle\,-\Lambda^{4}V+\frac{1}{2}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}\left(C_{ij}+2G_{ij}+J_{ij}\right)+\Lambda\square\phi^{j}E_{j}$
(227)
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\mu}\partial^{\nu}\phi^{j})}{\Lambda^{2}}\frac{A_{ij}}{2}+\frac{\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\mu}\phi^{j}\partial^{\nu}\phi^{k}}{\Lambda^{3}}\frac{B_{ijk}+B_{jik}+2F_{1,ijk}}{2}$
$\displaystyle+\frac{(\square\phi^{i})\partial_{\mu}\phi^{j}\partial^{\mu}\phi^{k}}{\Lambda^{3}}F_{2,ijk}(u)+\frac{(\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j})(\partial_{\nu}\phi^{k}\partial^{\nu}\phi^{l})}{\Lambda^{4}}\frac{D_{ijkl}+2H_{ijkl}+K_{ijkl}}{2}\,.$

This result is again formally equivalent to those obtained in §6.1 and §6.2,
the only difference being the presence of explicit field indices. The $E_{j}$
function is required to have a non-singular metric at the 2-derivatives level,
but it is redundant via IBP:313131As in the previous example, we find that for
any number of fields $N$, there are always as many independent IBPs as
operators with boxes. This implies that the set of all box-less operators
forms a complete and non-redundant Green’s basis.

$\displaystyle\square\phi^{j}E_{j}$
$\displaystyle=-\frac{\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}}{\Lambda}\frac{dE_{j}}{d(\phi^{i}/\Lambda)}\,.$
(228)

The term proportional to the function $F_{2,ijk}$ is also redundant and can be
recast as

$\displaystyle(\square\phi^{i})\partial_{\mu}\phi^{j}\partial^{\mu}\phi^{k}F_{2,ijk}$
$\displaystyle=-(\partial_{\mu}\partial_{\nu}\phi^{j})\partial^{\mu}\phi^{k}\partial^{\nu}\phi^{i}(F_{2,ijk}+F_{2,ikj})-\frac{(\partial_{\mu}\phi^{j}\partial^{\mu}\phi^{k})(\partial_{\nu}\phi^{i}\partial^{\nu}\phi^{l})}{\Lambda}\frac{dF_{2,ijk}}{d(\phi^{l}/\Lambda)}$
(229)

such that

$\displaystyle\mathcal{L}[\phi,g]=$ $\displaystyle\,\text{IBP
}\circ\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$ (230)
$\displaystyle=$
$\displaystyle\,-\Lambda^{4}V+\frac{1}{2}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}\left(C_{ij}+2G_{ij}+J_{ij}-2\frac{dE_{j}}{d(\phi^{i}/\Lambda)}\right)$
(231)
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\mu}\partial^{\nu}\phi^{j})}{\Lambda^{2}}\frac{A_{ij}}{2}+\frac{\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\mu}\phi^{j}\partial^{\nu}\phi^{k}}{\Lambda^{3}}\left(\frac{B_{ijk}+B_{jik}}{2}+F_{1,ijk}-F_{2,kij}-F_{2,kji}\right)$
$\displaystyle+\frac{(\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j})(\partial_{\nu}\phi^{k}\partial^{\nu}\phi^{l})}{\Lambda^{4}}\left(\frac{D_{ijkl}+K_{ijkl}}{2}+H_{ijkl}-\frac{dF_{2,kij}}{d(\phi^{l}/\Lambda)}\right)\,.$

This Lagrangian manifestly captures the complete Green’s basis in Eq. (217).

#### Equation-of-Motion redundancies.

We can repeat the same exercise illustrated in §6.2 in order to identify a
minimal operator basis with up to 4 derivatives, accounting for both IBP and
EOM reduction. The procedure is formally equivalent to the one illustrated
above, and leads to an analogous result, namely:

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{i}-\mathcal{V}(\phi)$
(232)
$\displaystyle+\frac{1}{2}\partial_{\mu}\phi^{2}\partial^{\mu}\phi^{2}\phi^{1}\phi^{1}\mathcal{F}^{0}_{2211}(\phi)+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j})(\partial_{\nu}\phi^{k}\partial^{\nu}\phi^{l})\mathcal{F}^{3}_{ijkl}(\phi)\,.$

For $N=2,3$ field insertions it is possible to remove all terms with up to 4
derivatives (except the two kinetic ones) and trade them for non-derivative
interactions in the potential. For $N\geq 4$, the structure
$(\partial\phi)^{4}$ is always independent. The main difference with the
previous examples is that, in addition, $(N-3)$ index assignments of the
2-derivative operator cannot be removed. Here we chose arbitrarily those
starting with $(2211)$.

The Lagrangian in Eq. (232) is a subset of (217), and therefore it can be
obtained pulling back a 1-jet bundle metric. Comparing to Eq. (231), we see
that such a metric should satisfy:

$\displaystyle A_{ij}$ $\displaystyle=0\,,$ (233) $\displaystyle
B_{ijk}+B_{jik}+2F_{1,ijk}-2F_{2,kij}-2F_{2,kji}$ $\displaystyle=0\,,$ (234)
$\displaystyle V$ $\displaystyle=\frac{\mathcal{V}(\phi)}{\Lambda^{4}}$ (235)
$\displaystyle C_{ij}+2G_{ij}+J_{ij}-2\frac{dE_{j}}{d(\phi^{i}/\Lambda)}$
$\displaystyle=\delta_{ij}+\delta_{i2}\delta_{j2}(\phi^{1})^{2}\mathcal{F}^{0}_{2211}(\phi)\,,$
(236)
$\displaystyle\frac{1}{2}\left(D_{ijkl}+K_{ijkl}\right)+H_{ijkl}-\frac{dF_{2,kij}}{d(\phi^{l}/\Lambda)}$
$\displaystyle=\mathcal{F}^{3}_{ijkl}(\phi)\,.$ (237)

##  7 Four scalars in 4d, with ${O(4)}$ internal symmetry

We now cover the case of a multiplet of 4 fields in Minkowski space, with an
internal $O(4)$ symmetry. This example is representative of SMEFT/HEFT with
exact custodial symmetry and in the absence of gauge and fermion fields. The
discussion in this section follows the same steps taken in the examples in §6.

###  7.1 The EFT Lagrangian up to 4 derivatives

We start by constructing a complete, IBP-non-redundant set of $O(4)$-symmetric
operators with up to 4 derivatives and arbitrary number of fields. We require
that no more than 2 derivatives act on each field and the absence of boxes. As
in the toy examples, this identifies a unique Lagrangian:

$\displaystyle{\cal L}=$
$\displaystyle\,\frac{1}{2}\partial_{\mu}\phi\cdot\partial^{\mu}\phi\mathcal{F}_{a}+(\partial_{\mu}\phi\cdot\phi)^{2}\mathcal{F}_{b}-\mathcal{V}$
(238)
$\displaystyle+\frac{1}{\Lambda^{2}}(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\partial^{\nu}\phi)\mathcal{F}_{1}+\frac{1}{\Lambda^{4}}(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\partial^{\nu}\phi\cdot\phi)\mathcal{F}_{2}+\frac{1}{\Lambda^{4}}(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\phi)(\partial^{\nu}\phi\cdot\phi)\mathcal{F}_{3}$
$\displaystyle+\frac{1}{\Lambda^{4}}(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\partial^{\nu}\phi)\mathcal{F}_{4}+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)^{2}\mathcal{F}_{5}+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)^{2}\mathcal{F}_{6}$
$\displaystyle+\frac{1}{\Lambda^{6}}(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)\mathcal{F}_{7}+\frac{1}{\Lambda^{6}}(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)(\partial_{\nu}\phi\cdot\phi)^{2}\mathcal{F}_{8}$
$\displaystyle+\frac{1}{\Lambda^{6}}(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)\mathcal{F}_{9}+\frac{1}{\Lambda^{8}}(\partial_{\mu}\phi\cdot\phi)^{4}\mathcal{F}_{10}\,.$

Here the dot indicates the contraction of two 4-vectors:
$\phi\cdot\phi=\phi^{i}\phi^{j}\delta_{ij}$ and $\mathcal{V},\mathcal{F}_{k}$
are generic functions of $(\phi\cdot\phi)/\Lambda^{2}$. The requirement of a
canonical kinetic term translates into ${\mathcal{F}_{a}(0)=1}$. As for the
examples in §6, the basis structure with and without EOM redundancies has been
cross-checked with BasisGen. Upon imposing preservation of the custodial
symmetry, the number of operators with 2 derivatives + 4 fields, and 4
derivatives + 2 fields in the Green’s basis of Eq. (238) matches the number in
Gherardi:2020det . The number of operators with 2 derivatives + 6 fields and 4
derivatives + 4 fields match those in Chala:2021pll .323232There are 3
custodial-preserving terms in Gherardi:2020det :
$O_{H\square},O^{\prime}_{HD},O_{DH}$. There are 8 in Chala:2021pll :
$O_{\phi^{6}}^{(1)}$, $O_{\phi^{6}}^{(3)}$, $O_{\phi^{4}}^{(3)}$,
$O_{\phi^{4}}^{(4)}$, $O_{\phi^{4}}^{(10)}$,
($O_{\phi^{4}}^{(1)}+O_{\phi^{4}}^{(2)}$),
($O_{\phi^{4}}^{(8)}+2O_{\phi^{4}}^{(11)})$,
($O_{\phi^{4}}^{(6)}+O_{\phi^{4}}^{(12)}$).

While the connection to (jet) bundle geometry is best studied considering the
4 scalars as the components of a real vector, it can be useful to make contact
with the more familiar form in terms of the $SU(2)$ Higgs doublet, by
exploiting the isomorphism

$O(4)\cong\left(\frac{SU(2)_{L}\times
SU(2)_{R}}{\mathbbm{Z}_{2}}\right)\rtimes\mathbbm{Z}_{2}\,,$ (239)

where the quotient by $\mathbb{Z}_{2}$ identifies the central elements of
$SU(2)_{L}$ and $SU(2)_{R}$; the part in parentheses is $SO(4)$, which recall
is a normal subgroup of $O(4)$.

Explicit expressions for the operators in (238) can be obtained
straightforwardly from the following dictionary:

$\displaystyle\partial_{\mu}\phi\cdot\phi$
$\displaystyle=\partial_{\mu}(H^{\dagger}H)\,,$ (240)
$\displaystyle\partial_{\mu}\phi\cdot\partial_{\nu}\phi$
$\displaystyle=(\partial_{\mu}H^{\dagger})(\partial_{\nu}H)+(\partial_{\nu}H^{\dagger})(\partial_{\mu}H)\,,$
(241) $\displaystyle\partial_{\mu}\partial_{\nu}\phi\cdot\phi$
$\displaystyle=(\partial_{\mu}\partial_{\nu}H^{\dagger})H+H^{\dagger}(\partial_{\mu}\partial_{\nu}H)\,,$
(242)
$\displaystyle\partial_{\mu}\partial_{\nu}\phi\cdot\partial_{\rho}\partial_{\sigma}\phi$
$\displaystyle=(\partial_{\mu}\partial_{\nu}H^{\dagger})(\partial_{\rho}\partial_{\sigma}H)+(\partial_{\rho}\partial_{\sigma}H^{\dagger})(\partial_{\mu}\partial_{\nu}H)\,,$
(243) $\displaystyle\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\phi$
$\displaystyle=(\partial_{\mu}\partial_{\nu}H^{\dagger})(\partial^{\mu}H)+(\partial^{\mu}H^{\dagger})(\partial_{\mu}\partial_{\nu}H)\,,$
(244) $\displaystyle\mathcal{F}(\phi\cdot\phi/\Lambda^{2})$
$\displaystyle=\mathcal{F}(H^{\dagger}H/\Lambda^{2})\,.$ (245)

#### Equation-of-Motion redundancies

If EOM redundancies are accounted for, the Lagrangian in Eq. (238) can be
further reduced. Applying the procedure described in §6.2, we find that a
possible IBP+EOM -non-redundant basis is:

$\displaystyle{\cal L}=$
$\displaystyle\,\frac{1}{2}\partial_{\mu}\phi\cdot\partial^{\mu}\phi+(\partial_{\mu}\phi\cdot\phi)^{2}\mathcal{F}_{b}-\mathcal{V}$
(246)
$\displaystyle+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)^{2}\mathcal{F}_{5}+\frac{1}{\Lambda^{4}}(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)^{2}\mathcal{F}_{6}$
$\displaystyle+\frac{1}{\Lambda^{6}}(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)(\partial_{\nu}\phi\cdot\phi)^{2}\mathcal{F}_{8}+\frac{1}{\Lambda^{6}}(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)\mathcal{F}_{9}$
$\displaystyle+\frac{1}{\Lambda^{8}}(\partial_{\mu}\phi\cdot\phi)^{4}\mathcal{F}_{10}\,.$

Notice that the function $\mathcal{F}_{a}$ accompanying the kinetic term has
now been removed. This is the case because, with $N\geq 4$ fields it is always
possible to map via IBP

$(\partial_{\mu}\phi\partial^{\mu}\phi)(\phi\cdot\phi)^{N-2}=(2-N)(\partial_{\mu}\phi\cdot\phi)^{2}(\phi\cdot\phi)^{N-4}-(\square\phi\cdot\phi)(\phi\cdot\phi)^{N-2}\,,$
(247)

and then remove the last term via EOM. Eq. (246) is consistent with the SMEFT
operator bases in the literature: for instance, the Warsaw basis of
dimension-6 operators Grzadkowski:2010es  contains 1 custodial-preserving
operator with 2 derivatives + 4 fields ($O_{H\square}\sim\mathcal{F}_{b}$),
and none with 4 derivatives + 2 fields. The Murphy basis of dimension-8
operators Murphy:2020rsh  also contains 1 operator with 2 derivatives + 6
fields ($Q_{H^{6}}^{(1)}\sim\mathcal{F}_{b}$), and 3 with 4 derivatives + 4
fields. However, only 2 combinations of those
($Q_{H^{4}}^{(6)}\sim\mathcal{F}_{5}$,
$(Q_{H^{4}}^{(1)}+Q_{H^{4}}^{(2)})\sim\mathcal{F}_{6}$) preserve the custodial
symmetry. The interactions in the last two lines of Eq (246) correspond to
dimension-10 and higher operators in SMEFT and were cross-checked against the
results of Ref. Harlander:2023psl .

###  7.2 The EFT Lagrangian from 1-jet bundle geometry

The full 4-derivative Lagrangian in Eq. (238) can be captured by pulling back
to spacetime a metric from the 1-jet bundle $J^{1}E$. This is a real
24-manifold, with local fibred coordinates

$\displaystyle y^{I}$ $\displaystyle=\\{x^{\mu},u^{i},u^{i}_{\mu}\\}\,.$ (248)

The metric components must be representations of $O(4)$,
$g^{\mu\nu},g^{\nu}_{\mu j},g^{\mu\nu}_{ij}$ are symmetric in $\mu\nu$ and
$g_{ij},g^{\mu\nu}_{ij}$ are symmetric in $ij$, while this is not enforced for
$g_{ij}^{\mu}$. The generic form is:

$\displaystyle g_{ij}^{\mu\nu}$
$\displaystyle=\eta^{\mu\nu}\delta_{ij}A_{0}(u)+\eta^{\mu\nu}\frac{u_{i}u_{j}}{\Lambda^{2}}A_{1}(u)\,,$
(249) $\displaystyle g_{ij}^{\mu}$
$\displaystyle=\frac{u^{\mu}_{i}u_{j}}{\Lambda^{3}}B_{0}(u)+\frac{u_{i}u^{\mu}_{j}}{\Lambda^{3}}B_{1}(u)+\delta_{ij}\frac{u\cdot
u^{\mu}}{\Lambda^{3}}B_{2}(u)+\frac{u_{i}u_{j}(u\cdot
u^{\mu})}{\Lambda^{5}}B_{3}(u)\,,$ (250) $\displaystyle g_{ij}$
$\displaystyle=\delta_{ij}C_{0}(u)+\frac{u_{i}u_{j}}{\Lambda^{2}}C_{1}(u)$
$\displaystyle+\delta_{ij}\frac{u_{\rho}\cdot
u^{\rho}}{\Lambda^{4}}D_{0}(u)+\frac{u_{i\rho}u_{j}^{\rho}}{\Lambda^{4}}D_{1}(u)+\left[\delta_{ij}\frac{(u\cdot
u^{\rho})^{2}}{\Lambda^{6}}+\frac{u_{i}u_{j}(u_{\rho}\cdot
u^{\rho})}{\Lambda^{6}}\right]\frac{D_{2}(u)}{2}$
$\displaystyle+\frac{(u_{i}u_{j\rho}+u_{j}u_{i\rho})(u\cdot
u^{\rho})}{2\Lambda^{6}}D_{3}(u)+\frac{u_{i}u_{j}(u\cdot
u_{\rho})^{2}}{\Lambda^{8}}D_{4}(u)\,,$ (251) $\displaystyle g_{\mu j}^{\nu}$
$\displaystyle=\delta^{\nu}_{\mu}\frac{u_{j}}{\Lambda}E(u)$
$\displaystyle+\frac{u_{j}(u^{\nu}\cdot
u_{\mu})}{\Lambda^{5}}F_{10}(u)+\frac{u_{j}^{\nu}(u\cdot u_{\mu})+(u\cdot
u^{\nu})u_{j\mu}}{2\Lambda^{5}}F_{11}(u)+\frac{u_{j}(u\cdot u^{\nu})(u\cdot
u_{\mu})}{\Lambda^{7}}F_{12}(u)$
$\displaystyle+\delta^{\nu}_{\mu}\frac{u_{j}(u_{\rho}\cdot
u^{\rho})}{\Lambda^{5}}F_{20}(u)+\delta^{\nu}_{\mu}\frac{u_{j\rho}(u\cdot
u^{\rho})}{\Lambda^{5}}F_{21}(u)+\delta^{\nu}_{\mu}\frac{u_{j}(u\cdot
u^{\rho})^{2}}{\Lambda^{7}}F_{22}(u)\,,$ (252) $\displaystyle g_{\mu j}$
$\displaystyle=\frac{u_{j\mu}}{\Lambda^{2}}G_{0}(u)+\frac{u_{j}(u\cdot
u_{\mu})}{\Lambda^{4}}G_{1}(u)$ $\displaystyle+\frac{u_{j\mu}(u_{\rho}\cdot
u^{\rho})}{\Lambda^{6}}H_{0}(u)+\frac{u_{j\rho}(u_{\mu}\cdot
u^{\rho})}{\Lambda^{6}}H_{1}(u)+\left[\frac{u_{j}(u\cdot
u_{\mu})(u_{\rho}\cdot u^{\rho})}{\Lambda^{8}}+\frac{u_{j\mu}(u\cdot
u_{\rho})^{2}}{\Lambda^{8}}\right]\frac{H_{2}(u)}{2}$
$\displaystyle+\left[\frac{u_{j}(u_{\mu}\cdot u_{\rho})(u\cdot
u^{\rho})}{\Lambda^{8}}+\frac{u_{j\rho}(u\cdot u_{\mu})(u\cdot
u^{\rho})}{\Lambda^{8}}\right]\frac{H_{3}(u)}{2}+\frac{u_{j}(u\cdot
u_{\mu})(u\cdot u_{\rho})^{2}}{\Lambda^{10}}H_{4}(u)\,,$ (253) $\displaystyle
g_{\mu\nu}$ $\displaystyle=-\frac{1}{2}\eta_{\mu\nu}V(u)$
$\displaystyle+\left[\frac{(u_{\mu}\cdot
u_{\nu})}{\Lambda^{4}}+\frac{\eta_{\mu\nu}}{4}\frac{(u_{\rho}\cdot
u^{\rho})}{\Lambda^{4}}\right]\frac{J_{0}(u)}{2}+\left[\frac{(u\cdot
u_{\mu})(u\cdot u_{\nu})}{\Lambda^{6}}+\frac{\eta_{\mu\nu}}{4}\frac{(u\cdot
u_{\rho})^{2}}{\Lambda^{6}}\right]\frac{J_{1}(u)}{2}$
$\displaystyle+\left[\frac{(u_{\mu}\cdot
u_{\nu})}{\Lambda^{4}}+\frac{\eta_{\mu\nu}}{4}\frac{(u_{\rho}\cdot
u^{\rho})}{\Lambda^{4}}\right]\frac{u_{\sigma}\cdot
u^{\sigma}}{\Lambda^{4}}\frac{K_{0}(u)}{2}+\left[\frac{(u_{\sigma}\cdot
u_{\mu})(u^{\sigma}\cdot
u_{\nu})}{\Lambda^{8}}+\frac{\eta_{\mu\nu}}{4}\frac{(u_{\sigma}\cdot
u_{\rho})^{2}}{\Lambda^{8}}\right]\frac{K_{1}(u)}{2}$
$\displaystyle+\left[\frac{(u\cdot u_{\mu})(u\cdot u_{\nu})(u_{\sigma}\cdot
u^{\sigma})}{\Lambda^{10}}+\frac{\eta_{\mu\nu}}{4}\frac{(u\cdot
u_{\rho})^{2}(u_{\sigma}\cdot u^{\sigma})}{\Lambda^{10}}+\frac{(u_{\mu}\cdot
u_{\nu})(u\cdot u_{\sigma})^{2}}{\Lambda^{10}}\right]\frac{K_{2}(u)}{3}$
$\displaystyle+\left[\frac{(u\cdot u_{\mu})(u^{\sigma}\cdot
u_{\nu})+(u^{\sigma}\cdot u_{\mu})(u\cdot
u_{\nu})}{2\Lambda^{7}}+\frac{\eta_{\mu\nu}}{4}\frac{(u\cdot
u_{\rho})(u^{\sigma}\cdot u^{\rho})}{\Lambda^{7}}\right]\frac{(u\cdot
u_{\sigma})}{\Lambda^{3}}\frac{K_{3}(u)}{2}$
$\displaystyle+\left[\frac{(u\cdot u_{\mu})(u\cdot
u_{\nu})}{\Lambda^{6}}+\frac{\eta_{\mu\nu}}{4}\frac{(u\cdot
u_{\rho})^{2}}{\Lambda^{6}}\right]\frac{(u\cdot
u_{\sigma})^{2}}{\Lambda^{6}}\frac{K_{4}(u)}{2}\,,$ (254)

where $V,A_{n}\dots K_{n}$ are functions of $(u\cdot u)/\Lambda^{2}$, we
implicitly defined $u_{i}=\delta_{ik}u^{k}$ and the dot denotes a contraction
of internal indices. We can also take $g_{\nu i}=g_{\mu j},g_{\nu
i}^{\mu}=g_{\mu j}^{\nu},g_{ij}^{\nu}=g_{ij}^{\mu}$ upon index relabeling and
we grouped together structures that yield identical results upon mapping to
the Lagrangian.

Pulling back along the section $j^{1}\phi$ and contracting with the inverse
metric

$\displaystyle\mathcal{L}[\phi,g]$
$\displaystyle=\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$
$\displaystyle\text{where}\quad\eta^{-1}$
$\displaystyle=\eta^{\rho\sigma}\partial_{\rho}\otimes\partial_{\sigma}\,,$
(255)

we get

$\displaystyle\mathcal{L}[\phi,g]=$
$\displaystyle\,\frac{\Lambda^{4}}{2}\eta^{\mu\nu}g_{\mu\nu}+\Lambda^{2}g_{\mu
j}\partial^{\mu}\phi^{j}+\Lambda g_{\mu
j}^{\nu}\partial^{\mu}\partial_{\nu}\phi^{j}+\frac{1}{2}g_{ij}\partial^{\mu}\phi^{i}\partial^{\mu}\phi^{j}$
(256)
$\displaystyle+\frac{1}{2}\frac{g_{ij}^{\mu}}{\Lambda}\partial_{\mu}\partial_{\nu}\phi^{i}\partial^{\nu}\phi^{j}+\frac{1}{2}\frac{g_{ij}^{\nu}}{\Lambda}\partial^{\mu}\phi^{i}\partial_{\mu}\partial_{\nu}\phi^{j}+\frac{1}{2}\frac{g_{ij}^{\mu\nu}}{\Lambda^{2}}\partial_{\rho}\partial_{\mu}\phi^{i}\partial^{\rho}\partial_{\nu}\phi^{j}$
$\displaystyle=$
$\displaystyle\,(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)\frac{C_{0}+2G_{0}+J_{0}}{2}+\frac{(\partial_{\mu}\phi\cdot\phi)^{2}}{\Lambda^{2}}\frac{C_{1}+2G_{1}+J_{1}}{2}+(\square\phi\cdot\phi)E-\Lambda^{4}V$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\partial^{\nu}\phi)}{\Lambda^{2}}\frac{A_{0}}{2}+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\partial^{\nu}\phi\cdot\phi)}{\Lambda^{4}}\frac{A_{1}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{4}}\frac{B_{0}+B_{1}+2B_{2}+2F_{11}}{2}+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\partial^{\nu}\phi)}{\Lambda^{4}}\frac{B_{0}+B_{1}+2F_{10}}{2}$
$\displaystyle+\frac{(\square\phi\cdot\phi)(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)}{\Lambda^{4}}F_{20}+\frac{(\square\phi\cdot\partial_{\mu}\phi)(\partial^{\mu}\phi\cdot\phi)}{\Lambda^{4}}F_{21}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)^{2}}{\Lambda^{4}}\frac{D_{0}+2H_{0}+K_{0}}{2}+\frac{(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)^{2}}{\Lambda^{4}}\frac{D_{1}+2H_{1}+K_{1}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{6}}(B_{3}+F_{12})+\frac{(\square\phi\cdot\phi)(\partial_{\mu}\phi\cdot\phi)^{2}}{\Lambda^{6}}F_{22}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)(\partial_{\nu}\phi\cdot\phi)^{2}}{\Lambda^{6}}\frac{D_{2}+2H_{2}+K_{2}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{6}}\frac{D_{3}+2H_{3}+K_{3}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\phi)^{4}}{\Lambda^{8}}\frac{D_{4}+2H_{4}+K_{4}}{2}\,.$
(257)

The Lagrangian obtained from the 1-jet metric manifestly matches the complete
Green’s basis of operators with up to 4 derivatives and arbitrary number of
fields, presented in Eq. (238). The only leftover IBP redundancies are in the
$E$, $F_{20},F_{21}$ and $F_{22}$ terms, that can be recast as333333As in all
the previous examples, we find as many independent IBP as operators with
boxes, at each fields and derivatives multiplicity (up to 4). Therefore the
set of all box-less operators forms a Green’s basis.

$\displaystyle(\square\phi\cdot\phi)E=$
$\displaystyle-(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)E-2\frac{(\partial_{\mu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\phi)}{\Lambda^{2}}\frac{dE}{d(\phi\cdot\phi/\Lambda^{2})}\,,$
(258)
$\displaystyle(\square\phi\cdot\phi)(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)F_{20}=$
$\displaystyle-(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)^{2}F_{20}-2(\partial_{\nu}\partial_{\mu}\phi\cdot\partial^{\mu}\phi)(\partial^{\nu}\phi\cdot\phi)F_{20}$
$\displaystyle-\frac{2(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)(\partial_{\nu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{2}}\frac{dF_{20}}{d(\phi\cdot\phi/\Lambda^{2})}$
(259)
$\displaystyle(\square\phi\cdot\partial_{\mu}\phi)(\partial^{\mu}\phi\cdot\phi)F_{21}=$
$\displaystyle-(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)F_{21}-(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\partial^{\nu}\phi)F_{21}$
$\displaystyle-(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)^{2}F_{21}-\frac{2(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{2}}\frac{dF_{21}}{d(\phi\cdot\phi/\Lambda^{2})}$
(260)
$\displaystyle(\square\phi\cdot\phi)(\partial_{\mu}\phi\cdot\phi)^{2}F_{22}=$
$\displaystyle-(\partial_{\nu}\phi\cdot\partial^{\nu}\phi)(\partial_{\mu}\phi\cdot\phi)^{2}F_{22}-2(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\phi)F_{22}$
$\displaystyle-2(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)F_{22}$
$\displaystyle-\frac{2(\partial_{\mu}\phi\cdot\phi)^{2}(\partial_{\nu}\phi\cdot\phi)^{2}}{\Lambda^{2}}\frac{dF_{22}}{d(\phi\cdot\phi/\Lambda^{2})}$
(261)

such that

$\displaystyle\mathcal{L}[\phi,g]=$ $\displaystyle\,\text{IBP
}\circ\frac{1}{2}\left\langle\eta^{-1},\,(j^{1}\phi)^{*}g\right\rangle$ (262)
$\displaystyle=$
$\displaystyle\,(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)\frac{C_{0}+2G_{0}+J_{0}-2E}{2}+\frac{(\partial_{\mu}\phi\cdot\phi)^{2}}{\Lambda^{2}}\frac{C_{1}+2G_{1}+J_{1}-4E^{\prime}}{2}-\Lambda^{4}V$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\partial^{\nu}\phi)}{\Lambda^{2}}\frac{A_{0}}{2}+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\partial^{\nu}\phi\cdot\phi)}{\Lambda^{4}}\frac{A_{1}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{4}}\frac{B_{0}+B_{1}+2B_{2}+2F_{11}-4F_{20}-2F_{21}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\partial^{\nu}\phi)}{\Lambda^{4}}\frac{B_{0}+B_{1}+2F_{10}-2F_{21}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)^{2}}{\Lambda^{4}}\frac{D_{0}+2H_{0}+K_{10}-2F_{20}}{2}+\frac{(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)^{2}}{\Lambda^{4}}\frac{D_{1}+2H_{1}+K_{1}-2F_{21}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\partial_{\nu}\phi\cdot\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{6}}(B_{3}+F_{12}-2F_{22})$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\partial^{\mu}\phi)(\partial_{\nu}\phi\cdot\phi)^{2}}{\Lambda^{6}}\frac{D_{2}+2H_{2}+K_{2}-4F_{20}^{\prime}-2F_{22}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\partial_{\nu}\phi)(\partial^{\mu}\phi\cdot\phi)(\partial^{\nu}\phi\cdot\phi)}{\Lambda^{6}}\frac{D_{3}+2H_{3}+K_{3}-4F_{21}^{\prime}-4F_{22}}{2}$
$\displaystyle+\frac{(\partial_{\mu}\phi\cdot\phi)^{4}}{\Lambda^{8}}\frac{D_{4}+2H_{4}+K_{4}-4F_{22}^{\prime}}{2}\,,$
(263)

where $E^{\prime}=dE/d(\phi\cdot\phi/\Lambda^{2})$ and analogously for the
other functions.

###  7.3 Beyond the custodial limit

In actual SMEFT/HEFT, the symmetry $O(4)$ is not exact: the electroweak vacuum
preserves only a $O(3)\sim SU(2)_{L+R}/\mathbbm{Z}_{2}$ subgroup, namely the
custodial symmetry. This is however broken explicitly by the gauging of the
hypercharge $U(1)$, which (in the absence of fermions) coincides with the
group spanned by the 3rd generator of $SU(2)_{R}$. While invariance under the
gauged $SU(2)_{L}\times U(1)$ subgroup must clearly be respected, it is
generally possible to include operators that are not invariant under the
groups spanned by the first two generators of $SU(2)_{R}$, and therefore break
the custodial $O(3)$.

We can ask whether custodial-violating effects can be described in the jet
bundle formalism. Indeed, custodial-violating terms can be added to the 1-jet
metric without posing any problem. More in general, any internal symmetry
structure of the Lagrangian can always be accommodated in the 1-jet bundle by
choosing suitable metric entries. Even situations with no symmetry at all can
be reproduced, as illustrated in the two scalars example in §6.3.

In the specific case of SMEFT/HEFT custodial violation, they can be introduced
allowing for contractions with the 3rd generator of $SU(2)_{R}$: given two
$O(4)$ vectors $u^{i},v^{j}$,

$\displaystyle u\cdot v=u^{i}v^{j}\delta_{ij}\quad$ $\displaystyle\text{is
custodial preserving},$ (264) $\displaystyle
u\,t^{3R}v=u^{i}v^{j}t^{3R}_{ij}\quad$ $\displaystyle\text{is custodial
violating}.$

If $O(4)$ vectors $\phi^{i},i=1\dots 4$ are defined such that the mapping to
$SU(2)$ doublets is

$H=\frac{1}{\sqrt{2}}\binom{\phi^{2}+i\phi^{1}}{\phi^{4}-i\phi^{3}}\,,$ (265)

then an explicit form for $t^{3R}_{ij}$ is

$\displaystyle t^{3R}=\frac{1}{2}\begin{pmatrix}&1&&\\\ -1&&&\\\ &&&-1\\\
&&1&\end{pmatrix}\,.$ (266)

While in this work we do not aim at constructing a complete and non-redundant
basis including this class of operators, a generalization would be
straightforward. As a practical example, the metric terms

$\displaystyle g_{ij}$
$\displaystyle=t^{3R}_{ik}t^{3R}_{jl}\,\frac{u^{k}u^{l}}{\Lambda^{2}}\,,$
$\displaystyle g_{\mu\nu}$
$\displaystyle=t^{3R}_{ik}t^{3R}_{jl}\,\frac{u^{i}_{\nu}u^{j\nu}u^{k}_{\rho}u^{l\rho}}{\Lambda^{8}}\,,$
(267)

pull back respectively to the custodial-violating dimension-6 and dimension-8
operators

$\displaystyle(\partial_{\mu}\phi\,t^{3R}\,\phi)^{2}$
$\displaystyle=(\partial_{\mu}H^{\dagger}H)(H^{\dagger}\partial^{\mu}H)\,,$
(268) $\displaystyle(\partial_{\mu}\phi\cdot
t^{3R}\cdot\partial_{\nu}\phi)^{2}$
$\displaystyle=(\partial_{\mu}H^{\dagger}\partial_{\nu}H-\partial_{\nu}H^{\dagger}\partial_{\mu}H)^{2}\,,$
(269)

after using our usual prescription for constructing the Lagrangian.

###  7.4 Topological terms

In §2.2 we mentioned the possibility of topological terms in scalar EFTs. The
Higgs EFTs we consider in this Section admit such topological terms (with a
similar story playing out for an $O(N)$-invariant theory of $N$ real scalars
in $d=N$ spacetime dimensions). We briefly digress in this Subsection to
sketch the construction of these topological terms, for the 4d
$O(4)$-invariant theory of 4 real scalars considered in this Section. Given
such terms are not the focus of our geometric jet bundle formalism – the
topological terms are, after all, those that explicitly do not require a
geometry – our discussion here is neither rigorous nor general. But we believe
it is sufficient for the particular example we consider, up to the omission of
possible torsion effects that cannot naïvely be captured by differential forms
(but can be captured, for example, by differential cohomology Davighi:2020vcm
).

To understand these topological terms, it is enough to stick with the field
space bundle $(E,\Sigma,\pi)$ introduced in §3, i.e. we do not need to pass to
the 1-jet bundle $J^{1}E$. Locally, we adopt our usual fibred coordinate
system $(x^{\mu},u^{i})$ and choose a section $\phi\in\Gamma_{x}(\pi)$ with
which to pull back objects. The topological term is proportional to the
following operator in the Lagrangian:343434Note that $\delta_{ij}$ and
$\epsilon_{ijkl}$ are the only $O(4)$ invariant tensors we can use to contract
fundamental $O(N)$ indices to form singlets. While the Kronecker-delta is used
ubiquitously in our constructions of invariant metric terms and EFT operators,
the epsilon tensor appears only in the topological terms discussed in this
Subsection, and so it does not appear when formulating our EFTs
perturbatively.

$\mathcal{O}_{\rm{top}}\propto\frac{1}{\Lambda^{4}}\epsilon_{\mu\nu\rho\sigma}\epsilon_{ijkl}\partial^{\mu}\phi^{i}\partial^{\nu}\phi^{j}\partial^{\rho}\phi^{k}\partial^{\sigma}\phi^{l}\,.$
(270)

The corresponding term in the action is ‘topological’ because it is in fact
the integral of an $O(4)$-invariant differential form. This means it is
independent of any metric – even that on spacetime which appears in all other
Lagrangian terms captured by our construction. Here the differential form in
question is a 4-form $A\in\Omega^{4}(E)$ that is closed ($dA=0$) but need not
be exact (its integral over a closed submanifold can be non-zero), which can
be written in our local coordinate chart as

$A|_{\mathcal{U}_{\Sigma}\times\mathcal{F}_{M}}=du^{1}\wedge du^{2}\wedge
du^{3}\wedge du^{4}\,.$ (271)

The corresponding action is obtained by pulling back along the section $\phi$
and integrating,

$S_{\rm{top}}=\frac{\theta}{2\pi
V}\int_{\Sigma}\phi^{\ast}A,\qquad\theta\in\mathbb{R}/2\pi\mathbb{Z}\,,$ (272)

which will give the operator structure $\mathcal{O}_{\rm{top}}$. Here
$V\propto\Lambda^{4}$ is an appropriate normalisation factor whose precise
value is subtle, but which can be roughly thought of as the (dimensionful)
‘volume’ of a typical fibre $M$, such that the action is dimensionless and the
coupling constant $\theta$ is $2\pi$ periodic, analogous to a ‘theta-angle’ in
gauge theories such as QCD.

Because the 4-form $A$ that we pull back and integrate is closed, it is
locally (but not necessarily global) exact by the Poincaré lemma. This means
that, on our local coordinate patch, the Lagrangian term
$\mathcal{O}_{\rm{top}}$ is a total derivative. This term in the action
therefore gives no contribution to the classical equations of motion, nor does
it contribute to any Feynman diagrams. We therefore choose to neglect such
terms in our main discussion – but we caution that this does not preclude
their having physical effects non-perturbatively.353535Indeed, depending on
the topology of the field space fibre $M$, and because $A$ is not exact, the
topological term can seemingly be activated by instantons, corresponding to
maps in non-trivial homotopy classes in $\pi_{4}(M)$. For example, if
$M=S^{4}$ as in the Minimal Composite Higgs model Agashe:2004rs , we have
$\pi_{4}(S^{4})=\mathbb{Z}$ characterised by maps of different integer winding
number. A scaling argument (Davighi:2018xwn, , Appendix A) suggests these
instanton effects are expected to be important close to the UV cut-off scale
(in contrast to the more familiar story for 4d gauge theory, where instantons
give important effects in the IR).

The same is true for a (seemingly) more general class of terms that we can try
to build using the $\epsilon$ tensor. For example, consider a more general
operator structure of the form

$\displaystyle\mathcal{O}_{\rm
top}^{(1)}=\frac{1}{\Lambda^{4}}\,f\left(\frac{\phi\cdot\phi}{\Lambda^{2}}\right)\,\epsilon_{\mu\nu\rho\sigma}\epsilon_{ijkl}\,\partial^{\mu}\phi^{i}\partial^{\nu}\phi^{j}\partial^{\rho}\phi^{k}\partial^{\sigma}\phi^{l}\,,$
(273)

for any suitably smooth function with compact support $f(x)$. This is again
topological, obtained by pulling back a differential form $A^{(1)}$ along the
section $\phi$, where $A^{(1)}$ has the (local) form

$A^{(1)}|_{\mathcal{U}_{\Sigma}\times\mathcal{F}_{M}}=f\left(\frac{u\cdot
u}{\Lambda^{2}}\right)\,du^{1}\wedge du^{2}\wedge du^{3}\wedge du^{4}\,.$
(274)

Again, it is easy to verify that the differential form $A^{(1)}$ is closed,
viz. $dA^{(1)}\propto f^{\prime}u^{i}du^{i}\wedge A=0$, where
$f^{\prime}=df/dx$, and where the last equality holds because $du^{i}\wedge
du^{i}$ (no sum) vanishes for any $i\in\\{1,...,4\\}$. Therefore, again by the
Poincaré lemma, any operator like $\mathcal{O}_{\rm top}^{(1)}$ is also a
total derivative.363636One might ask if there are any other contractions we
can play with using the $\epsilon$ tensor that give terms which are not total
derivatives, and the answer is no. For example, one might consider an operator
structure $\mathcal{O}_{\rm
top}^{(2)}=\epsilon_{\mu\nu\rho\sigma}\epsilon_{ijkl}\,\phi^{i}\partial^{\nu}\phi^{j}\partial^{\rho}\phi^{k}\partial^{\sigma}\phi^{l}\partial^{\mu}\phi^{m}\,\phi^{m}$.
But $\mathcal{O}_{\rm
top}^{(2)}d^{4}x\propto\phi^{\ast}(\epsilon_{ijkl}u^{i}du^{j}\wedge
du^{k}\wedge du^{l})\wedge du^{m}u^{m}=(u^{1}du^{2}\wedge du^{3}\wedge
du^{4})\wedge du^{1}u^{1}+\text{perms}\propto(u\cdot u)\,du^{1}\wedge
du^{2}\wedge du^{3}\wedge du^{4}$, and so this is the same form as $A^{(1)}$.
Finally, we remark that a differential form like $A^{(1)}$, for non-constant
function $f$, will likely be not just closed but also exact, in which case it
has no effects even in the presence of instanton configurations, since it
evaluates identically to zero on any closed manifold.

##  8 From Jets to Amplitudes

To conclude the paper, in this Section we seek to connect the higher-
derivative EFT Lagrangians that we have constructed using geometry on 1-jet
bundles with physical observables – namely, some basic amplitudes in these
scalar theories. Our study here is far from comprehensive, but we nonetheless
already see some advantages of going to the jet bundle geometry.

###  8.1 Normal coordinates on the jet bundle (and why they fail)

In Section 2.4 we reviewed the use of normal coordinates in the geometric
formulation of EFTs, showcasing their ability to relate amplitudes to
geometric invariants by offering an efficient coordinate system in which to
expand the metric around a particular point.

In this paper, we have re-formulated scalar EFTs first using bundles
$(E,\Sigma,\pi)$, which (besides other general advantages) allows one to
incorporate the 0-derivative potential terms into the metric, alongside the
2-derivative terms. Additionally, the bundle construction provides a starting
point for passing to higher jet bundles which allow one to further incorporate
higher-derivative terms into geometry. In particular, we have seen in detail
that geometry on the 1-jet bundle $J^{1}E$ allows one to capture the full
4-derivative EFT. A natural question to ask is whether normal coordinates can
be used to expand our jet bundle metric, in a similar fashion to the
2-derivative case considered in §2.4, in an effort to obtain generalised
formulae relating $n$-point correlators to components of the jet bundle
Riemann tensor.

It turns out that such an attempt would be doomed to fail, in general. This is
because, even though normal coordinates still exist (since the total spaces of
the various bundles we consider are themselves pseudo-Riemannian manifolds),
the map to normal coordinates is not guaranteed to respect the bundle
structure of $(E,\Sigma,\pi)$,373737 A simple but technical explanation lies
in the fact that the map to normal coordinates around a point $m\in E$ is
defined by the exponential from some neighbourhood $V\subset T_{m}E$ to
$U\subset E$, as described above in footnote 9. But for the transformation to
preserve the fibres, we require $V\subset\pi^{*}(T_{\pi(m)}\Sigma)$, which is
not necessarily true. likewise of the 1-jet bundle. The additional structure
of a bundle $(E,\Sigma,\pi)$ is crucial to our formulation of the EFT, and
this bundle structure imposes some restrictions. In particular, the fields are
defined to be sections of the bundle, $\phi\in\Gamma(\pi)$. Any physically
meaningful transformation that we can do must therefore map the bundle to
another bundle; if not, then we can no longer even define fields (because we
will have lost the notion of sections). For example, we have already discussed
how field redefinitions, including derivative field redefinitions, may be
defined as suitable changes of section, $\phi\to\psi$ with
$\phi,\psi\in\Gamma(\pi)$; the non-derivative field redefinitions can moreover
be identified with bundle morphisms $(E,\Sigma,\pi)\to(F,\Omega,\rho)$, which
induce maps $\Gamma(\pi)\to\Gamma(\rho)$ at the level of sections. In either
case, it is straightforward to see that the condition
$x^{\mu}\circ\phi=x^{\mu}$ on our local fibred coordinate system
$(x^{\mu},u^{i})$, and the simple requirement that any section is locally an
inverse of the projection $\pi$, restricts us to transformations that act on
coordinates as

$\displaystyle x^{\mu}\mapsto x^{\prime\mu}(x^{\nu})\,,$ (275) $\displaystyle
u^{i}\mapsto u^{\prime i}(x^{\nu},u^{j})\,.$

This is manifestly true in the case of non-derivative redefinitions, as can be
seen from the commutative diagram in (45), but should hold more generally for
any physical ‘transformation’ of the bundle $(E,\Sigma,\pi)$ that we use to
define our EFT.

The map to normal coordinates can violate the basic requirement (275) for
preserving the (essential) bundle structure. To illustrate this, we consider
two counter-examples in which one cannot satisfy the conditions for normal
coordinates while preserving the bundle structure, both in the case of a
single real scalar in 4d (the subject of §6.2).

##### Counter-example: condition 1 for normal coordinates.

Recall from §2.4 that the first condition (35) for normal coordinates is that
the metric is diagonal (with entries further normalised to $\pm 1$,
determining the signature). Given an initial set of coordinates
$x^{I}=(x^{\mu},u,u_{\mu})$ on the 1-jet bundle, there are choices of
invariant metrics written in those coordinates that are not diagonal (with
off-diagonal elements indeed being necessary to match onto certain EFTs via
our construction of the Lagrangian). This means that to pass to coordinates
satisfying condition 1, we need to first do a rotation, and this rotation can
mix $x^{\mu}$ with the other coordinates, violating (275).

For an explicit example, using the general expressions (154–159) for the
invariant metric in the case of one real scalar in 4d, and evaluating at a
point $p\in J^{1}E$ such that $u(p)=0=u_{\mu}(p)$, we see that the metric
tensor $g_{IJ}(p)$ takes the following block form

$g_{IJ}=\begin{pmatrix}\eta_{\mu\nu}V(0)&0&\delta^{\mu}_{\nu}E(0)\\\
0&C(0)&0\\\ \delta^{\mu}_{\nu}E(0)&0&\eta^{\mu\nu}A(0)\end{pmatrix}$ (276)

Assuming that none of the component functions vanish, it is clear that
diagonalizing the metric requires a rotation of the form
$(x^{\mu},u_{\mu})\mapsto O(x^{\mu},u_{\mu})$ which mixes $x^{\mu}$ with
$u_{\mu}$, thus violating the bundle condition (275).

Now, the attentive reader might retort that, for the EFT described by this
metric (with all functions non-zero), the function $E(0)$ appearing on the
off-diagonal is actually redundant, in that it pulls back to the $\phi$
operator that is equivalent to the kinetic term after using IBPs; so if we
take $E(0)$ to zero, the metric is already (block) diagonal without the need
to mix $x^{\mu}$ with other coordinates. However, there are EFTs where we need
to keep the $E(0)$ function; in particular, if $A(0)=0$, in which case the EFT
is purely 2-derivative and the $E(0)$ term is required for invertibility of
the metric. In that case, diagonalising the metric clearly requires a non-
trivial rotation $(x^{\mu},u_{\mu})\mapsto O(x^{\mu},u_{\mu})$, violating the
bundle condition.

##### Counter-example: condition 3 for normal coordinates.

We now suppose that one has a metric for which condition 1 is already
satisfied, for example by taking the metric (276) and setting $E(0)=0$ and
$A(0)\neq 0$, and try to thence find coordinates that further satisfy
condition 3 (37). The determination of the explicit coordinate transformation
that takes us to normal coordinates on the total space can be found order-by-
order via an iterative procedure, but for our demonstration it will suffice to
stop at the first order, at which point the obstruction shall already arise.

We start with a local fibred coordinate system $x^{I}=(x^{\mu},u,u_{\mu})$ on
a local patch of the 1-jet manifold. To try to construct normal coordinates in
a perturbative fashion, consider a new coordinate system
$y^{I}=(y^{\mu},v,v_{\mu})$, defined by $x^{I}=y^{I}+a^{I}_{JK}y^{J}y^{K}$,
where $a^{I}_{JK}$ is a tensor of constants satisfying
$a^{I}_{JK}=a^{I}_{KJ}$. When evaluating the metric components $\bar{g}_{IJ}$
in the $y_{I}$ coordinate system, at the origin in these coordinates, we have

$\displaystyle\frac{\partial}{\partial y^{K}}\bar{g}_{IJ}(0)$
$\displaystyle=\frac{\partial}{\partial y^{K}}\bigg{(}\frac{\partial
x^{R}}{\partial y^{I}}\frac{\partial x^{S}}{\partial
y^{J}}g_{RS}\bigg{)}\bigg{|}_{y=0}$ (277)
$\displaystyle=a^{R}_{IK}g_{RJ}(0)+a^{S}_{JK}g_{IS}(0)+\frac{\partial}{\partial
x^{K}}g_{IJ}(0)$

We see that the number of indices and the symmetries of $a^{I}_{JK}$ and
$\frac{\partial}{\partial x^{K}}g_{IJ}(0)$ match. This implies it is possible
to choose the components $a^{I}_{JK}$ such that $\frac{\partial}{\partial
y^{K}}\bar{g}_{IJ}(0)=0$, thus satisfying a necessary condition for normal
coordinates (see §2.4), by requiring

$\frac{\partial}{\partial
x^{K}}g_{IJ}(0)=-\bigg{(}a^{R}_{IK}g_{RJ}(0)+a^{R}_{JK}g_{RI}(0)\bigg{)}\,.$
(278)

From (278) we see that $a^{I}_{JK}=-\Gamma^{I}_{JK}(0)$, where the symmetries
dictate that there are 18 different structures. The Christoffel symbols (with
respect to the original $x^{I}$ coordinates) are as usual given by

$\Gamma^{I}_{JK}=\frac{1}{2}g^{IM}(g_{MJ,K}+g_{MK,J}-g_{JK,M})\,.$ (279)

For our particular example, using the general expressions in Eqs. (154–159)
for the metric components and lowering all indices of the Christoffel symbols,
we find

$\displaystyle\Gamma_{\rho\mu\nu}(0)=0,\quad\Gamma_{\rho\mu
u}(0)=-\frac{1}{2}\partial_{u}V(0),\quad\Gamma_{\rho uu}(0)=0,$ (280)
$\displaystyle\Gamma_{\rho\mu u}^{\;\;\;\;\nu}(0)=0,\quad\Gamma_{\rho
uu}^{\;\;\;\;\mu}(0)=\frac{1}{2}\delta^{\mu}_{\rho}\bigg{(}\frac{1}{\Lambda^{2}}G(0)+\partial_{u}E(0)\bigg{)},\quad\Gamma_{\rho
uu}^{\;\;\mu\nu}(0)=0,$
$\displaystyle\Gamma_{uuu}(0)=\frac{1}{2}\partial_{\mu}C(0),\quad\Gamma_{uu\mu}(0)=0,\quad\Gamma_{u\mu\nu}=\frac{1}{2}\partial_{u}V(0),$
$\displaystyle\Gamma_{uuu}^{\;\;\;\;\mu}(0)=0,\quad\Gamma_{u\mu
u}^{\;\;\;\;\nu}(0)=\frac{1}{2}\delta^{\nu}_{\mu}\bigg{(}\frac{1}{\Lambda^{2}}G(0)-\partial_{u}E(0)\bigg{)},\quad\Gamma_{uuu}^{\;\;\mu\nu}(0)=\eta^{\mu\nu}\bigg{(}\frac{1}{\Lambda^{2}}B(0)-\frac{1}{2}\partial_{u}A(0)\bigg{)},$
$\displaystyle\Gamma_{uuu}^{\rho\mu\nu}(0)=0,\quad\Gamma_{uuu}^{\mu\nu}=\frac{1}{2}\eta^{\mu\nu}\partial_{u}A(0),\quad\Gamma_{uu\rho}^{\mu\nu}(0)=0$
$\displaystyle\Gamma_{uuu}^{\mu}(0)=0,\quad\Gamma_{u\nu\rho}^{\mu}(0)=0,\quad\Gamma_{uu\nu}^{\mu}(0)=\frac{1}{2}\delta^{\mu}_{\nu}\bigg{(}\frac{1}{\Lambda^{2}}B(0)+\partial_{u}E(0)-\frac{1}{\Lambda^{2}}G(0)\bigg{)}\,.$

Now let us consider a section $j^{1}\psi$, obtained by prolongation of a
section $\psi\in\Gamma(\pi)$, with

$\displaystyle y^{\mu}\circ j^{1}\psi$ $\displaystyle=y^{\mu}$ (281)
$\displaystyle v\circ j^{1}\psi$ $\displaystyle=\psi$ $\displaystyle
v_{\mu}\circ j^{1}\psi$ $\displaystyle=\frac{\partial\psi}{\partial y^{\mu}}$

But when evaluated on the coordinate map $x^{\mu}$, this section yields

$\displaystyle x^{\mu}\circ j^{1}\psi$
$\displaystyle=(y^{\mu}-\frac{1}{2}g^{\mu L}(0)\Gamma_{LJK}(0)y^{J}y^{K})\circ
j^{1}\psi$ (282) $\displaystyle=(y^{\mu}-\frac{1}{2}(g^{\mu\nu}(0)\Gamma_{\nu
JK}(0)+g^{\mu u}_{\;\;\nu}(0)\Gamma_{uJK}^{\nu}(0))y^{J}y^{K})\circ j^{1}\psi$
$\displaystyle=(y^{\mu}-(g^{\mu\nu}(0)\Gamma_{\nu\rho
u}(0)y^{\rho}v+g^{\mu\nu}(0)\Gamma_{\nu uu}^{\;\;\rho}(0)v_{\rho}v$
$\displaystyle\,\,\,\,\,+g^{\mu
u}_{\;\;\nu}(0)\Gamma_{uuu}^{\nu\rho}(0))vv_{\rho}+g^{\mu
u}_{\;\;\nu}(0)\Gamma_{uu\rho}^{\nu}(0))y^{\rho}v)\circ j^{1}\psi$
$\displaystyle=y^{\mu}-(g^{\rho\nu}(0)\Gamma_{\nu\mu
u}(0)y^{\mu}\psi+g^{\rho\nu}(0)\Gamma_{\nu
uu}^{\;\;\mu}(0)\psi\partial_{\mu}\psi$ $\displaystyle\,\,\,\,\,+g^{\rho
u}_{\;\;\nu}(0)\Gamma_{uuu}^{\nu\mu}(0))\psi\partial_{\mu}\psi+g^{\rho
u}_{\;\;\nu}(0)\Gamma_{uu\mu}^{\nu}(0))y^{\mu}\psi)\,,$

thus failing to satisfy the condition (275).

This means that the change of coordinates we wish to implement does not
preserve the bundle structure that is essential in formulating the
EFT.383838Lest there is any confusion here, thanks to the coordinate
independent nature of the metric tensor it is possible to use normal
coordinates to obtain the same result as in any other coordinate chart, since
the transformation $g_{IJ}(x)\to\bar{g}_{IJ}(y)$ is compensated by
$dx^{I}\otimes dx^{J}\to dy^{I}\otimes dy^{J}$, such that the combination
preserves the fibres, although there is little value in using normal
coordinates in this way. The map to normal coordinates not being fibre
preserving only causes an issue if we try to use it as a smooth map, since
then there is no compensating factor. Of course, one can always find example
geometries for which going to normal coordinates does happen to be consistent
with the bundle structure; from Eq. 282 it is easy to see that if the
Christoffel symbols or metric components appearing were set to zero, then the
map to normal coordinates would indeed satisfy (275). But this situation is
far from generic, meaning that normal coordinates are not especially useful in
our formalism.

Another important reason why normal coordinates are not generally useful in
our formalism is that passing to normal coordinates would be inconsistent with
the inclusion of any non-constant potential $V$. This can be seen by supposing
that we have gone to normal coordinates, and expanding the metric up to second
order, giving

$g=g_{IJ}(u)dx^{I}\otimes
dx^{J}=\bigg{(}\bar{g}_{IJ}+\frac{1}{3}R_{IJKL}(m)x^{K}x^{L}+\dots\bigg{)}dx^{I}\otimes
dx^{J}\,.$ (283)

If we wish to impose Poincaré invariance, then a term such as $R_{i\mu\nu
j}x^{\mu}x^{\nu}du^{i}\otimes du^{j}$ must be set to zero. However, the
symmetries of the Riemann tensor state that $R_{i\mu\nu j}=R_{\mu ij\nu}$ and
$R_{\mu ij\nu}u^{i}u^{j}dx^{\mu}\otimes dx^{\nu}$ corresponds to the mass term
in the Lagrangian. Therefore, the map to normal coordinates in conjunction
with Poincaré invariance requires the mass term to vanish and it is easy to
see that all terms in the potential are forced to vanish beyond the leading
order in the expansion.

Before moving on, we wish to make one last comment about normal coordinates in
relation to our jet bundle formalism, which is that the motivation for passing
to normal coordinates is also weaker than it was in the geometric approach
that we reviewed in §2. An important motivation for passing to normal
coordinates comes from the inclusion of the potential $V$ contributions to the
geometric formulae for amplitudes: the problem is that the second derivative
of the potential, which enters for example in four point amplitudes, does not
in general transform like a tensor on field space manifold $M$ due to the
appearance of Christoffel symbols in the expression. Normal coordinates
circumvent this non-tensorial behaviour because the Christoffel symbols
vanish. In contrast, in our bundle-based reformulation of EFTs the potential
is itself coming from components of the metric tensor, rather than being added
in by hand, so we never face this issue.

###  8.2 Expansion of the jet bundle metric

We now show how the metric on the jet bundle may be expanded, explicitly not
in normal coordinates, but in a way that is consistent with the iterated fibre
bundle structure. As usual, we work with the local fibred coordinate system
$\\{x^{\mu},u^{i},u^{i}_{\mu}\\}$ on a general jet manifold $J^{1}E$.393939To
avoid any confusion in this Section, we remind the reader that the coordinates
$\\{x^{\mu},u^{i},u^{i}_{\mu}\\}$ are all independent on the jet manifold; in
particular $\frac{\partial}{\partial x^{\mu}}u^{i}=0$ (with no relation to
$u_{\mu}^{i}$). The choice of origin point in the 1-jet manifold around which
we expand the metric will be determined by the section $j^{1}\phi$ according
to

$\displaystyle u^{i}(j^{1}\phi(p))=0\,,$ (284) $\displaystyle
u^{i}_{\mu}(j^{1}\phi(p))=0\,,$

i.e. $p\in\Sigma$ is a point in spacetime where the scalar field values
$\phi^{i}$, evaluated in our choice of local fibred coordinates, vanish. We
can consider a completely general scalar theory, with arbitrary spacetime
dimensions $d$ and an arbitrary number of fields. While we do not impose any
internal symmetries, we will impose Poincaré invariance, which simplifies the
expansion substantially since:

  * •

None of the metric tensor components may depend explicitly on $x^{\mu}$, i.e.
$\frac{\partial}{\partial x^{\mu}}g_{IJ}=0$ $\forall I,J$.

  * •

All spacetime indices must be contracted into singlets, including those
labelling derivative coordinates thanks to (112), thus any terms in the
expansion with an odd number of spacetime indices must vanish.

We adopt the following notational conventions for the expansion

$\bar{g}_{IJ}:=g_{IJ}(p)\quad\textrm{and}\quad
g_{IJ,K}:=\frac{\partial}{\partial x^{K}}g_{IJ}\,.$ (285)

By Taylor expanding the metric and keeping all contributions to operators with
up to four derivatives, we find

$\displaystyle g_{IJ}(x^{K})$ $\displaystyle dx^{I}\otimes
dx^{J}=\sum_{n=0}^{\infty}\frac{1}{n!}\bigg{(}\bar{g}_{\mu\nu,k_{1}\dots
k_{n}}u^{k_{1}}\dots u^{k_{n}}+\bar{g}_{\mu\nu,qrk_{1}\dots
k_{n-2}}^{\;\;\;\;\>\rho\sigma}u^{q}_{\rho}u^{r}_{\sigma}u^{k_{1}}\dots
u^{k_{n-2}}$ (286) $\displaystyle+\bar{g}_{\mu\nu,qrstk_{1}\dots
k_{n-4}}^{\;\;\;\;\>\rho\sigma\alpha\beta}u^{q}_{\rho}u^{r}_{\sigma}u^{s}_{\alpha}u^{t}_{\beta}u^{k_{1}}\dots
u^{k_{n-4}}\bigg{)}dx^{\mu}\otimes dx^{\nu}$
$\displaystyle+\sum_{n=0}^{\infty}\frac{1}{n!}\bigg{(}\bar{g}_{ij,k_{1}\dots
k_{n}}u^{k_{1}}\dots u^{k_{n}}+\bar{g}_{ij,qrk_{1}\dots
k_{n-2}}^{\;\;\;\;\rho\sigma}u^{q}_{\rho}u^{r}_{\sigma}u^{k_{1}}\dots
u^{k_{n-2}}\bigg{)}du^{i}\otimes du^{j}$
$\displaystyle+\sum_{n=0}^{\infty}\frac{1}{n!}\bar{g}_{ij,k_{1}\dots
k_{n}}^{\mu\nu}u^{k_{1}}\dots u^{k_{n}}du^{i}_{\mu}\otimes du^{j}_{\nu}$
$\displaystyle+2\sum_{n=0}^{\infty}\frac{1}{n!}\bigg{(}\bar{g}_{i\mu,qk_{1}\dots
k_{n-1}}^{\;\;\;\;\>\rho}u^{q}_{\rho}u^{k_{1}}\dots
u^{k_{n-1}}+\bar{g}_{i\mu,qrsk_{1}\dots
k_{n-3}}^{\;\;\;\;\>\rho\sigma\alpha}u^{q}_{\rho}u^{r}_{\sigma}u^{s}_{\alpha}u^{k_{1}}\dots
u^{k_{n-3}}\bigg{)}du^{i}\otimes dx^{\mu}$
$\displaystyle+2\sum_{n=0}^{\infty}\frac{1}{n!}\bigg{(}\bar{g}_{i\mu,k_{1}\dots
k_{n}}^{\nu}u^{k_{1}}\dots u^{k_{n}}+\bar{g}_{i\mu,qrk_{1}\dots
k_{n-2}}^{\nu\;\;\>\rho\sigma}u^{q}_{\rho}u^{r}_{\sigma}u^{k_{1}}\dots
u^{k_{n-2}}\bigg{)}du^{i}_{\nu}\otimes dx^{\mu}$
$\displaystyle+2\sum_{n=0}^{\infty}\frac{1}{n!}\bar{g}_{ij,qk_{1}\dots
k_{n-1}}^{\mu\;\;\rho}u^{q}_{\rho}u^{k_{1}}\dots
u^{k_{n-1}}du^{i}_{\mu}\otimes du^{j}$

where the factors of 2 appearing in front of off-diagonal blocks match our
previous conventions.

We re-iterate that, in the field space approach, the potential poses a problem
in such an expansion because it does not transform like a tensor. But within
the jet bundle approach, the potential is included in a component of the
metric tensor, and thus is packaged into an object that transforms
covariantly. The derivatives of the metric that appear in the expansion may be
re-expressed in terms of Christoffel symbols and curvature invariants if we so
desire, but the translation would be tedious and quickly becomes unwieldy as
we pass to higher jets.

###  8.3 Geometrizing amplitudes on the jet bundle

We are now almost ready to extract the lowest degree $n$-point amplitudes from
the expansion (286) by recalling our definition of the Lagrangian

$\mathcal{L}=\frac{1}{2}\langle\eta^{-1},(j^{1}\phi)^{*}g\rangle$ (287)

where composing the section $j^{1}\phi$ with the coordinate charts yields

$\displaystyle x^{\mu}\circ j^{1}\phi=x^{\mu}\,,$ (288) $\displaystyle
u^{i}\circ j^{1}\phi=\phi^{i}\,,$ $\displaystyle u^{i}_{\mu}\circ
j^{1}\phi=\partial_{\mu}\phi^{i}\,.$

For the propagator we find

$i$$j$
$\displaystyle=\frac{i}{\bar{g}_{ij}p^{2}+\frac{\eta^{\mu\nu}}{2}\bar{g}_{\mu\nu,ij}^{\;\;\;\;\>\rho\sigma}p_{\rho}p_{\sigma}+2\left(\bar{g}_{i\mu,j}^{\;\;\;\;\nu}-\bar{g}_{i\mu,j}^{\nu}\right)p_{\nu}p^{\mu}+\frac{\eta^{\mu\nu}}{2}\bar{g}_{\mu\nu,ij}+\bar{g}_{ij}^{\mu\nu}p^{2}p_{\mu}p_{\nu}}$
(289)

Requiring a canonical 2-derivative term enforces the relation

$\bar{g}_{ij}+\frac{1}{2d}\eta^{\mu\nu}\eta_{\rho\sigma}\bar{g}_{\mu\nu,ij}^{\;\;\;\;\>\rho\sigma}+\frac{2}{d}\delta^{\mu}_{\nu}\left(\bar{g}_{i\mu,j}^{\;\;\;\;\nu}-\bar{g}_{i\mu,j}^{\nu}\right)=\delta_{ij}\,,$
(290)

reproducing e.g. (194), which can be thought of as a physics constraint on the
jet bundle geometry. More precisely, for a metric that describes a physical
EFT, we must be able to find a section $\phi$ such that the kinetic term is
canonically normalised in this way, and it is implicit that we have already
performed this change of section.

For $\bar{g}_{ij}^{\mu\nu}\neq 0$, avoiding tachyonic instabilities in the
propagator Brivio:2014  additionally requires that

$\eta_{\mu\nu}\bar{g}_{ij}^{\mu\nu}=\begin{pmatrix}\lambda_{1}&0&\dots&0\\\
0&\lambda_{2}&\dots&0\\\ \dots&\dots&\dots&\dots\\\
0&0&\dots&\lambda_{n}\end{pmatrix},\quad\lambda_{i}<0\,\,\,\forall
i\in\\{1,\dots,n=\text{dim}(M)\\}\,,$ (291)

Again, it is implicit that we have done a derivative field redefinition, or
change of section, to further bring $\eta_{\mu\nu}\bar{g}_{ij}^{\mu\nu}$ into
diagonal form, which is always possible up to order $p^{6}$ corrections. This
inequality partly fixes the signature of the jet bundle metric (in the
derivative-coordinate directions), and is the manifestation of positivity in
our geometric formulation. For example, in the case of a single real scalar in
4d (§6.2), and using the parametrization (177), the above conditions become
simply

$\displaystyle\mathcal{F}_{0}(0)$ $\displaystyle=1\quad\text{(2-derivative
term)},$ (292) $\displaystyle\mathcal{F}_{1}(0)$
$\displaystyle<0\quad\text{(4-derivative term)}\,,$ (293)

with the latter being a known positivity bound Brivio:2014 .

Continuing, the $3$-point amplitude is404040 In these expressions for the
amplitudes, a ‘dot product’ denotes a contraction in spacetime indices, i.e.
$p_{q}\cdot p_{r}:=\eta^{\mu\nu}(p_{q})_{\mu}(p_{r})_{\nu}$. Furthermore,
Lorentz symmetry, together with the symmetry properties of the metric, fixes
the space-time structure of metric components. For example,
$\bar{g}_{\mu\nu,ijk}^{\;\;\;\;\;\alpha\beta}\eta^{\mu\nu}\sim\eta^{\alpha\beta}$
and $\bar{g}_{i\mu,jk}^{\;\;\;\;\alpha}\sim\delta_{\mu}^{\alpha}$.

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman [inline = (b.base)]
\vertex(a) {1}; \vertex[below= 1cm of a] (f); \vertex[right=1 cm of f] (b);
\vertex[right= 1cm of b] (c) {2} ; \vertex[below = 2cm of a] (d) {3} ;
\par\par\diagram*{ (a) -- [scalar] (b) -- [scalar] (c), (d) -- [scalar] (b) };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\begin{aligned}
&i\bigg{(}\frac{1}{12}\bar{g}_{\mu\nu,ijk}\eta^{\mu\nu}-\frac{1}{12}\bar{g}_{\mu\nu,ijk}^{\;\;\;\;\;\alpha\beta}\eta^{\mu\nu}\sum_{q,r}(p_{q})_{\alpha}(p_{r})_{\beta}-\frac{1}{2}\bar{g}_{ij,k}\sum_{q,r}p_{q}\cdot
p_{r}\\\
&-\frac{1}{2}\bar{g}_{i\mu,jk}^{\;\;\;\;\alpha}\sum_{q,r}(p_{q})_{\alpha}(p_{r})^{\mu}-\frac{1}{2}\bar{g}_{i\mu,jk}^{\nu}\sum_{q}(p_{q})_{\nu}(p_{q})^{\mu}\\\
&+\frac{1}{2}\bar{g}_{ij,k}^{\mu\nu}\sum_{q,r}(p_{q}\cdot
p_{r})(p_{q})_{\mu}(p_{r})_{\nu}+\bar{g}_{ij,k}^{\mu\;\;\nu}\sum_{q,r,s}(p_{q}\cdot
p_{r})(p_{q})_{\mu}(p_{s})_{\nu}\\\
&+\frac{1}{2}\bar{g}_{i\mu,jk}^{\nu\;\;\;\alpha\beta}\sum_{q,r,s}(p_{q})_{\alpha}(p_{r})_{\beta}(p_{s})_{\nu}(p_{s})^{\mu}\bigg{)}\,.\end{aligned}$
(294)

The $4$-point amplitude is

$\displaystyle\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman [inline = (b.base)]
\vertex(a) {1}; \vertex[below= 1cm of a] (f); \vertex[right=1 cm of f] (b);
\vertex[right= 1cm of b] (c) ; \vertex[below = 2cm of a] (d) {2} ;
\vertex[right = 2cm of a] (h) {3} ; \vertex[below = 2cm of h] (g) {4} ;
\par\diagram*{ (a) -- [scalar] (b) -- [scalar] (g), (d) -- [scalar] (b) --
[scalar] (h), }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\begin{aligned}
&i\bigg{(}\frac{1}{48}\bar{g}_{\mu\nu,ijlk}\eta^{\mu\nu}-\frac{1}{48}\bar{g}_{\mu\nu,ijkl}^{\;\;\;\;\;\alpha\beta}\eta^{\mu\nu}\sum_{q,r}(p_{q})_{\alpha}(p_{r})_{\beta}-\frac{1}{4}\bar{g}_{ij,kl}\sum_{q,r}p_{q}\cdot
p_{r}\\\
&-\frac{1}{6}\bar{g}_{i\mu,jkl}^{\;\;\;\;\,\nu}\sum_{q,r}(p_{q})^{\mu}(p_{r})_{\nu}-\frac{1}{6}\bar{g}_{i\mu,jkl}^{\nu}\sum_{q}(p_{q})^{\mu}(p_{q})_{\nu}\\\
&+\frac{1}{4}\bar{g}_{ij,kl}^{\mu\nu}\sum_{q,r}(p_{q}\cdot
p_{r})(p_{q})_{\mu}(p_{r})_{\nu}+\frac{1}{48}\bar{g}_{\mu\nu,ijkl}^{\;\;\;\;\gamma\delta\alpha\beta}\eta^{\mu\nu}\sum_{q,r,s,t}(p_{q})_{\gamma}(p_{r})_{\delta}(p_{s})_{\alpha}(p_{t})_{\beta}\\\
&+\frac{1}{4}\bar{g}_{ij,kl}^{\;\;\;\alpha\beta}\sum_{q,r,s,t}(p_{q}\cdot
p_{r})(p_{s})_{\alpha}(p_{t})_{\beta}+\frac{1}{6}\bar{g}_{i\mu,jkl}^{\;\;\;\;\,\nu\alpha\beta}\sum_{q,r,s,t}(p_{q})^{\mu}(p_{r})_{\nu}(p_{s})_{\alpha}(p_{t})_{\beta}\\\
&+\frac{1}{6}\bar{g}_{i\mu,jkl}^{\nu\;\;\,\alpha\beta}\sum_{q,r,s}(p_{q})^{\mu}(p_{q})_{\nu}(p_{r})_{\alpha}(p_{s})_{\beta}+\frac{1}{2}\bar{g}_{ij,kl}^{\mu\;\;\nu}\sum_{q,r,s}(p_{q}\cdot
p_{r})(p_{q})_{\mu}(p_{s})_{\nu}\bigg{)}\,.\end{aligned}$ (295)

To explain a little our notation here: sums over two indices run over values
$1\leq q<r\leq n$, where $n$ is the number of ‘points’ in the amplitude; a sum
over $n$ indices runs over all possible permutations, e.g.
$(q,r,s)\in\sigma(123)$ when appearing in $3$-point amplitudes, and
$(q,r,s,t)\in\sigma(1234)$ in $4$-point amplitudes; lastly, the sums over
three indices in Eq. (295) for the $4$-point amplitude run over all possible
combinations of three unique values out of four objects (a total of 12
combinations).

###  8.4 Comment on the role of geometric invariants

We conclude with a brief discussion of the role of curvature invariants on the
jet manifold, and their possible relevance to scattering amplitudes. For any
(pseudo)-Riemannian metric that we define on the jet manifold we can calculate
the curvature tensors associated to its Levi-Civita connection. The Riemann
tensor, Ricci tensor, and Ricci scalar of a given metric are all objects we
can compute on the jet manifold. All of them can only be related to physical
fields after being pulled back along a section $j^{1}\phi$ of the jet bundle,
to form a Lagrangian functional of $\phi$.

Since a smooth map moves points around on the manifold, it should be clear
that none of these object are invariant under one. We have seen many examples
of this throughout the paper. For the Riemann and Ricci tensors, the story is
the same one we saw back in §2, whereby a smooth map changes the metric tensor
$g$ (see e.g. Eq. (5.5.3)). The story for the Ricci scalar (which has no
‘components’) is perhaps more subtle and thus deserving of a closer look. The
isometry invariance of the Ricci scalar (as is relevant, for example, in
showing that it is invariant under coordinate transformations) follows simply
from the manipulations

$f^{*}R((f^{*})^{-1}g)=R(f^{*}(f^{*})^{-1}g)=R(g)\,.$ (296)

In contrast, under a smooth map $f$ (that actually moves points, rather than
just relabelling them) we do not have the compensating factor of
$(f^{*})^{-1}$, thus giving

$f^{*}R(g)=R(f^{*}g)\neq R(g)\,,$ (297)

unless, of course, the smooth map $f$ is itself an isometry with respect to
the metric $g$ (in this case it would be a ‘non-trivial isometry’, like the
internal $O(4)$ symmetry we have discussed in the context of the Higgs EFTs).
If we further take the smooth map $f$ to be a diffeomorphism, which we have
seen in preceding Sections can replicate a non-derivative field redefinition,
then $f$ will generally change the values of the curvature invariants – but
the requirement that $f$ and $f^{-1}$ be well defined and differentiable
ensures that a singularity cannot be introduced by $f$. This helps to make
sense of the criterion suggested in Ref. Cohen:2020xca  for determining
whether SMEFT is enough.

More generally, a field redefinition can be viewed as a change of section,
which is more general than a diffeomorphism (indeed, it cannot in general be
implemented as a bundle morphism, as discussed quite generally in §3.3.5).
Such a derivative change of section can introduce a singularity into any of
the curvature invariants, which is precisely the case in the example discussed
in Appendix E of Cohen:2020xca . Finally, we observe that a change of basis at
the level of the Lagrangian is typically done through a field redefinition
involving derivatives (such as the ‘$\Box\phi$’-dependent redefinitions we
frequently encountered in this paper), which can completely change the values
of curvature invariants, and potentially even introduce singularities. This
makes the use of curvature invariants a somewhat unreliable (i.e.
fundamentally basis-dependent) way of comparing theories at the level of the
Lagrangian.414141We nonetheless offer some related comments concerning
curvature invariants, and their role in detecting a certain subset of
redundancies in our metric to Lagrangian maps, in Appendix A.

##  9 Conclusion

In this paper we used geometry on jet bundles, which are a sequence of
manifolds that one can construct given a set of scalar fields $\phi^{i}$ for
which spacetime derivatives $\partial_{\mu_{1}\dots\mu_{r}}\phi^{i}$ are
treated as independent coordinates, as a source of Lagrangians defining scalar
field theories. In particular, we showed that Lagrangians containing operators
with up to 4 spacetime derivatives can be obtained by pulling back a metric on
the appropriate 1-jet bundle given the field content of the theory. The jet
bundle geometry is taken to be the most general geometry that is invariant
under the symmetries of the EFT, which we saw are naturally extended (or
‘prolongated’) to the jet bundle.

In all the examples we have explored (§6 and §7), the Lagrangian obtained by
pulling back the 1-jet bundle metric contains a complete basis of operators up
to 4 derivatives, where all IBP redundancies have been removed. The IBP
redundancies are accounted for automatically in the formalism, meaning that
the jet bundle metric naturally pulls back to a Lagrangian without
$\square\phi$ terms – one could say that the jet bundle metric naturally takes
us to the most general possible EFT Lagrangian (with up to 4-derivatives)
expressed in a Green’s basis. On the other hand, equation-of-motion
redundancies, which stem from invariance of the $S$-matrix under certain
derivative field redefinitions, can be identified and implemented, but in a
more ad hoc fashion. An important thing to stress, in our view, is that the
map from 1-jet metrics to 4-derivative EFT Lagrangians surjects, in the sense
that one can find metrics that map to every possible EFT. We suspect that this
statement can be generalized to invariant Lagrangians with up to $2(r+1)$
derivatives, in correspondence with invariant metrics on the $r$-jet bundle –
but we leave a proof to future work.

Along the way, we hope to have clarified various issues related to field
redefinitions, which we understand to be smooth changes of (possibly
prolongated) section of the (jet) bundle. This applies to both non-derivative
and derivative field redefinitions. In the former case, one is afforded an
equivalent description in terms of bundle morphisms, which means the field
redefinition can be implemented by doing a diffeomorphism on the jet bundle
metric. We find no such correspondence between derivative changes of section
and bundle morphisms.

Finally, we expand our metric in the vicinity of a point in the 1-jet bundle.
Translating this to the Lagrangian, we organise an expansion of ${\cal L}$ in
terms higher- and higher-point interactions, inspired by the ideas of
Cohen:2021ucp , keeping all interactions with up to 4 derivatives. This
facilitates a connection between components of the 1-jet bundle metric (and
its derivatives) and propagators/amplitudes of the EFT. We do not elucidate
this correspondence comprehensively in this paper, but we already highlight
some nice features – for example, we obtain the lowest-point correlation
functions explicitly, and infer e.g. how positivity (unitarity) constrains the
signature of the jet bundle metric.

In the future, we aim to investigate more systematically the structure of
amplitudes in this jet-bundle formalism, especially in the important case of
the Higgs EFTs. Other elements that we plan to incorporate into our jet bundle
formalism, to describe higher-derivative EFTs geometrically, include the
incorporation of fermions, and the (partial) gauging of symmetries – both of
which are necessary to connect with the rich phenomenology of the EFTs that
describe elementary particles at the energy frontier.

###### Acknowledgements.

JD is grateful to Ben Gripaios and Joseph Tooby-Smith for passing on their
knowledge of jet bundles a number of years ago. JD further thanks J. Tooby-
Smith for comments on the first version of this manuscript, and Nakarin
Lohitsiri for discussions. IB thanks J.C. Criado, R.V. Harlander and M. Schaaf
for their support with the validation of the dimension-10 and 12 SMEFT bases.
We thank Javier Lizana, Ken Mimasu, Dave Sutherland, Tevong You, and other
participants of the HEFT 2023 workshop, for helpful discussions. We are also
grateful to Nathaniel Craig and Yu-Tse Lee for their patience in waiting for
us to complete a draft of our manuscript, and to coordinate with their release
of Craig:2023hhp . The work of JD is funded by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme under grant agreement 833280 (FLAY), and by the Swiss National
Science Foundation (SNF) under contract 200020-204428. IB and MA acknowledge
funding from SNF through the PRIMA grant no. 201508.

##  Appendix A Diagnosing redundancies in the metric using invariants

Although the geometric invariants are unlikely to be useful when it comes to
distinguishing Lagrangians, they do have an application when it comes to
redundancies at the level of the metric.

Our expression for the Lagrangian is coordinate free, which has the benefit of
allowing us to express the metric in our preferred coordinate system, but
introduces ambiguities as it is difficult at a first glance to determine
whether two metrics $g$ and $g^{\prime}$ will yield the same Lagrangian. We
saw earlier that a coordinate transformation leaves the Lagrangian unchanged,
thus if we can show that $g$ and $g^{\prime}$ are related by a coordinate
transformation, then we know that they will generate the same Lagrangian.
Writing down the most general coordinate transformation is quite challenging
and quickly becomes unreasonable as the dimension of our jet manifold
increases. Comparing scalar geometric invariants gives an approach to see if
two metrics are dissimilar Mergulhao:2020awr .

Scalar geometric invariants are coordinate independent, but they are typically
expressed in coordinates making them difficult to compare directly. However,
an alternative approach exists. One can start by calculating several scalar
invariants, such as:

  * •

Ricci scalar: $\mathcal{R}_{1}=g^{\mu\nu}g^{\rho\sigma}R_{\mu\nu\rho\sigma}$

  * •

Kretschmann scalar: $\mathcal{R}_{2}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}$

  * •

Cubic curvature invariant:
$\mathcal{R}_{3}=R^{\mu\nu}_{\alpha\beta}R^{\alpha\beta}_{\rho\sigma}R^{\rho\sigma}_{\mu\nu}$

Then one can establish functional relations among these invariants. If at any
point the functional relations of the two metrics disagree, then we conclude
that they are not related by a coordinate transformation. However, agreement
in all computed functional relations is not proof that the metrics are related
by a transformation, since there is an infinite number of scalar invariants.

In Eq. (137) we saw that several components of the metric pullback to the same
term resulting in redundancies in the final expression. To identify the terms
that could be removed by a coordinate transformation it is not enough to
compare scalar invariants, rather we need a way to prove that two metrics
generate the same geometry, which can be done using the Cartan-Karlhede
algorithm cartan1951leccons ; brans1965invariant ; karlhede1980review ;
Mergulhao:2020awr ; McNutt:2017zxm .

The first step in applying the formalism is to choose a symmetric metric with
constant signature (e.g. $\delta_{ij},\eta_{ij},\dots$) and introduce a frame
of vector fields $\\{e_{a}=e^{i}_{a}\partial_{i}\\}$ such that

$g_{ij}e^{i}_{a}e^{j}_{b}=\delta_{ab}$ (298)

the choice of frame is not unique since

$e^{a}_{i}\to e^{\prime a}_{i}=\Lambda^{a}_{b}e^{b}_{i}$ (299)

with

$\delta_{ab}\Lambda^{a}_{c}\Lambda^{b}_{d}=\delta_{cd}$ (300)

is also a valid frame of vector fields.

The matrices $\Lambda$ form a Lie group with dimension $\frac{n(n-1)}{2}$. Now
we compute the Riemann tensor and project it along these frames looking for
the subgroup $H_{0}$ of matrices that preserve the form of the Riemann tensor

$R_{abcd}=R^{\prime}_{\alpha\beta\gamma\delta}$ (301)

and determine the number $\tau_{0}$ of independent parameters upon which the
components of the Riemann tensor depend.

Afterwards we repeat the procedure for the covariant derivatives of the
Riemann tensor and determine subgroups $H_{1},H_{2},\dots$ as well as numbers
of independent variables $\tau_{1},\tau_{2},\dots$ stopping once we have
reached an order $q$ at which $H_{q}=H_{q-1}$ and $\tau_{q}=\tau_{q-1}$. If a
frame exists at this order such that all the derivatives of Riemann tensor for
each of the two metrics $g$ and $g^{\prime}$ agree then we conclude that they
are equivalent and produce the same geometry and thus yield the same
Lagrangian upon pullback.

The redundancies captured by this algorithm are only the ones that could be
removed by a coordinate transformation. Redundancies that need to be removed
via integration by parts relations and field redefinitions are only seen after
pullback and thus are not related to coordinate transformations.

##  Appendix B Higher jet bundles for higher derivatives: a completeness proof

One of the main points of this paper is that scalar EFTs with up to 4
derivatives admit a geometric interpretation on a 1-jet bundle. More
precisely, we showed in a number of explicit examples that the Lagrangian
obtained pulling back to spacetime a 1-jet bundle metric always contains a
redundant set of effective operators with 0, 2 and 4 derivatives, that can be
reduced down to a minimal basis by applying a small number of IBP (and EOM, if
desired) relations.

In this Appendix we provide a general proof that the Lagrangian obtained by
pulling back to spacetime the most general $r$-jet bundle metric always
contains a redundant set of effective operators with up to $2(r+1)$
derivatives and arbitrary number of field insertions. Moreover, within this
set, it is always possible to identify a complete and non-redundant Green’s
basis that does not contain operators with boxes.

This result proves that the empirical observations in §§6, 7 generalize to any
scalar theory, and that operators with 6 or more derivatives also admit a
geometric interpretation in terms of higher-jet bundle metrics. The proof
formally holds for a Lorentz-invariant scalar EFT in any number of spacetime
dimensions (with Lorentz indices being everywhere contracted via the Minkowski
metric, as justified generally in §5.5.2), although the interpretation of the
EFT expansion is of course different in different dimensions.424242For
example, in the special case of 2d EFTs, the scalar field is dimensionless and
so the counting of derivatives coincides precisely with the mass dimension
counting of EFT operators.

Our argument consists of two points, that we prove individually below:

  1. 1.

Consider pure scalar interactions among $N$ fields and with a total of
$2(r+1)$ derivatives. It is always possible to construct a complete and IBP-
non-redundant basis of operators at this order, that does not contain
operators with boxes, nor operators with more than $(r+1)$ derivatives acting
on a single field.

  2. 2.

Any operator containing an arbitrary number of scalar fields $N$ and up to
$2(r+1)$ derivatives, such that no more than $(r+1)$ derivatives act on one
field and that boxes are absent, can be obtained by pulling back to spacetime
the most general metric of a $r$-jet bundle.

Put together, these statements imply that a non-redundant Green’s basis can
always be found within the set of box-less operators with $N$ fields and up to
$2(r+1)$ derivatives produced by the $r$-jet bundle metric. Vice versa, any
arbitrary scalar EFT Lagrangian always admits a geometric interpretation on
the appropriate jet bundle.

In practice, the metric will always produce a redundant Lagrangian, that
contains all possible box-less structures with up to $2(r+1)$ derivatives,
plus some (but not all!) of those with boxes. In general, there will be
several options to reduce the resulting operator set down to a minimal basis.
Here we choose, as an algorithmic rule, to remove all operators with boxes.
This choice is convenient for a number of reasons: (i) statement 1. above
guarantees that boxes can always be safely removed in any scalar theory at any
order; (ii) statement 2. above ensures that a box-less basis will be obtained
directly from the jet bundle metric, without requiring manipulations at the
Lagrangian level; (iii) empirically we find that this choice removes most
ambiguities in the basis construction. In fact, in all the examples
considered, there turned out to be a unique basis satisfying the box-less
requirement; (iv) any alternative choice aimed at retaining operators with
boxes would need to distinguish between box operators that can and cannot be
obtained from the metric, making the construction more cumbersome.

In the statements above we are also requesting the absence of operators with
more than half of the derivatives $(r+1)$ acting on a single field. This
condition is strictly necessary in order to obtain the Lagrangian from a
$r$-jet bundle. In principle one could have evaded this rule by choosing a
higher-jet bundle, which however would have also produced terms with more than
$2(r+1)$ derivatives. It is interesting that the minimal (and more natural)
choice of a $r$-jet bundle suffices to produce a complete basis.

In the next two subsections we provide proofs of points 1. and 2. above. The
results in this Appendix hold independently of the number of scalar flavours
present and on whether an internal symmetry is imposed or not, as they are
insensitive to the presence of flavour indices on the scalar fields. Arbitrary
contractions of internal indices can always be inserted appropriately in the
metric functions.

###  B.1 Proof of point 1.

Operators with more than $(r+1)$ derivatives on a single field can always be
removed by trivially moving derivatives across the operator via IBP, until
there is a maximum of $(r+1)$ derivatives on each field.434343Of course, this
requires at least 2 fields to be inserted in one operator. On the other hand,
a term with 1 field only would automatically be a total derivative. Therefore
they are always redundant.

Operators with boxes acting on one or more fields can also be removed
similarly, by using IBP to move one of the two derivatives contracted in a box
to other fields. This operation requires to apply, in sequence, an IBP for
each box, _e.g._

$\displaystyle(\partial_{\mu}\square\phi^{i_{1}})(\square\phi^{i_{2}})(\partial^{\mu}\phi^{i_{3}})A(\phi)$
$\displaystyle=-(\partial_{\mu}\partial_{\rho}\phi^{i_{1}})(\square\phi^{i_{2}})(\partial^{\mu}\phi^{i_{3}})(\partial^{\rho}A(\phi))+\dots$
(302)
$\displaystyle=(\partial_{\sigma}\partial_{\mu}\partial_{\rho}\phi^{i_{1}})(\partial^{\sigma}\phi^{i_{2}})(\partial^{\mu}\phi^{i_{3}})(\partial^{\rho}A(\phi))+\dots$

where $A$ is some analytic function of the fields, $i_{k}$ is a potential
flavour index carried by the $k$-th field, and the dots stand for other terms
stemming from the IBPs. A key point is that this operation can never
reintroduce boxes because, at each step, it moves a derivative with a
different Lorentz index. Therefore operators with boxes can always be fully
removed from an operator basis.

To conclude our proof, we need to check that the two redundancy classes can be
treated independently, _i.e._ that the IBPs used to remove boxes do not re-
introduce operators with more than $(r+1)$ derivatives on a single field. Let
us consider an operator with $N$ fields and $2(r+1)$ derivatives, such that
$d_{k}$ derivatives act on the $k$-th field, and they are contracted to form
$b_{k}$ boxes, with $2b_{k}\leq d_{k}$.

The boxes can be removed with a sequence of IBPs as shown above. They send

$\displaystyle d_{k}$ $\displaystyle\to d_{k}-b_{k}+s\,,$
$\displaystyle\text{with}\quad 0$ $\displaystyle\leq s\leq\sum_{j\neq
k}b_{j}\,.$ (303)

The term $-b_{k}$ accounts for the fact that, for every box on the $k$-th
field, one derivative is removed and placed elsewhere. In addition, the $k$-th
field receives $s$ derivatives coming from other boxes in the product. IBPs
replace the initial operator with a sum over several terms, in which $s$ takes
all possible values in a range from 0 to the total number of boxes originally
acting on fields other than the $k$-th. The latter number is bounded by

$\displaystyle\sum_{j\neq k}b_{j}$
$\displaystyle\leq\frac{2(r+1)-d_{k}-(d_{k}-2b_{k})}{2}=r-d_{k}+b_{k}+1\,.$
(304)

The numerator on the righthand side counts how many Lorentz indices on
$\phi_{j\neq k}$ can be contracted into boxes: there are at most
$2(r+1)-d_{k}$, from which we need to subtract $(d_{k}-2b_{k})$ that must be
contracted with open derivatives on $\phi_{k}$. Putting together (303) and
(304) we see that

$\displaystyle d_{k}-b_{k}+s\leq d_{k}-b_{k}+\sum_{j\neq k}b_{j}\leq r+1\,.$
(305)

This proves that the IBP required to remove boxes can never reintroduce terms