aainstitutetext: Physik Institut, Universität Zürich,
Switzerlandbbinstitutetext: Dipartimento di Fisica e Astronomia, Università di
Bologna, Italy

# Jet Bundle Geometry of Scalar Field Theories

Mohammad Alminawi a,b Ilaria Brivio a and Joe Davighi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For scalar field theories, such as those EFTs describing the Higgs, it is
well-known that the 2-derivative Lagrangian is captured by geometry. That is,
the set of operators with exactly 2 derivatives can be obtained by pulling
back a metric from a field space manifold $M$ to spacetime $\Sigma$. We here
generalise this geometric understanding of scalar field theories to higher-
(and lower-) derivative Lagrangians. We show how the entire EFT Lagrangian
with up to 4-derivatives can be obtained from geometry by pulling back a
metric to $\Sigma$ from the _1-jet bundle_ that is (roughly) associated with
maps from $\Sigma$ to $M$. More precisely, our starting point is to trade the
field space $M$ for a fibre bundle $\pi:E\to\Sigma$, with fibre $M$, of which
the scalar field $\phi$ is a local section. We discuss symmetries and field
redefinitions in this bundle formalism, before showing how everything can be
‘prolongated’ to the 1-jet bundle $J^{1}E$ which, as a manifold, is the space
of sections $\phi$ that agree in their zeroth and first derivatives above each
spacetime point. Equipped with a notion of (spacetime and internal) symmetry
on $J^{1}E$, the idea is that one can write down the most general metric on
$J^{1}E$ consistent with symmetries, in the spirit of the effective field
theorist, and pull it back to spacetime to build an invariant Lagrangian;
because $J^{1}E$ has ‘derivative coordinates’, one naturally obtains operators
with more than 2-derivatives from this geometry. We apply this formalism to
various examples, including a single real scalar in 4d and a quartet of real
scalars with $O(4)$ symmetry that describes the Higgs EFTs. We show how an
entire non-redundant basis of 0-, 2-, and 4-derivative operators is obtained
from jet bundle geometry in this way. Finally, we study the connection to
amplitudes and the role of geometric invariants.

##  1 Introduction

Effective Field Theories (EFTs) have long played a crucial role in guiding our
understanding of fundamental physics to higher and higher energies. In the
past decade, EFTs have been increasingly regarded as essential tools to
explore physics beyond the Standard Model (SM), becoming prominent in both the
theoretical and experimental programs of new physics searches Brivio:2017vri ;
Isidori:2023pyp . Two EFTs are particularly pertinent in this context: the so-
called Standard Model EFT (SMEFT) Buchmuller:1985jz ; Grzadkowski:2010es  and
the Higgs EFT (HEFT) Feruglio:1992wf ; Buchalla:2012qq ; Alonso:2012px ;
Buchalla:2013rka ; Brivio:2016fzo ; Dong:2022jru . Both theories extend the SM
Lagrangian with higher-dimensional interactions, but they differ in the choice
of scalar fields and in the power-counting adopted to organize the EFT series.
While the SMEFT is formulated in terms of the $SU(2)$ Higgs doublet, that
transforms linearly under the (gauge) symmetries, the HEFT builds upon the
formalism of the electroweak chiral Lagrangian, whereby the three Goldstone
bosons are embedded in a chiral field that transforms non-linearly
Coleman:1969sm ; Appelquist:1974tg ; Longhitano:1980iz ; Longhitano:1980tm ,
and the physical Higgs particle is treated as a singlet excitation
Grinstein:2007iv . This difference is expected to capture different dynamics
underlying the electroweak symmetry breaking (EWSB) mechanism, which gives
great significance to the question of which EFT should be used to go beyond
the SM Falkowski:2019tft .

The phenomenological characterization of SMEFT and HEFT is a long-standing
open challenge (see e.g. Isidori:2013cla ; Brivio:2013pma ; Ellis:2013lra ;
Liu:2019rce ; Eboli:2021unw ; Gomez-Ambrosio:2022qsi ; Domenech:2022uud ;
Dawson:2023ebe ). The main obstacle to overcome in realising this program
stems from ambiguities in the formulation of these EFTs; namely, the fact that
the two can be partially mapped onto each other via field redefinitions,
indicating that at least a fraction of the phenomenological differences
emerging in an order-by-order comparison might be unphysical. In particular,
some of these differences might be washed out upon (partially) resumming the
infinite tower of EFT operators.

Geometrical formulations of these EFTs Burgess:2010zq ; Alonso:2015fsp ;
Alonso:2016oah ; Helset:2018fgq ; Corbett:2019cwl ; Helset:2020yio  were
introduced in recent years partly to address this issue, and have already
proven very powerful in characterizing HEFT and SMEFT at a more fundamental
level Alonso:2017tdy ; Hays:2020scx ; Cohen:2020xca ; Cohen:2021ucp ;
Alonso:2021rac ; Alonso:2022ffe ; Helset:2022pde ; Talbert:2022unj ;
Helset:2022tlf ; see Alonso:2023upf  for a pedagogical overview. The core idea
is that the (generalised) kinetic term of the scalar sector, by which we mean
the set of all 2-derivative operators, could be seen as defining a metric on a
real 4-manifold $M$ spanned (locally) by the four scalar fields. In this
language, the invariance of physical scattering amplitudes under field
redefinitions has been interpreted in terms of the invariance of geometric
quantities (specifically, curvature invariants) of the field space under
coordinate transformations. This approach allowed a more rigorous (and
representation-independent) formulation of the differences between HEFT and
SMEFT, and of the conditions under which each EFT is appropriate. Because the
scalar manifold geometry emerges as a remnant of the decoupling of heavy BSM
particles, these conditions can also be related to specific properties of the
UV sector Alonso:2016oah ; Cohen:2020xca ; Banta:2023prj . For instance, a
SMEFT expansion cannot be constructed if the scalar geometry is singular at
the point where the EW symmetry is restored. This can be associated to the
presence of so-called ‘loryons’, heavy particles that have been integrated out
in the EFT but which take at least ‘half’ of their mass from the Higgs
mechanism Banta:2021dek .

Arguably, this geometric approach has two important limitations when applied
to EFTs: (i) the scalar manifold geometry can only capture interaction terms
with exactly 2 derivatives, and (ii) there is no geometric understanding of
invariance under field redefinitions with derivatives, which, from the field
theory perspective, are entirely reasonable ‘symmetries’ of the path integral.
Point (i) is problematic because higher-derivative operators obviously play an
important role in EFTs, and carry information that is in principle independent
of that encoded in the 2-derivative terms. Point (ii) implies that geometric
quantities are in fact basis-dependent. For instance, upon applying derivative
field redefinitions, the curvature on $M$ changes, in general. First steps
towards obviating these issues were taken recently in Cohen:2022uuw ;
Craig:2023wni .

Of particular relevance to limitation (i), and to the present paper, Ref.
Craig:2023wni  recently introduced the idea of ‘Lagrange spaces’ as a
generalisation of the field space geometry, wherein directional derivatives of
the field space coordinates are treated as independent coordinates. Lagrangian
functions can be built with these coordinates, which encode a subset of
operators with more than 2 derivatives in a covariant formalism – in
particular, because different spacetime derivatives $\partial_{\mu}$ cannot be
distinguished, only operators with specific ‘flavour’ symmetries can be
realised using these derivative coordinates. The present paper is in a similar
spirit, in that we also seek a geometric formulation of scalar field theories
that extends to higher-derivatives. We approach the problem in a largely
orthogonal direction, using, instead, pure geometry on ‘jet bundles’ to
formulate scalar EFT Lagrangians.

In differential geometry, jet bundles provide a coordinate-free way to
describe higher-derivatives of maps between manifolds; in the context of our
EFTs, jet bundles are spaces on which derivatives of the scalar field are
treated as extra coordinates in a consistent fashion. Roughly speaking, given
a spacetime $\Sigma$, a field space $M$, and an integer $r$, one can construct
an ‘$r$-jet bundle’ which is itself a manifold, with local coordinates
corresponding to spacetime coordinates $x^{\mu}$, field space coordinates
$u^{i}$, and ‘$r$-derivative coordinates’ $u^{i}_{\mu}$,
$u^{i}_{\mu_{1}\mu_{2}}$, …, $u^{i}_{\mu_{1}\dots\mu_{r}}$. This set of
manifolds offers a promising arena for extending the geometric approach to
EFTs to Lagrangians with more than 2 derivatives – suggesting a route for
overcoming limitation (i) described above, which we aim to explore in this
paper.

Figure 1: A schematic illustration of three key players in this paper: a
spacetime manifold $(\Sigma,\eta)$ with local coordinates $x^{\mu}$ on a patch
thereof; a ‘field space bundle’ $E$ whose base space is $\Sigma$ and fibre is
$M$, which has fibre coordinates $u^{i}$; and lastly the 1-jet manifold
$J^{1}E$ of the bundle $E$, which is itself a fibre bundle over both $E$ and
over $\Sigma$. A ‘scalar field’ configuration is a section $\phi$ of the
bundle $E$, and the composition $u^{i}\circ\phi=\phi^{i}(x)$ returns its
‘field value’, given the choice of local fibre coordinates $u^{i}$, at $x$.
From $\phi$ one can construct a section $j^{1}\phi$ of the bundle
$j^{1}E\to\Sigma$ by ‘prolongation’. The 1-jet bundle also admits a local
fibred coordinate system $(x^{\mu},u^{i},u^{i}_{\mu})$, where the extra
$\\{u_{\mu}^{i}\\}$ are ‘derivative coordinates that, when evaluated on a
section $j^{1}\phi$, agree with the first derivatives $\partial_{\mu}\phi(x)$.
Finally, metrics $g^{(0)}$ and $g^{(1)}$ can be defined on $E$ and $J^{1}E$
such that, upon pulling back to $\Sigma$ along $\phi$ or $j^{1}\phi$
respectively, one can obtain the most general EFT Lagrangian for that scalar
field theory with up to 2- or 4-derivative operators (and any number of
fields).

To that end, we formulate scalar field theory Lagrangians using geometry on
the 1-jet bundle. The geometries of the various manifolds involved in our
discussion, and maps between them, are illustrated in Figure 1. Our strategy
is to write down the most general metric on the 1-jet bundle consistent with
the symmetries of the EFT, which are themselves naturally extended to (or,
more precisely, ‘prolongated’ to) the jet bundle. An invariant EFT Lagrangian
function is then formed by pulling back this metric all the way to spacetime
$\Sigma$ along a section of the jet bundle (which is itself obtained by
prolonging a scalar field configuration $\phi$), and contracting with the
inverse spacetime metric. One immediately obtains terms with more than 2
derivatives from this recipe; for example, if we pull back a metric component
$\delta_{ij}\eta^{\mu\nu}du^{i}_{\mu}\,du^{j}_{\nu}$ along a prolongation of
$\phi$, and then contract with $\eta^{\mu\nu}$, we obtain the 4-derivative
term $(\partial_{\mu}\partial_{\nu}\phi)^{2}$. We show that, in the case of
the SMEFT and HEFT, one obtains a complete basis of non-redundant EFT
operators with 0-, 2-, and 4-derivatives in this way, starting from a 1-jet
bundle geometry that is invariant under Poincaré and $O(4)$ custodial
symmetry.

The conclusion, that one can construct a complete EFT basis with up to
4-derivatives from a jet bundle metric, remains unchanged upon allowing a
breaking of custodial symmetry (§7.3), as occurs in the real-world electroweak
theory. In fact, it generalises to scalar Lagrangians with arbitrary ‘flavour’
structure, relying in no way on the assumption of an internal symmetry. Nor is
this an accident that occurs only at the 1-jet bundle order. We prove in
Appendix B that, following a similar recipe, the most general, Poincaré
invariant $r$-jet bundle metric can be used to construct an EFT Lagrangian
that contains a complete basis of operators with up to $2(r+1)$ derivatives
and arbitrary number of field insertions. In this sense, we suggest that jet
bundles offer a natural way of constructing generic scalar EFTs (barring only
topological terms) using only geometry – albeit on an increasingly high-
dimensional space if we want to cover an increasingly large number of
derivatives.

As well as extending the geometric formulation of scalar field theories to
higher-derivatives, we try to develop an understanding of field redefinitions
in this picture that includes derivative field redefinitions. An important
realisation is that the scalar field $\phi(x)$ is not really identified with
coordinates on the field space; rather, as the notation suggests, $\phi(x)$ is
the map from spacetime $\Sigma$ to field space $M$ (or, better still, $\phi$
is a local section of a bundle $\pi:E\to\Sigma$ with fibre $M$, which we take
as our starting point in this work). The image of this map (or section) is of
course evaluated in the coordinate chart, for each $x$, to return the ‘field
value’ at $x$. A field redefinition is then understood to be a change of map,
or better a change of section, rather than just a change of coordinates (the
latter leaves any metric $g$ trivially invariant, simply by virtue of $g$
being a tensor). Given that the maps (sections) ‘know about’ the spacetime
source manifold $\Sigma$, it is not surprising that they can be differentiated
with respect to $x$, whereas a coordinate on the target space cannot.
Therefore a change of section can depend on derivatives of the original
section, offering a more general understanding of field redefinitions – and
one that is, in essence, geometric.

In certain cases, in particular those a physicist would call ‘non-derivative
field redefinitions’, the change of section can be equivalently described by
doing a morphism on the bundle (or field space). All these notions lift to the
1-jet bundle formalism in a natural and consistent way. Lastly, derivative
field redefinitions can, in a very limited range of examples, be replicated by
(jet) bundle morphisms, but these are essentially accidents; indeed, there are
mathematical arguments (see §3.3.5) that suggest ‘derivative changes of
section’ cannot be induced by a (jet) bundle morphism. We try to elucidate
these various points throughout this paper.

We remark that jet bundles have been recently used in the study of scalar
field theories in Ref. Gripaios:2021hsc  by Gripaios and Tooby-Smith, but in
an altogether different context. There, the authors use jet manifolds to
rigorously formulate so-called ‘inverse Higgs constraints’. The use of jet
bundles is natural (and fruitful) in that context, because the inverse Higgs
constraint equations relate fields to their derivatives, and so can be
embedded as submanifolds of the 1-jet bundle that are invariant under the
action of the symmetry. While we were completing this paper, we became aware
of the forthcoming work Craig:2023hhp  by Craig and Lee, that builds on
Craig:2023wni  and also utilizes the jet bundle formalism in the context of
the Higgs EFTs.

### Summary of results

In this work we elucidate a geometric description of EFTs with up to four
derivatives, in terms of geometry on the 1-jet bundle associated to our
original field space maps. In doing so, here are some things we achieve (the
various concepts and terminology we use will be duly explained in the main
text):

  1. 1.

We construct a map from the space of 1-jet bundle metrics $g$ to 4-derivative
Lagrangians ${\cal L}$, in a natural way. Namely, given a section $\phi$ of
the field space bundle, a.k.a. a scalar field configuration, one pulls back
the jet bundle metric $g$ to spacetime along the associated section
$j^{1}\phi$ of the jet bundle, and then contracts with the inverse metric
$\eta^{-1}$ on spacetime to get an ordinary function or ‘Lagrangian’:

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

This map does not inject to the space of physically inequivalent Lagrangians,
as one might expect given the target space does not know about e.g.
integration by parts. So, lastly (and somewhat naïvely in our formalism), one
can implement integration by parts (IBP) identities to reduce the Lagrangian
to a particular basis.

  2. 2.

We prove that this map surjects, in the sense that every Lagrangian with up to
4-derivatives can be obtained by pulling back a metric on the 1-jet bundle. We
emphasize that all terms with 0-, 2-, or 4-derivatives are obtained in this
way, so that even the potential is captured ‘geometrically’. We show this
result explicitly in a number of toy examples, and we also prove that it holds
at any order, for a map from the space of $r$-jet bundle metrics into the
space of scalar Lagrangians with up to $2(r+1)$ derivatives.

  3. 3.

Along the way, we discuss how the symmetries of, say, the Higgs EFTs, can be
implemented in this (jet) bundle formalism. The internal $O(4)$ symmetry and
the spacetime $SO(1,3)$ Poincaré symmetry are described by different kinds of
bundle morphisms in this picture (depending on whether they move points in the
base space or not). Both kinds of symmetry naturally carry over to the 1-jet
bundle, and we enforce these symmetries on our metrics (hence Lagrangians).

  4. 4.

We give a geometric meaning to the notion of general (including derivative)
field redefinitions in terms of a (suitably smooth) change of section, which
induces a change of section of the 1-jet bundle after prolongation.

  5. 5.

In doing so, we also hope to clarify the geometric meaning of non-derivative
field redefinitions. These are a special class of change of section that can
be equivalently described by doing a bundle morphism (under which the metric
changes), before pulling back to spacetime along the same section. Such an
equivalent description is not available for changes of section that involve
derivative maps.

  6. 6.

Thanks to the components of the metric being smooth functions on the jet
bundle, we are able to expand the metric around a point, which after
specifying a section allows us to directly extract $n$-point amplitudes, here
done for $n=2,\,3,\,4$, in terms of products of momenta and the components of
the metric evaluated at that point.

Here are some things that we do not do in this paper and that are left for
future work:

  1. 1.

We do not describe how to implement gauging of a subgroup of the global
symmetry in this jet bundle formulation.

  2. 2.

We do not show how to couple this jet bundle geometry to fermions. For
2-derivative Lagrangian terms with two fermion insertions, this issue has been
tackled recently in Refs. Finn:2020nvn ; Assi:2023zid ; Gattus:2023gep .

  3. 3.

We do not go into any details concerning phenomenology.

The paper is structured as follows: in §2 we review and refine the formulation
of 2-derivative EFT Lagrangians using field space geometry. In particular, we
hope to clarify a little the notion of (non-derivative) field redefinitions.
In §3 we recast this formalism using a field space bundle, which is more
general (and motivated by locality). We again revisit the issue of field
redefinitions, now equipped with the notion of a bundle morphism. Jet bundles
are introduced in §4 and their relation to scalar Lagrangians is discussed in
great detail in §5. Section 6 contains a few toy examples that will be useful
to see how 1-jet bundle geometry captures operator bases including up to 4
derivatives; these examples also invite us to take a first look at the
redundancies present in the description. In §7 we turn to the SMEFT/HEFT case
of four scalars respecting a custodial $O(4)$ symmetry. Finally, in §8 we
comment on the interplay with scattering amplitudes, before concluding in §9.

##  2 Preliminaries

Our goal is to eventually describe the higher-derivative expansion of the
Higgs EFTs, namely ‘HEFT’ or ‘SMEFT’, using geometry. We begin by reviewing
the basic idea for geometrically formulating the Higgs EFTs.

The Higgs sector of the electroweak theory, by which we refer to the part of
the Lagrangian involving only scalar fields, is described by a 4d sigma model
with a target space $(M,g)$, which is a smooth manifold of real dimension 4,
with Riemannian metric $g$. This means that, given also a 4d spacetime
manifold with a (pseudo-)Riemannian metric, $(\Sigma,\eta)$, the degrees of
freedom of the QFT are smooth ($C^{\infty}$) maps

$\phi(x):\Sigma\to M\,,$ (2)

with dynamics described by an action that is a particular functional
$S[\phi]$. On a patch (open set) $\mathcal{F}_{M}$ of $M$ one can provide
local coordinates $u^{i}:\mathcal{F}_{M}\to\mathbb{R}^{4}$, where
$i=1,\dots,4$, and on a patch $\mathcal{U}_{\Sigma}$ of $\Sigma$ one can
provide local coordinates $x^{\mu}$. In this paper, we always take $\Sigma$ to
be a fixed non-dynamical background manifold, with a fixed metric – usually
flat Minkowski space $\mathbb{R}^{3,1}$, with
$\eta=\text{diag}(1,-1,-1,-1)$.111Extensions of scalar field theories of the
kind we consider to include dynamical gravity, so-called ‘scalar-tensor
theories’, can also be described using a covariant geometric formalism
Finn:2019aip .

Of course this geometric picture is not reserved for the particular 4d
theories that describe the Higgs, but equally describes scalar fields in any
spacetime dimension $d$,222The source manifold $\Sigma$ need not be identified
with spacetime, either; more generally it is the ‘worldvolume’ of the theory.
The most familiar example in which $\Sigma$ is not identified with spacetime
occurs in string theory, where one usually identifies $\Sigma$ with the 2d
string worldsheet, and the target space $M$ with the physical spacetime in
which the string moves. A similar description can be used for any extended
object. with any target space geometry. In §6.1, for example, we consider a
quantum mechanical example i.e. a scalar field theory in $0+1$ dimensions
(related examples have been recently discussed in Gripaios:2015pfa ;
Davighi:2019ffp ). An important class of examples in physics is those
(pseudo-)scalars arising from spontaneous symmetry breaking $G\to H$, for
$d>2$ Coleman:1973ci . In this case the scalar fields, here pseudo Nambu
Goldstone bosons (or pNGBs), are known to parametrize the coset space $M\cong
G/H$. In addition to the chiral Lagrangian Weinberg:1966fm  that describes QCD
pions, arguably a pillar of theoretical particle physics, there exist
physically important examples of such pNGB Lagrangians in condensed matter
physics, e.g. describing fluids Gripaios:2014yha ; Dersy:2022kjd  and more
exotic phases Nicolis:2013lma ; Nicolis:2015sra , and even cosmology
Nicolis:2008in ; Piazza:2017bsd .

##### Example: Standard Model (SM).

For the special case of the SM Higgs sector, we can take
$M\cong\mathbb{R}^{4}$ with globally defined Cartesian coordinates $u^{i}$,
and a flat metric.

### Symmetries of SMEFT and HEFT

An essential bit of data required to define these Higgs EFTs is an internal
symmetry, by which we mean (for now) that the target space $M$ is equipped
with the smooth action $\sigma$ by a Lie group $G$. This induces a group
action on the fields $\phi(x)$ which are (for now) maps into $M$, under which
we require that the field theory action functional $S[\phi]$, or more
precisely the phase $e^{iS[\phi]}$), be invariant. It is in this Lie group
action where the difference between HEFT and SMEFT lies. For both cases the
group is the same, namely $G=O(4)$ in the custodial limit333Actually, in
neither SMEFT nor HEFT do we know the global structure of the custodial
symmetry group $G$, only that its Lie algebra is $\mathfrak{so(4)}$. We take
$SO(4)$ to be enlarged by a parity symmetry e.g. $u^{4}\to-u^{4}$ to an
$O(4)\cong SO(4)\rtimes\mathbb{Z}/2$ symmetry; in other literature, the
symmetry is taken to be the universal cover $SU(2)_{L}\times
SU(2)_{R}\cong\text{Spin}(4)$, which is a non-isomorphic central extension of
$SO(4)$ also by $\mathbb{Z}/2$. This global ambiguity is unsurprising given
that even the electroweak gauge symmetry is ambiguous, being either
$SU(2)\times U(1)$ or $U(2)$ (see e.g. Tong:2017oea ; Davighi:2019rcd ).
(which we assume throughout most of this paper), but its action $\sigma$ on
$M$ is different.

This distinction is easy to express given a particular set of local
coordinates on a patch, but is harder to formulate in a coordinate-free (or
indeed field-redefinition-invariant) way. One of the successes of the
geometric formulation of Higgs EFTs has been to provide coordinate-free
criteria for detecting an $O(4)$ action that is decisively non-SMEFT
Falkowski:2019tft ; Cohen:2020xca . For now let us use particular local
coordinates to get started.

In the case of SMEFT, and for a particular set of local coordinates
$u^{i}=h^{i}$ on some open set $\mathcal{F}_{M}\subset M$, the group action
$\sigma_{\text{S}}$ is simply

$\displaystyle\sigma_{\text{S}}:O(4)\times M\to M:\,\,(O,h^{i})\mapsto
Oh^{i}\,,$ (3)

where $O\in O(4)$ acts on 4-component real vectors by matrix
multiplication.444We are being a little cavalier in expressing the group
action, which is well-defined on all of $M$, only by its action on points in
the patch $\mathcal{F}_{M}$ that is coordinatized by $\\{h^{i}\\}$. In
principle, one should provide an atlas of coordinate charts covering $M$ and
define the group action everywhere. Also note that the fixed patch
$\mathcal{U}$ will itself be moved by the group action, and so there can be
points ‘either side of the edge’ of the patch for which Eq. (3) cannot be
used. We invite the reader to look past such subtleties here.  Evidently, this
means that electroweak symmetry $SU(2)_{L}\times U(1)_{Y}\subset G$ acts
linearly. A consequence of this linear group action is that the patch
$\mathcal{F}_{M}$ always contains a fixed point of the group action, namely
the origin $o$ ($h^{i}=\vec{0}$), meaning that $\sigma_{\text{S}}:(O,o)\mapsto
o$ $\forall O\in O(4)$. This is the electroweak symmetry preserving point. As
shown in Alonso:2016oah , the converse is also true, namely if $M$ contains an
$O(4)$ fixed point then, in the vicinity of that point, there is always a
coordinate chart $u^{i}=h^{i}$ in which $O(4)$ acts linearly on $h^{i}$.
Moving away from this fixed point, any other point on $\mathcal{F}_{M}$,
written in the local coordinates $h^{i}$, preserves an $O(3)\subset O(4)$
subgroup. The vacuum of our Universe is such a symmetry breaking point
$h^{i}\neq\vec{0}$, with length squared $\delta_{ij}h^{i}h^{j}=v^{2}$.

For HEFT, the symmetry breaking is made manifest by the choice of group
action. Let us write our local coordinates in the form $u^{i}=(h,\pi^{a}/v)$,
where $a=1,\dots,3$. HEFT is characterised by a non-linear group action
$\sigma_{\text{H}}$ in these local coordinates

$\displaystyle\sigma_{\text{H}}:O(4)\times M\to
M:\,\,(O,h,\pi^{a}/v)\mapsto\left(h,O\left(\pi^{a}/v,\sqrt{1-\pi\cdot\pi/v^{2}}\right)^{T}\right)\,.$
(4)

In words, the radial mode $h$ is invariant under the $O(4)$ group action,
while the Goldstone modes $\pi^{a}$ transform non-linearly, via the usual
action of $O(4)$ on a unit vector spanning $S^{3}\cong O(4)/O(3)$, the coset
space associated with the electroweak symmetry breaking.

Spacetime $(\Sigma,\eta)$ is also equipped with a symmetry, which is a smooth
action $\rho$ by the Poincaré group $\cong\mathbb{R}^{1,3}\rtimes O(1,3)$. For
our coordinates $x^{\mu}$ on Minkowski spacetime $\mathbb{R}^{1,3}$, the (non-
linear) group action is

$\rho:\text{Poinc}\times\Sigma\to\Sigma\,\,:\,\,(a^{\mu},L^{\mu}_{\nu},x^{\mu})\mapsto
L^{\mu}_{\nu}x^{\nu}+a^{\mu}$ (5)

Throughout this paper we only consider Poincaré-invariant theories (including,
but not only, SMEFT and HEFT).

###  2.1 Two-derivative Lagrangians from field space geometry

The dynamics of such scalar field theories lend themselves to a geometric
description. This means that certain terms in the Lagrangian can be built
using geometric structures on the target space, which we have already
stipulated is a Riemannian manifold (and sometimes, more basically, a topology
alone is sufficient).

The most important and natural geometric construction is of the kinetic term
or, more generally, the Lagrangian terms featuring exactly two derivatives.
This requires a geometry on $M$, in the form of the metric $g$, which recall
is a symmetric and everywhere non-degenerate $(0,2)$-tensor on $M$. The two-
derivative Lagrangian terms are then

${\cal
L}_{2\partial}[\phi,g]=\frac{1}{2}\,\langle\eta^{-1},\,\phi^{\ast}(g)\rangle\,.$
(6)

That is, we pull back the metric $g$ along the map $\phi$,555We use the
notation $f^{\ast}(\cdot)$, standard in differential geometry, to denote the
pullback of an object along the map $f$. Likewise, pushforwards are denoted
$f_{\ast}(\cdot)$. In order to avoid confusion, any notion of ‘complex
conjugation’ e.g. of a scalar Higgs field, will be denoted with a dagger, viz.
$\phi^{\dagger}$.  to obtain a $(0,2)$ tensor on spacetime, which we contract
with the inverse spacetime metric to get an ordinary function. Note that this
formula for the Lagrangian is ‘coordinate-free’; the map $\phi:\Sigma\to M$ is
defined irrespective of our choices of coordinate chart, and both $\eta^{-1}$
and $g$ are tensors. The Lagrangian will be moreover invariant under the
symmetry of our theory if the metric $g$ is $\sigma$-invariant,666Note that
the stipulation of ‘invariance’ refers to the specific group action, which
recall differs for HEFT vs SMEFT, and which we emphasize by writing
‘$\sigma$-invariant’ rather than just ‘$G$-invariant’. assuming also that
$\eta$ is Poincaré invariant. Our notation ${\cal L}_{2\partial}[\phi,g]$
emphasizes that the Lagrangian is a functional of the map (field) $\phi$,
given $g$. To obtain the action $S[\phi,g]$, we integrate ${\cal
L}_{2\partial}[\phi,g]$ over $\Sigma$ using the canonical volume form
$\omega_{\Sigma}:=\sqrt{|\eta|}d^{4}x$, as we would for any Lagrangian ‘term’
that is a function (rather than a form).

The compact, coordinate-free formula (6) can be written in a notation more
familiar to physicists by using our local coordinates $x^{\mu}$ (on
$\mathcal{U}_{\Sigma}\subset\Sigma$) and $u^{i}$ (on $\mathcal{F}_{M}\subset
M$). In these local coordinates, the metrics are

$\displaystyle\eta$ $\displaystyle=\eta_{\mu\nu}\,dx^{\mu}\otimes
dx^{\nu}\,,\qquad\text{(flat)},$ (7) $\displaystyle g$
$\displaystyle=g_{ij}(u^{i})\,du^{i}\otimes du^{j}\,,$ (8)

and then Lagrangian (6) is

${\cal
L}_{2\partial}[\phi,g]=\frac{1}{2}\,\eta^{\mu\nu}\,g_{ij}\left(\phi^{i}(x)\right)\,\partial_{\mu}\phi^{i}(x)\partial_{\nu}\phi^{j}(x)\,,$
(9)

where

$\phi^{i}(x):=(u^{i}\circ\phi)(x)\,.$ (10)

In words, the quantity $\phi^{i}(x)$ returns the values of the local
coordinates $u^{i}$ evaluated at the image of spacetime point $x$ under the
map $\phi$ – that is, the ‘field value’ at $x$.

##### Example (ctd.): SM.

Take $M=\mathbb{R}^{4}$ and $u^{i}$ to be global Cartesian coordinates
thereon. The SM construction takes only the subset of Lagrangian terms that
are renormalisable. With this extra condition, symmetry dictates that
$g=\delta_{ij}du^{i}du^{j}$ is just the flat metric on $\mathbb{R}^{4}$, as
anticipated above. This pulls back under $\phi(x)$ to give the usual Higgs
kinetic term,

${\cal
L}_{\text{kin}}=\frac{1}{2}\eta^{\mu\nu}\delta_{ij}\partial_{\mu}\phi^{i}(x)\partial_{\nu}\phi^{j}(x)\,,$

upon evaluating (6).

##### Example: SMEFT.

In the case of SMEFT the most general $\sigma_{\text{S}}$-invariant metric can
be expressed in terms of two arbitrary functions $A(z)$ and $B(z)$, via
Alonso:2015fsp ; Alonso:2016oah

$g_{ij}(u^{i})=A\left(\frac{u\cdot
u}{\Lambda^{2}}\right)\,\delta_{ij}+B\left(\frac{u\cdot
u}{\Lambda^{2}}\right)\,\delta_{ik}\delta_{jl}\frac{u^{k}u^{l}}{\Lambda^{2}}\,,$

in our chosen SMEFT coordinate chart, where $v\cdot w:=\delta_{ij}v^{i}w^{j}$.
We have introduced appropriate factors of the formal power counting parameter
$\Lambda$, which will be identified with the EFT cut-off scale in a physical
theory. Here, $A(0)=1$ and $B(0)=0$ in order to recover the SM in the limit
$\Lambda\to\infty$. Substituting this metric into (9), or equivalently (6),
yields the most general set of $\sigma_{\text{S}}$-invariant SMEFT operators
with exactly two derivatives.

###  2.2 Other terms in the scalar EFT

We emphasize that this procedure constructs only a subset of the non-
renormalisable EFT Lagrangian, namely all operators with exactly two
derivatives (but any number of field insertions), from the metric $g$ on field
space $M$. Other terms in the EFT are, in this picture, included ‘by hand’,
and can be considered as extra structures on $M$ beyond the metric. These
other interactions fall into three classes:

  1. 1.

The ‘potential’ is the part of the Lagrangian with zero derivatives; it is the
pullback of a $\sigma$-invariant function $V(\phi):M\to\mathbb{R}$ from $M$ to
$\Sigma$, which is again integrated on $\Sigma$ using $\omega_{\Sigma}$ to
obtain the action contribution; that is,
$S[\phi]\supset\int_{\Sigma}\phi^{\ast}(V)\omega_{\Sigma}$.

  2. 2.

There are all the local operators appearing in the EFT construction of Callan,
Coleman, Wess, and Zumino Coleman:1969sm ; Callan:1969sn  with more than two
derivatives. Capturing these operators systematically using geometry is a
primary purpose of the present paper.

  3. 3.

The action might also admit topological terms like the Wess–Zumino–Witten term
Wess:1971yu ; Witten:1983tw , that are the pullback of (possibly locally-
defined) differential forms from $M$ to $\Sigma$, then integrated directly on
$\Sigma$. These require only a topology on each space. Invariance under
$\sigma$ is rather more subtle here than for local operators Davighi:2018inx ,
especially when the map $\phi$ is topologically non-trivial; a large class of
such terms are given by $\sigma$-invariant differential cocycles
Davighi:2020vcm  on $M$, of degree $d+1$ where $d=\text{dim}(\Sigma)$ in
general. Examples of such terms abound in Composite Higgs models (which are
SMEFT) Gripaios:2009pe ; Davighi:2018xwn , and e.g. in the condensed matter
examples mentioned previously Dubovsky:2011sk ; Delacretaz:2014jka .

We largely neglect terms of type (3) in the rest of this paper, focussing
rather on the non-topological (i.e. metric-dependent) sector of the EFT. We
briefly comment on topological terms in the Higgs EFTs in §7.4.

###  2.3 What is a (non-derivative) field redefinition?

One central feature of the geometric formulation of Higgs EFTs has been the
identification of (some subset of) field redefinitions, which are symmetries
of the partition function and thence of observables (see e.g. Criado:2018sdb
), with coordinate transformations on field space $M$. Specifically, ‘non-
derivative field redefinitions’ of the form $\phi(x)\mapsto\psi(\phi(x))$ have
been identified with changes of coordinates. It is not clear how to think
about derivative field redefinitions, whereby
$\phi\mapsto\psi(\phi,\partial\phi,\dots)$, in this geometric picture, since
the target space $M$ knows nothing about spacetime derivatives. At this stage,
we want to revisit this established lore concerning non-derivative field
redefinitions. Our discussion will be fairly pedantic, but in doing so will
open the way to a simple geometric picture for more general smooth field
redefinitions, even before we consider the jet bundle extension of field space
in later Sections.

To begin, we emphasize that we have been careful, so far, to distinguish local
coordinates $u^{i}$ on $\mathcal{F}_{M}$ from the field variable $\phi$, which
is a map from $\Sigma$ into $M$. This is again distinguished from
$\phi^{i}=u^{i}\circ\phi$, which evaluates the local coordinate values of the
field for each spacetime point. A ‘change of coordinates’, a.k.a. a
‘coordinate transformation’ or a ‘change of chart’, is, when restricted to
$\mathcal{F}_{M}$, a map $u^{i}\to u^{i\prime}(u^{i})$. This is not obviously
related to a redefinition of the field, which we would more readily identify
with some transformation on the map $\phi$ (inducing a transformation on the
$\phi^{i}$). In this Subsection we contrast how these two kinds of
transformation, namely a change in coordinates $u^{i}\to u^{i\prime}$ vs. a
change in map $\phi\to\psi$, affect the Lagrangian that we construct from the
metric on $M$ via (6). The main concepts presented in this Subsection are
illustrated in Figure 2.

####  2.3.1 Change of coordinates

The first key point is simply that the field space metric
$g=g_{ij}(u^{i})du^{i}\otimes du^{j}$ does not change at all under a change of
coordinates, when evaluated at the same point $m$ in $M$. The components
change via appropriate factors of the Jacobian matrix, viz.

$g_{ij}=\frac{\partial u^{k\prime}}{\partial u^{i}}\frac{\partial
u^{l\prime}}{\partial u^{j}}g_{kl}^{\prime}\,,$ (11)

which is compensated by the variation of the 1-forms

$du^{\prime i}=\frac{\partial u^{\prime i}}{\partial u^{k}}du^{k}\,.$ (12)

This is just the statement that the metric is a tensor, and so $g(m)$
evaluated at any point $m\in M$ is independent of the choice of the coordinate
chart $\\{u^{i}\\}$. Once we pullback the metric along the map $\phi(x)$ and
form the Lagrangian ${\cal L}=(1/2)\langle\eta^{-1},\,\phi^{\ast}(g)\rangle$,
one obtains exactly the same function of the field variable $\phi(x)$,
regardless of having done the change of coordinates.

If this argument sounds a little thin, we can make the point totally explicit
with an example. Consider a 1-dimensional field space manifold $\mathbb{R}$,
on which we define a local coordinate $\\{u\\}$. In this coordinate chart,
consider a (flat) metric $g=du\otimes du$; in components, $g_{uu}=1$. Choose a
map $\phi:\Sigma\to\mathbb{R}$, namely a scalar field configuration. The
Lagrangian formed as in Eq. (6) is then ${\cal
L}=\frac{1}{2}\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)$, the kinetic energy
for a real scalar.777We slightly abuse notation here and let $\phi(x)$ denote
both the map itself and the coordinate value $u\circ\phi$; there is no
component index because the target space is 1-dimensional. Now, let us do a
change of coordinates in the target space, and send

$u\mapsto u^{\prime}=u^{2}\,.$ (13)

In the illustration of Fig. 2, this corresponds to reading off coordinates
using the scale on the right side of the plot, rather than the left, for the
_same_ points on $M$. The metric expressed in the new coordinate system, but
at the same point, is

$g^{\prime}=\frac{1}{4u^{\prime}}du^{\prime}du^{\prime}(=g)\,,$ (14)

and so its components are $g_{u^{\prime}u^{\prime}}=1/(4u^{\prime})$.

To form the ‘new’ Lagrangian after doing the coordinate transformation, we
then pullback the metric, in the new coordinates, along the same map
$\phi(x)$. Note that the map $\phi(x)$ is defined independently of coordinates
– it takes points in $\Sigma$ to points in $M$ – but that we evaluate its
image in the given coordinate chart. We find the Lagrangian

$\displaystyle{\cal L}[\phi,g]\mapsto{\cal L}[\phi,g^{\prime}]$
$\displaystyle=\frac{1}{2}\langle\eta^{-1},\,\phi^{\ast}(g^{\prime})\rangle$
$\displaystyle=\frac{1}{2}\eta^{\rho\sigma}\left\langle{\partial_{\rho}}\otimes{\partial_{\sigma}},\,\frac{1}{4(u^{\prime}\circ\phi)}\partial_{\mu}(u^{\prime}\circ\phi)\partial_{\nu}(u^{\prime}\circ\phi)\,dx^{\mu}\otimes
dx^{\nu}\right\rangle$
$\displaystyle=\frac{1}{2}\eta^{\rho\sigma}\left\langle\partial_{\rho}\otimes\partial_{\sigma},\,dx^{\mu}\otimes
dx^{\nu}\right\rangle\,\,\frac{1}{4\phi^{2}}\partial_{\mu}(\phi^{2})\partial_{\nu}(\phi^{2})$
$\displaystyle=\frac{1}{2}\eta^{\mu\nu}\,\frac{1}{4\phi^{2}}\,(2\phi\partial_{\mu}\phi)(2\phi\partial_{\nu}\phi)$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi={\cal
L}[\phi,g]\,,$ (15)

the same as before. The tensorial nature of the metric means that, in general,
the Lagrangian (6) is not changed (as a functional of its argument) by any
change of coordinates on $M$. The freedom to change coordinate charts is akin
to a gauge redundancy in this geometric formulation of the theory.

Figure 2: Illustration of the notion of non-derivative field redefinitions in
a toy example describing a single real scalar (take $M=\mathbb{R}$, with
coordinate $u$) in 1d spacetime (take $\Sigma=\mathbb{R}$, with coordinate
$x$). A field configuration is a map from $\Sigma$ to $M$; here $\phi$ (red-
solid) and $\psi$ (blue-dashed) denote two such maps. A change of field space
coordinates, such as $u\mapsto u^{\prime}=u^{2}$, simply corresponds to
reading off coordinate values using the right axis rather than the left axis.
It does not move points in $M$, nor change the metric $g$ (or any other
tensor) evaluated at any point $m\in M$; if we pull back $g$ to $\Sigma$ using
a fixed map $\phi$ to form a Lagrangian, we obtain the same result regardless
of whether we used the $u$ or $u^{\prime}$ coordinate chart on $M$. A (non-
derivative) field redefinition is rather a particular change of maps such that
$u\circ\psi$ can be expressed as a function of $u\circ\phi$. In the figure,
$u\circ\psi=(u\circ\phi)^{2}$, such that evaluating $\psi(x)$ is equivalent to
first evaluating $\phi(x)$, then doing a diffeomorphism $f$ on $M$ that sends
a point in $M$ with coordinate $u$ to a different point with coordinate
$u^{2}$, expressed in a fixed coordinate chart. Reading off the value of the
curve $\psi(x)$ on the left is equivalent to reading off the value of
$\phi(x)$ on the right. The figure also illustrates that if $\phi$ sends
multiple points on $\Sigma$ to the same point on $M$, the change of map can be
equivalent to a diffeomorphism on $M$ only if $\psi$ preserves this feature.

####  2.3.2 Change of maps

Let us instead consider what happens to the Lagrangian (6) when we change
field space maps, from $\phi(x)$ to a different map $\psi(x):\Sigma\to M$. We
assume the change in map $\phi\to\psi$ is smooth. Recall that
$\phi^{i}(x)=(u^{i}\circ\phi)(x)$ denotes the local coordinates of the image
of point $x$ under the original map $\phi$. A new map $\psi$ sends each $x$ to
a different point $\psi(x)\in M$, for which $\psi^{i}(x)=(u^{i}\circ\psi)(x)$
are its local coordinates. The two different maps are indicated in Fig. 2 by
the red-solid and blue-dashed curves.

We again use the simple example of a real scalar field for illustration, with
field space $M=\mathbb{R}$, local coordinate $u$, and flat metric $g=du\otimes
du$. Consider the particular change of map, defined in terms of coordinate
values (in fixed chart $\\{u\\}$) of field space points,

$\phi(x)\mapsto\psi(x)\quad|\quad u\circ\psi=(u\circ\phi)^{2}\,.$ (16)

(If we continue to abuse notation as above, we would write this simply as
$\psi=\phi^{2}$, denoting both the map and the coordinate value by the same
symbol $\phi$.) Now, when we pull back the metric $g$ along this new map
$\psi$, the Lagrangian (6) is not necessarily the same (even though the metric
$g$ is unchanged) because $\phi^{\ast}(g)\neq\psi^{\ast}(g)$. We obtain

$\displaystyle{\cal L}[\phi,g]\mapsto{\cal L}[\psi,g]$
$\displaystyle=\frac{1}{2}\langle\eta^{-1},\,\psi^{\ast}(g)\rangle$
$\displaystyle=\frac{1}{2}\eta^{\rho\sigma}\left\langle\partial_{\rho}\otimes\partial_{\sigma},\,dx^{\mu}\otimes
dx^{\nu}\right\rangle\,\,\partial_{\mu}(u\circ\psi)\partial_{\nu}(u\circ\psi)$
$\displaystyle=\frac{1}{2}\eta^{\mu\nu}\,\,\partial_{\mu}(\phi^{2})\partial_{\nu}(\phi^{2})=2\phi^{2}\partial_{\mu}\phi\partial^{\mu}\phi\,.$
(17)

When expressed in terms of the original field variable, the functional form of
the Lagrangian has indeed changed; the key difference to the manipulations in
(15) is that there is no compensating factor of $1/4\phi^{2}$ coming from the
Jacobian.

Intuitively, the reason we get a different answer is because we pulled back
the metric from different points in field space, namely those in the image of
$\psi$ rather than $\phi$; this is not true for a simple change of
coordinates, which does not move points in $M$ but only relabels them. The
transformation (17) of ${\cal L}$ under change of section $\phi\to\psi$ is
what one would usually call a ‘field redefinition’.

####  2.3.3 Field space diffeomorphisms

The change of maps $\phi\to\psi$ that we have performed is, in this case,
precisely equivalent to doing a (local) field space diffeomorphism $f$ before
pulling back the transformed metric $f^{\ast}(g)$ along the same section
$\phi$. Recall that a diffeomorphism is a smooth invertible function between
manifolds $M$ and $N$:

$f:M\to N\,.$ (18)

In words, every point in the manifold $N$ can be related uniquely to a point
in the manifold $M$; if $f$ moves the points hit by $\phi$ to the points hit
by $\psi$, then it exactly replicates the effect of changing maps. To relate
two maps $\phi$ and $\psi$ we try to construct a diffeomorphism $f$ to fill in
the commutative diagram:

${\Sigma}$${(M,g)}$${(N,g)}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$$\scriptstyle{f}$
(19)

so that

$\psi=f\circ\phi\,.$ (20)

When considering field redefinitions, as we do here, we can take $M$ and $N$
to be the same manifold, and so $f$ becomes a map from $M$ to itself.

It is clear that such a diffeomorphism $f$ cannot be found for any pair $\phi$
and $\psi$ of field space maps. So, when can a change of maps (a.k.a. a field
redefinition) be realised as a field space diffeomorphism? Consider a map
$\phi$ that sends two spacetime points $x$ and $y$ to the same point in field
space, viz. $\phi(x)=\phi(y)=m_{x}$. Then, via (20), we can only change to a
new map $\psi$ that satisfies $\psi(x)=f(\phi(x))=f(m_{x})$ and
$\psi(y)=f(\phi(y))=f(m_{x})$, thus $\psi(x)$ and $\psi(y)$ must also agree
(see Fig. 2 for illustration). This means that $\psi$ must be a function of
$\phi$. Explicitly, evaluating Eq. (20) on our coordinate chart $\\{u^{i}\\}$
on $M$,

$u^{i}\circ\psi=u^{i}\circ f\circ\phi=u^{i}\circ f(\phi)\,,$ (21)

thus the particular change of map induced by a smooth map $f$ can be expressed
as

$\phi^{i}\mapsto\psi^{i}=f^{i}(\phi^{j})\,.$ (22)

This is precisely what physicists call a ‘non-derivative field redefinition’,
and applies to the example (16) considered above. If we additionally can write

$\psi^{i}\mapsto\phi^{i}=(f^{-1})^{i}(\psi^{i})$ (23)

then the change of maps is a diffeomorphism and the theories are equivalent at
the level of the manifolds on which they live.

Indeed, if we reconsider the example considered in the previous Subsection, we
can see explicitly how one recovers the shift (17) in the Lagrangian that we
obtained by changing map $\phi\to\psi$, by instead doing a diffeomorphism
$f:M\to M$ before pulling back along the same map $\phi$. The required
diffeomorphism888If the manifold contains a point $m\in M$ with $u(m)=0$, then
this map is only a local diffeomorphism, i.e. a diffeomorphism on a patch that
does not include this point, because $f^{-1}:u\to\sqrt{u}$ is not
differentiable at $m$. Alternatively, the map $f$ can be ‘promoted’ to a
proper diffeomorphism by instead taking $M=\mathbb{R}_{>0}$, the positive
half-line.  is one that sends field space point $m\in\mathbb{R}$ with local
coordinate $u$ to the point $m^{\prime}$ with coordinate $u^{2}$ (written in
the same coordinate chart); in local coordinates, we might simply write this
as $f:u\mapsto u^{2}$. In Fig. 2, this can be visualised by interpreting the
right axis of the plot as the image of the left axis under $f$, such that the
two axes represent _different_ sets of points on $M$. Under this
diffeomorphism the metric changes, as

$g=du\otimes du\mapsto f^{\ast}g=\bigg{(}\frac{\partial u\circ f}{\partial
u}\bigg{)}^{2}du\otimes du=4u^{2}du\otimes du\,.$ (24)

Pulling back $f^{\ast}g$ along $\phi$ then gives the transformed Lagrangian

$\displaystyle\mathcal{L}[\phi,g]\mapsto{\cal L}[\phi,f^{\ast}(g)]$
$\displaystyle=\frac{1}{2}\langle\eta^{-1},\,\phi^{\ast}(f^{\ast}(g))\rangle$
$\displaystyle=\frac{1}{2}\eta^{\mu\nu}4(u\circ\phi)^{2}\partial_{\mu}(u\circ\phi)\partial_{\nu}(u\circ\phi)$
$\displaystyle=2\phi^{2}\partial_{\mu}\phi\partial^{\mu}\phi={\cal
L}[\psi,g]\,,$ (25)

matching the result (17) that we obtained by changing maps.

In a similar way, any non-derivative field redefinition of the form (22) can
be implemented as a smooth map on the field space. The transformed Lagrangian
can be equivalently expressed in terms of the transformed map $\psi$, or in
terms of the smooth map $f$:

${\cal
L}\mapsto\frac{1}{2}\langle\eta^{-1},\,\psi^{\ast}(g)\rangle=\frac{1}{2}\langle\eta^{-1},\,\phi^{\ast}(f^{\ast}(g))\rangle\,.$
(26)

At risk of being overly pedantic, we wish to distinguish a diffeomorphism on
$M$ from a change of coordinates, although the two notions are intimately
related. A diffeomorphism involves a change of coordinates plus a pushforward
to the new point; equivalently, a change of coordinates means doing a
diffeomorphism, but then pulling back the tensor to be evaluated at the
original point.

Finally, we stress that a change of maps $\phi(x)\to\psi(x)$ is more general
than the special case (22) that can be captured by field space diffeomorphisms
(or even general smooth maps on field space). Differentials of the smooth map
$\phi$ can be used to construct, at least infinitesimally, what one would call
‘derivative field redefinitions’.

####  2.3.4 Local field space diffeomorphisms, and HEFT vs. SMEFT

There is an important subtlety in this discussion that we have so far glossed
over, that was hinted at in footnote 8. Namely, it may be the case that the
diffeomorphism $f$ is only defined locally, meaning in words that not all
points in $M$ correspond to points in $N$. A particularly relevant physics
example of this situation occurs for the map from SMEFT to HEFT, as follows.
Let us start with a coordinate system $\\{u^{1},u^{2},u^{3},u^{4}\\}$ on a
patch $\mathcal{F}_{M}\subset M$ in which the metric is

$g_{ij}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{pmatrix}\,.$ (27)

Pulling this back to $\Sigma$ along a section $\phi$ whose components in the
chart are $u^{i}\circ\phi=\phi^{i}(x)$, we find the familiar 2-derivative
Lagrangian

$\mathcal{L}=\frac{1}{2}\langle\eta^{-1},\phi^{*}g\rangle=\frac{1}{2}\delta_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}\,$
(28)

of a canonically normalised set of Higgs fields.

Now consider the map $f:\mathcal{F}_{M}\to\mathcal{F}_{N}\subset N$ defined by

$f:\begin{pmatrix}u^{1}\\\ u^{2}\\\ u^{3}\\\ u^{4}\\\
\end{pmatrix}\to\left(1+\frac{u^{4}}{v}\right)\begin{pmatrix}u^{1}\\\ u^{2}\\\
u^{3}\\\ \sqrt{v^{2}-(u^{1})^{2}-(u^{2})^{2}-(u^{3})^{2}}\end{pmatrix}\ $ (29)

expressed in the same coordinate chart $\\{u^{i}\\}$. After doing this smooth
mapping, we compute the new metric to be

$f^{*}g_{ij}=\begin{pmatrix}\left(1+\frac{u^{4}}{v}\right)^{2}\left(1+\frac{(u^{1})^{2}}{v^{2}-\vec{u}^{2}}\right)&\left(1+\frac{u^{4}}{v}\right)^{2}\frac{u^{1}u^{2}}{v^{2}-\vec{u}^{2}}&\left(1+\frac{u^{4}}{v}\right)^{2}\frac{u^{1}u^{3}}{v^{2}-\vec{u}^{2}}&0\\\\[5.69054pt]
\left(1+\frac{u^{4}}{v}\right)^{2}\frac{u^{1}u^{2}}{v^{2}-\vec{u}^{2}}&\left(1+\frac{u^{4}}{v}\right)^{2}\left(1+\frac{(u^{2})^{2}}{v^{2}-\vec{u}^{2}}\right)&\left(1+\frac{u^{4}}{v}\right)^{2}\frac{u^{2}u^{3}}{1-\vec{u}^{2}}&0\\\\[5.69054pt]
\left(1+\frac{u^{4}}{v}\right)^{2}\frac{u^{1}u^{3}}{v^{2}-\vec{u}^{2}}&\left(1+\frac{u^{4}}{v}\right)^{2}\frac{u^{2}u^{3}}{1-\vec{u}^{2}}&\left(1+\frac{u^{4}}{v}\right)^{2}\left(1+\frac{(u^{3})^{2}}{1-\vec{u}^{2}}\right)&0\\\\[5.69054pt]
0&0&0&1\end{pmatrix}\,,$ (30)

where $\vec{u}=(u^{1},u^{2},u^{3})$. Once again we can pullback along a
section $\phi$ and contract with $\eta^{-1}$ to obtain the Lagrangian function

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi^{4}\partial^{\mu}\phi^{4}+\frac{1}{2}\left(1+\frac{\phi^{4}}{v}\right)^{2}\bigg{(}\partial_{\mu}\phi^{1}\partial^{\mu}\phi^{1}+\partial_{\mu}\phi^{2}\partial^{\mu}\phi^{2}+\partial_{\mu}\phi^{3}\partial^{\mu}\phi^{3}\bigg{)}$
(31)
$\displaystyle+\frac{(1+\phi^{4}/v)^{2}}{v^{2}-(\phi^{1})^{2}-(\phi^{2})^{2}-(\phi^{3})^{2}}\bigg{(}\phi^{1}\partial_{\mu}\phi^{1}\phi^{2}\partial^{\mu}\phi^{2}+\phi^{1}\partial_{\mu}\phi^{1}\phi^{3}\partial^{\mu}\phi^{3}+\phi^{2}\partial_{\mu}\phi^{2}\phi^{3}\partial^{\mu}\phi^{3}\bigg{)}$
$\displaystyle+\frac{1}{2}\frac{(1+\phi^{4}/v)^{2}}{v^{2}-(\phi^{1})^{2}-(\phi^{2})^{2}-(\phi^{3})^{2}}\bigg{(}(\phi^{1})^{2}\partial_{\mu}\phi^{1}\partial^{\mu}\phi^{1}+(\phi^{2})^{2}\partial_{\mu}\phi^{2}\partial^{\mu}\phi^{2}+(\phi^{3})^{2}\partial_{\mu}\phi^{3}\partial^{\mu}\phi^{3}\bigg{)}\,.$

Introducing the notation $\vec{\phi}=(\phi^{1},\phi^{2},\phi^{3})$, this can
be written in the more compact form:

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\partial_{\mu}\phi^{4}\partial^{\mu}\phi^{4}+\frac{1}{2}\left(1+\frac{\phi^{4}}{v}\right)^{2}\left[\partial_{\mu}\vec{\phi}\cdot\partial^{\mu}\vec{\phi}+\frac{1}{v^{2}-\vec{\phi}^{2}}\bigg{(}\vec{\phi}\cdot\partial_{\mu}\vec{\phi}\bigg{)}^{2}\right]\,.$
(32)

If we identify $\phi^{4}$ with the physical Higgs boson field and the other
$\phi^{i}$ with the Goldstones, we see that the Higgs has a canonical kinetic
term and does not enter any of the scalar products. Eq. (32) is in agreement
with the results in Refs. Gomez-Ambrosio:2022qsi ; Alonso2016:smn . The map
from SMEFT to HEFT is known to always exist and be smooth at the level of the
Lagrangian.

On the other hand, the inverse map (that would take one from HEFT to SMEFT)
may have a singularity. The inverse map $f^{-1}$ would be

$f^{-1}:\begin{pmatrix}u^{1}\\\ u^{2}\\\ u^{3}\\\ u^{4}\\\
\end{pmatrix}\to\frac{v}{\sqrt{\mathbf{u}^{2}}}\begin{pmatrix}u^{1}\\\
u^{2}\\\ u^{3}\\\
\frac{\sqrt{\mathbf{u}^{2}}}{v}\left(\sqrt{\mathbf{u}^{2}}-v\right)\end{pmatrix}\,,$
(33)

where $\mathbf{u}=(u^{1},u^{2},u^{3},u^{4})$. This is clearly only well
defined if $\mathbf{u}^{2}\neq 0$.

The map from this Lagrangian back to (28) may have a singularity at the
origin, in which case $f$ is simply a smooth map on the whole manifold and
only becomes a diffeomorphism (which must be invertible) when restricted to a
patch that does not contain the origin (the $O(4)$ fixed point).
Mathematically speaking, this means that the two manifolds on which SMEFT and
HEFT are formulated are indistinguishable away from this point. When SMEFT is
not enough Cohen:2020xca , we really only have a ‘smooth map’ $f$ from SMEFT
to HEFT, which cannot be inverted.

###  2.4 Geometrizing amplitudes

As we saw in §2.3.1, a change of coordinates does not change the Lagrangian
(whereas, we reiterate, a smooth map does – and the latter is what we mean by
a non-derivative field redefinition). The coordinate invariance of the
Lagrangian nevertheless remains a useful redundancy to exploit, because it
means we are free to pick any valid coordinate system on a given local patch.

In particular, this opens up the choice of ‘normal coordinates’ in the
vicinity of some point $m$ in patch $\mathcal{F}_{M}\subset M$ of field space,
defined by the following properties:999Normal coordinates always exist under
the conditions we assume. This is because, for any pseudo-Riemannian manifold
$(M,g)$ and any point $m\in M$, there exists a neighbourhood $U\subset M$ and
a neighbourhood $V\subset T_{m}M$ such that the map  $\exp_{m}:V\to U$ (34) is
a diffeomorphism. Using a torsion free connection (such as the Levi-Civita
connection) we can construct an isomorphism $T_{m}M\sim\mathbb{R}^{n}$
allowing us to consider the triple $\\{\exp_{m}^{-1},V,U\\}$ as a coordinate
chart, which we refer to as normal coordinates (see e.g. Petersen:2006riemann
). The chart is unique up to the choice of isomorphism.

  1. 1.

For all $1\leq i,j\leq n$ we have

$g_{ij}(m)=\bar{g}_{ij},$ (35)

which is a diagonal matrix whose entries are either $+1$ or $-1$, depending on
the signature of the metric;

  2. 2.

For all $1\leq i,j,k\leq n$ we have

$\Gamma_{ij}^{k}(m)=0;$ (36)

  3. 3.

For all $1\leq i,j,k\leq n$ we have

$\partial_{k}g_{ij}(m)=0.$ (37)

Using this system of normal coordinates, the metric tensor $g_{ij}$ admits the
following expansion around point $m\in\mathcal{F}_{M}$

$g=g_{ij}(u)du^{i}\otimes
du^{j}=\bigg{(}\bar{g}_{ij}+\frac{1}{3}R_{iklj}(m)u^{k}u^{l}+\frac{1}{6}R_{iklj;r}(m)u^{k}u^{l}u^{r}+\dots\bigg{)}du^{i}\otimes
du^{j}$ (38)

where the normalization of the first term in the expansion naturally leads to
a canonical kinetic term once we fix the signature of the metric.

The choice of normal coordinates is useful because the expansion then
naturally organizes itself based on the number of fields, thus allowing for an
easy relation to be established between components of the Riemann tensor and
$n$-point amplitudes Cohen:2021ucp ; Nagai2019:nc :

  * •

$\delta_{ij}du^{i}\otimes du^{i}$ enters in the propagator;

  * •

$\frac{1}{3}R_{iklj}(m)u^{k}u^{l}du^{i}\otimes du^{j}$ enters in four point
amplitudes;

  * •

$\frac{1}{6}R_{iklj;r}(m)u^{k}u^{l}u^{r}du^{i}\otimes du^{j}$ enters in five
point amplitudes.

In this field space picture, a potential (0-derivative) contribution to the
Lagrangian is added to the Lagrangian by hand (see §2.2), independently of the
field space geometry. Choosing coordinates $u^{i}$ on $\mathcal{F}_{M}$ such
that $u^{i}\circ m_{\text{vac}}=\vec{0}$, where $m_{\text{vac}}\in M$ is the
vacuum point, we have

$\displaystyle\mathcal{L}$
$\displaystyle=\bigg{(}\delta_{ij}+\frac{1}{3}R_{iklj}(0)\phi^{k}\phi^{l}+\frac{1}{6}R_{iklj;r}(0)\phi^{k}\phi^{l}\phi^{r}+\dots\bigg{)}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}$
(39)
$\displaystyle-\bigg{(}\frac{1}{2}\frac{\partial^{2}V}{\partial\phi^{k}\partial\phi^{l}}(0)\phi^{k}\phi^{l}+\frac{1}{6}\frac{\partial^{3}V}{\partial\phi^{k}\partial\phi^{l}\partial\phi^{r}}(0)\phi^{k}\phi^{l}\phi^{r}+\dots\bigg{)}$

It can be seen that the contribution to the three point amplitude at this
point comes entirely from the potential as a result of the vanishing of the
Christoffel symbols Cohen:2021ucp . When we eventually pass to the 1-jet
bundle, we will see (§8) how to adapt this story to incorporate higher-
derivative terms (and also how, in the jet bundle picture, the potential
contribution also comes from components of the metric).

##  3 From Field Space to Bundles

Before we get into the main business of this paper, namely our consideration
of EFT terms with $>2$ derivatives, we find it helpful to first generalise
slightly the above account of EFTs in terms of field space geometry. The
generalisation, which is motivated by locality, is to replace the description
in terms of maps (2) to a field space $M$ by one in terms of fibre bundles,
which is almost equivalent but a bit more general.101010JD is grateful to Ben
Gripaios and Joseph Tooby-Smith for past discussion and correspondence related
to ideas in this Section. See also Ref. Gripaios:2021hsc . While the
generalisation to bundles might seem like a technicality here, we will
describe several advantages over the field space picture in this Section;
moreover, the bundle picture provides the starting point for passing to jet
bundles when we turn to describe higher-derivative Lagrangians.

###  3.1 Fields as sections of fibre bundles

The idea is simple, but first let us recap some basic concepts. Recall that a
fibre bundle is a triple $(E,\Sigma,\pi)$, where

  * •

$E$ is called the total space of the bundle;

  * •

$\Sigma$ is called the base space. Both $E$ and $\Sigma$ are smooth manifolds;

  * •

The projection map $\pi:E\to\Sigma$ is a surjective submersion;

  * •

Local trivializations: for any point $x\in\Sigma$ there exists an open
neighbourhood $\mathcal{U}_{x}$ and a diffeomorphism
$\varphi:\pi^{-1}(\mathcal{U}_{x})\to\mathcal{U}_{x}\times F$, where $F$ is
itself a manifold called the ‘fibre’, such that the projection $\pi$ agrees
with projection from $\mathcal{U}_{x}\times F$ onto its first factor. In other
words, locally the total space $E$ looks like a direct product space, but this
need not be true globally.

A section of the fibre bundle is then a smooth map $\phi:\Sigma\to E$ such
that $\pi\circ\phi=\text{id}_{\Sigma}$. A section is like an ‘inverse’ of the
projection map $\pi$, taking you from the base space into the total space of
the bundle, while preserving the base space point. Identifying the base
$\Sigma$ with spacetime, what a physicist calls a ‘field’ is thus a section of
some bundle over $\Sigma$. Sections will moreover play a starring role in
defining jet bundles. We denote the set of all sections of $\pi$ by
$\Gamma(\pi)$, and the set of local sections whose domains include a point
$x\in\Sigma$ by $\Gamma_{x}(\pi)$. With these definitions, let us return to
scalar field theory on target space $M$.

The sigma model map $\phi(x):\Sigma\to M$ that we encountered before trivially
defines a section $\phi$ of a particular fibre bundle $(\Sigma\times
M,\Sigma,\pi)$ with fibre $F=M$ coinciding with the ‘field space’ of the
previous picture, and where the projection is trivially $\pi(x,m)=x$ $\forall
m\in M$, $x\in\Sigma$. This bundle can moreover be equipped with a product
metric $\eta\oplus g$, given the metrics $\eta$ and $g$ on $\Sigma$ and $M$
respectively (7, 8). Such a bundle that is globally a direct product is called
a trivial bundle, for which $\phi$ is a global section.

#### Locality

This field space formulation described in §2 implicitly assumes that $\phi$ is
a global section of a trivial bundle; but this restriction is unecessarily
strong. Given that we, as local physicists exploring our patch of the
Universe, can only know about the structure of field space on some open set
$\mathcal{U}_{\Sigma}\subset\Sigma$, there is no reason to enforce that the
field space bundle, which locally has a product structure
$\cong\mathcal{U}_{\Sigma}\times M$, has a direct product structure
$=\Sigma\times M$ globally. More generally, therefore, one can (and should)
allow fibre bundles $(E,\Sigma,\pi)$ with fibre $F=M$ that are topologically
non-trivial.

We should therefore pass from a description in which the field $\phi(x)$ is a
map from $\Sigma$ to $E$ to a slightly more general picture, in
which:111111This viewpoint is emphasized in Ref. Gripaios:2021hsc , which
adopts an even more general setup, again motivated by locality arguments,
whereby the fibre bundle is replaced by a fibred manifold. For a fibred
manifold, the fibres $\pi^{-1}(x)$ above each spacetime point $x$, which the
reader can think of as the ‘field space’ associated to the point $x$, are not
even required to be diffeomorphic to one another. We are happy to limit
ourselves to plain old fibre bundles in the present work.

$\text{A scalar field $\phi\in\Gamma_{x}(\pi)$ is a local section of a fibre
bundle $(E,\Sigma,\pi)$ with fibre $M$}.$ (40)

Given a patch $\mathcal{U}_{\Sigma}\subset\Sigma$ with local coordinates
$x^{\mu}$, one can introduce a local fibred coordinate system
$(x^{\mu},u^{i})$ on $\pi^{-1}(\mathcal{U}_{\Sigma})\subset E$, that matches
our previous choice. To guide the reader, we illustrate the basic ‘local
product’ structure of a fibre bundle and the notion of a section in Fig. 3
(left), for a toy example in which the total space $E$ has dimension two.

#### Lower derivative Lagrangian terms from geometry

In these coordinates, and starting from metrics $\eta$ and $g$ on $\Sigma$ and
$M$ respectively (in the field space picture), a natural choice of geometry on
$E$ is to take the metric on $E$ to be locally the product metric,
$g_{E}=\eta\oplus g$. Pulling $g_{E}$ back to $\Sigma$ along a section $\phi$,
and contracting with the inverse metric $\eta^{-1}$ on $\Sigma$, via
essentially the same formula as (6), we get the same 2-derivative Lagrangian
(9) that we obtained from the metric $g$ in the ‘field space maps’ picture of
§2, up to a constant shift coming from pulling back the
$\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ part of $g_{E}$.

Even though this geometry is a natural choice, one can consider more general
geometries on the field space bundle; in particular, consider a metric on the
field space bundle $E$ of the form

$g_{E}=-\frac{2}{d}V(u)\eta_{\mu\nu}\,dx^{\mu}\otimes
dx^{\nu}+g_{ij}(u)du^{i}\otimes du^{j}\,,$ (41)

where $d$ is the spacetime dimension (say, $d=4$ for the Higgs EFTs). Upon
pulling this metric back to spacetime $(\Sigma,\eta)$, which we emphasize is
still equipped with the flat geometry of Minkowski space, and contracting with
the inverse spacetime metric, we get the Lagrangian

${\cal
L}[\phi,g_{E}]=\frac{1}{2}\,\langle\eta^{-1},\,\phi^{\ast}(g_{E})\rangle=\frac{1}{2}g_{ij}(\phi)\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}-V(\phi)\,.$
(42)

We thus recover the most general scalar EFT Lagrangian with up to
2-derivatives, but including also the 0-derivative potential terms, by passing
from geometry on a target space $M$ to a geometry on the fibre bundle $E$. As
an example, this is already enough to capture the whole (renormalisable) Higgs
sector of the SM Lagrangian via geometry. We will revisit this feature when we
pass to jet bundles later in the paper.

Figure 3: Left: Illustration of a fibre bundle $(E,\Sigma,\pi)$ with fibre
$M$, which here we take to be a trivial bundle ($E\cong\Sigma\times M$) for
ease of illustration, on which we have a fibred coordinate system $(x,u)$.
This toy example describes a single real scalar (e.g. take $M=\mathbb{R}$) in
1d spacetime (e.g. take $\Sigma=\mathbb{R}$). The field $\phi$ is a _section_
of such a fibre bundle, which in general need not be a trivial bundle. Right:
bundle morphisms are pairs of maps $(f,\bar{f})$ that generically move points
on $E$ and $\Sigma$ respectively, in a way that is compatible with the bundle
projection maps. A field redefinition is a change of section. A non-derivative
field redefinition is equivalent to a bundle morphism with $\bar{f}={\rm
id}_{\Sigma}$, as indicated in our sketch by the map $f_{E}$, for which the
arrows are directed ‘vertically’.

#### Symmetries on the field space bundle

For the case of SMEFT or HEFT, in this fibred coordinate system
$(x^{\mu},u^{i})$ there is also a natural implementation of the symmetries
described in §2 at the level of field space bundles $(E,\Sigma,\pi)$. The
group action on the bundle is, at least locally, simply

$(a^{\mu},L^{\mu}_{\nu},O;\,x^{\mu},u^{i})\mapsto(L^{\mu}_{\nu}x^{\nu}+a^{\mu},\sigma(O,u^{i}))\,,$
(43)

where one can insert the group action $\sigma=\sigma_{\text{S}}$ or
$\sigma_{\text{H}}$ for SMEFT or HEFT respectively. It is, however,
instructive to still think of the spacetime and internal symmetries as two
distinct group actions, both in this case of Higgs EFTs and more generally. As
we will see in §3.3.1, we can think of these symmetries as bundle morphisms,
and the identification of a symmetry as ‘internal’ can be given a natural
definition in these terms.

###  3.2 Field redefinitions are changes of section

In addition to the motivation by locality that we have described, plus the
straightforward inclusion of lower-derivative Lagrangian terms, passing to
this ‘field space bundle’ formulation of scalar EFTs affords a technical
benefit when it comes to describing field redefinitions geometrically. Recall
from §2.3 that, in the ‘maps to field space’ picture, a non-derivative field
redefinition (or change of map $\phi^{i}\mapsto\psi^{i}(\phi^{j})$) is
equivalent to doing a local diffeomorphism on the target space.

Regarding a scalar field to instead be a section $\phi$ of the bundle
$(E,\Sigma,\pi)$, a general field redefinition will be implemented as a smooth
change of (local) section

$\Gamma_{x}(\pi)\ni\phi\mapsto\psi\in\Gamma_{x}(\pi)$ (44)

This is itself a geometric notion of what it means to do a general field
redefinition. A simple example of this is given in §3.3.4 below. For
derivative field redefinitions, however, we are not guaranteed an equivalent
description in terms of morphisms of the target space (or bundle), as we study
more carefully in the next Section.

###  3.3 Morphisms on the field space bundle

In this ‘field space bundle’ picture, a more general class of field
redefinitions can be described in terms of morphisms of the bundle, than we
saw in the case of field space smooth maps, above. To unpack this statement
requires a more-or-less formal definition of bundle morphisms, which also
provide a natural language for describing e.g. symmetries of the theory.

Suppose we have a fibre bundle $\pi:E\to\Sigma$, in the notation of §3, and
$\phi$ is a (local) section of that bundle. Given another bundle
$\rho:F\to\Omega$, a bundle morphism is a pair of maps $f:E\to F$ and
$\bar{f}:\Sigma\to\Omega$ such that

${E}$${F}$${\Sigma}$${\Omega}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$$\scriptstyle{f}$$\scriptstyle{\bar{f}}$
(45)

is a commutative diagram. (We could, more generally, consider only a local
bundle morphism, whereby every element in the commutative diagram is replaced
by open submanifolds, but such generality will not add much to the
discussion.)

To describe both symmetries and field redefinitions, we are interested in
fixing both bundles to be the same, $(E,\Sigma,\pi)$. Fig. 3 (right)
illustrates how a bundle morphism acts in this case, in a 1d example. We are
also sometimes interested in fixing the map from the base space to itself to
be

$\bar{f}=\text{id}_{\Sigma}\,.$ (46)

(The obvious exception to this will be when we discuss spacetime symmetries
below.) The map $f_{E}$ along the top is an automorphism from the bundle
$(E,\Sigma,\pi)$ to itself, $f_{E}:E\to E$. The commutative diagram becomes

${E}$${E}$${\Sigma}$$\scriptstyle{f_{E}}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$
(47)

We thus have the relation of maps

$\pi=\rho\circ f_{E}\,.$ (48)

This, which follows from assumption (46), means that $f_{E}$ does not move
points in the base space, but only moves points in the fibre. In our local
fibred coordinate system $(x^{\mu},u^{i})$ on $E$, we have

$f_{E}:(x^{\mu},u^{i})\mapsto(x^{\mu},f^{i}(x^{\mu},u^{j}))\,.$ (49)

The map $f^{i}(x^{\mu},u^{j})$ on the fibre coordinates can be regarded as a
family of diffeomorphisms on fibres $\\{M_{x}\\}$, for each value of the base
point $x$, that is free to vary (smoothly) with $x$.

A pair of local sections $\phi\in\Gamma_{x}(\pi)$ and
$\psi\in\Gamma_{x}(\rho)$ for these two bundles will also be related by the
bundle morphism $f_{E}$:

$\psi=f_{E}\circ\phi\,.$ (50)

In our fixed local fibred coordinate system $(x^{\mu},u^{i})$, Eq. (50)
implies

$\psi^{i}(x)=f^{i}(x,\phi^{j}(x))\,.$ (51)

This is clearly a generalisation of (22) because now the functions $f_{x}^{i}$
can vary continuously (and smoothly) with the base point $x$. However, it is
perhaps not an especially useful generalisation, given our primary interest is
in describing Poincaré-invariant quantum field theory actions – the more
general change of section expressed in (51) allows for field redefinitions
that have explicit spacetime dependence, which typically violate Poincaré
symmetry. Next, we discuss how this notion of bundle morphisms applies first
to symmetries, and then turn to field redefinitions.

####  3.3.1 Internal vs spacetime symmetries

We already offered a description of symmetries in the ‘field space bundle’
formulation of Higgs EFTs, in terms of a group action by $\text{Poinc}\times
O(4)$ on the bundle $E$. Generally, a group action of Lie group $G$ on a
manifold $E$ is equivalent to specifying a diffeomorphism of $E$ for each
group element $g\in G$. A symmetry that can be described via such a group
action can thus be naturally interpreted in terms of bundle morphisms;
moreover, the bundle structure offers an obvious notion of what we mean by
saying that a symmetry is ‘internal’ or otherwise.

In this language, we might say rather abstractly that a symmetry is a
particular bundle morphism $(f,\bar{f})$ such that the Lagrangian ${\cal
L}=(1/2)\langle\eta^{-1},\phi^{\ast}(g)\rangle$ shifts by at most a total
derivative:121212We also implicitly assume an invariance condition
Davighi:2018inx  is satisfied for the topological term, if present.

$\langle\eta^{-1},\phi^{\ast}(f^{\ast}-1)g\rangle=\partial_{\mu}K^{\mu}\,.$
(52)

Such a broad definition would account for the fact that (i) there can be many
redundancies in the map from metric to Lagrangian (as will become more obvious
when we pass to 1-jet bundles, especially in the examples of §6 and §7), and
(ii) we wish to identify Lagrangians that differ by total derivatives. Both
these facts mean that requiring invariance of the metric itself, viz.

$f^{\ast}g=g\,,$ (53)

is too strong a condition for a symmetry. Nonetheless, the more general
condition (52) is rather nasty to implement in practice – indeed, it is not
even obvious why the set of morphisms satisfying (52) should form a
group,131313Moreover, a definition of ‘symmetry’ like Eq. (52) is still far
from being comprehensive; indeed, finding all symmetries can be an almost
overwhelming task even for very simple theories, if a symmetry is broadly
defined to be any ‘thing that does not change observables’. This would include
transformations that act completely trivially, such as the change of
coordinates considered in §2.3.1 above as well as more general redundancies of
description such as gauge symmetries, right through to the huge symmetry under
field redefinitions that is not seen at the Lagrangian level, but only after
doing the path integral. The symmetries we consider here, namely symmetries of
the metric on $E$ that can be captured by bundle morphisms, lie somewhere
between these two extremes. Finally, we remark that in recent years yet
further ‘generalised’ notions of symmetry have been appreciated, which are not
seen by their action on any local operators but by their action on extended
objects like Wilson lines (see Bah:2022wot ; Cordova:2022ruw  for recent
surveys).  whereas (diffeo)morphisms satisfying (53) clearly do form a group.
In practice, the particular symmetries that we consider will be implemented by
bundle morphisms that are symmetries of the metric (isometries), satisfying
(53), so we are happy to proceed with this definition of (a certain class of)
symmetry.

Continuing, an internal symmetry is then a symmetry that is described by a
‘base-point preserving’ bundle morphism $(f,\bar{f})$ of type (47), namely for
which $\bar{f}=\text{id}_{\Sigma}$, which means the group action does not move
points in spacetime. More precisely, it moves points in a given fibre $M_{x}$
to other points in that same fibre $M_{x}$. In the case of SMEFT, say, the
internal symmetry is unsurprisingly the $O(4)$ group action:

$\sigma_{\text{S}}:O(4)\times E\to
E:(O,x^{\mu},u^{i})\mapsto(x^{\mu},Ou^{i}),$ (54)

in our local fibred coordinates. This defines a set of base-point preserving
bundle morphisms satisfying both the commutative diagram (47) and the group
multiplication law. In general, an internal symmetry described by a Lie group
action by $G$ on the bundle $E$ corresponds to the set

$\\{(f_{g},\bar{f}=\text{id}_{\Sigma})\text{~{}a bundle morphism}\,\,\forall
g\in G\,\,|\,\,f_{g_{1}}\circ f_{g_{2}}=f_{g_{1}g_{2}}\,\,\forall
g_{1},\,g_{2}\in G\\}\,.$ (55)

In our SMEFT example, the morphisms are

$f_{O}:(x^{\mu},u^{i})\to(x^{\mu},Ou^{i})\,\,\forall O\in O(4)\,.$ (56)

On the other hand, a spacetime symmetry is a map $\bar{f}$ such that
$(f,\bar{f})$ is a bundle morphism that is a symmetry. In our HEFT and SMEFT
examples, which have $(3+1)$-d Poincaré invariance, we can take this spacetime
symmetry to be defined by the group action on the bundle

$\rho:\text{Poinc}\times E\to
E:(a^{\mu},L^{\mu}_{\nu},x^{\mu},u^{i})\mapsto(L^{\mu}_{\nu}x^{\nu}+a^{\mu},u^{i})\,.$
(57)

Again this defines a collection of bundle morphisms, where the important thing
is the map on the base of the bundle, $\bar{f}_{a,L}:x^{\mu}\mapsto
L^{\mu}_{\nu}x^{\nu}+a^{\mu}$ for every $a^{\mu}$ and $L^{\mu}_{\nu}$.

It is interesting that, as a collection of bundle morphisms, the internal
symmetry is naturally implemented locally. The general formula for an
‘internal morphism’ in the bundle picture is
$\sigma:(x^{\mu},u^{i})\mapsto(x^{\mu},O(x).u^{i})$, where $O(x)$ is an
$O(4)$-valued function on spacetime. This suggests there should be a natural
notion of gauging in this language (indeed, gauge fields are perhaps the most
familiar example of quantum fields being sections on bundles over spacetime).
We save a proper treatment of the gauging of internal symmetries, which is of
course a crucial element in the electroweak theory, for future work.

####  3.3.2 Non-derivative field redefinitions (again)

Having discussed symmetries, we now return to the issue of field
redefinitions, and ask when such transformations can be described by the
bundle morphisms defined in §3.3 above.

Firstly, if we take the maps $f^{i}(x^{\mu},u^{j})$ in e.g. Eq. (49) to be
independent of $x$, then we have that $u^{i}\circ\psi=u^{i}\circ f(\phi)$,
equivalent to doing the same ‘field space diffeomorphism’ everywhere in
spacetime, i.e. at every fibre. This special case of bundle morphisms can
describe any non-derivative field redefinition, as considered in §2.3.2.
Later, in §5.5.3 and §5.5.4, we will see how this is implemented in the more
general jet bundle context.

####  3.3.3 More general field redefinitions?

Consider again the general (base-point preserving) bundle morphism described
by Eq. (51). We now examine, for completeness, what kind of field
redefinitions can be captured by such bundle morphisms, and whether we get
‘anything more’ than non-derivative field redefinitions. Recall that we
already have a geometric description of arbitrary derivative field
redefinitions as arbitrary changes of section $\phi\to\psi$. We see an
explicit example of this below in §3.3.4.

To study this, let us start from the more general class of Lagrangian captured
by our field space bundle geometry, including 0-derivative (potential) terms.
We take

$g_{E}=-\frac{2}{d}V(u)\eta_{\mu\nu}\,dx^{\mu}\otimes
dx^{\nu}+g_{ij}(u)du^{i}\otimes du^{j}\,$ (58)

as in (41), which, pulling back along a section $\phi:\Sigma\to E$, gives the
general 2-derivative Lagrangian ${\cal
L}[\phi,g_{E}]=\frac{1}{2}g_{ij}(\phi)\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}-V(\phi)$.
If we do an ‘internal diffeomorphism’ on the bundle of the kind (47), which
acts as $f_{E}:(x^{\mu},u^{i})\to(x^{\mu},f^{i}(x,u))$ in our local fibred
coordinate system, then the metric $g_{E}$ transforms as

$\displaystyle g_{E}\to f_{E}^{\ast}g_{E}$
$\displaystyle=\,\,-\frac{2}{d}V(u\circ
f_{E})\eta_{\mu\nu}\,dx^{\mu}dx^{\nu}+g_{ij}(u\circ
f_{E})\frac{\partial(u^{i}\circ f_{E})}{\partial
u^{k}}\bigg{|}_{x,u}\frac{\partial(u^{j}\circ f_{E})}{\partial
u^{m}}\bigg{|}_{x,u}du^{k}du^{m}$ $\displaystyle\qquad\qquad+g_{ij}(u\circ
f_{E})\frac{\partial(u^{i}\circ f_{E})}{\partial
x^{\mu}}\bigg{|}_{x,u}\frac{\partial(u^{j}\circ f_{E})}{\partial
x^{\nu}}\bigg{|}_{x,u}dx^{\mu}dx^{\nu}$ (59)

Upon pulling back along the (same) section $\phi$ and forming the new
Lagrangian, we get

$\displaystyle{\cal L}[\phi,g_{E}]\mapsto{\cal L}[\phi,f^{\ast}_{E}g_{E}]$
(60)
$\displaystyle=\frac{1}{2}g_{ij}(f(x,\phi))\left[\partial_{\mu}f^{i}\bigg{|}_{x,u=\phi}\partial^{\mu}f^{j}\bigg{|}_{x,u=\phi}+\frac{\partial
f^{i}}{\partial u^{k}}\bigg{|}_{x,u=\phi}\frac{\partial f^{j}}{\partial
u^{m}}\bigg{|}_{x,u=\phi}\partial_{\mu}\phi^{k}\partial^{\mu}\phi^{m}\right]-V(f(x,\phi))$

While formulae (3.3.3) and (60) may look elaborate, they are essentially the
result of applying the chain rule.

The result (60) generalises a little the effect of ‘field space smooth maps’
considered in §2. There are two main differences: first, the possible
dependence of the transformed $g_{ij}$ and $V$ functions on spacetime
coordinates $x^{\mu}$; second, the first term on the RHS which comes from
partial differentiation of the functions $f^{i}$ with respect to spacetime. If
we restrict to bundle morphisms that respect Poincaré symmetry, then neither
of these generalisations are relevant.

If we further assume the morphism is a ‘perturbation’ around the trivial
morphism, i.e. that it takes the form

$f^{i}(x^{\mu},u^{j})=u^{i}+\epsilon\lambda^{i}(x^{\mu},u^{j})\,,\qquad\epsilon\ll
1\,,$ (61)

then we can expand the various functions $g_{ij}(f)$ and $V(f)$ around their
‘old values’. For the potential, for example, we can do an ordinary Taylor
expansion

$V(u)\mapsto V(f(x,u))=V(u)+\epsilon\lambda^{i}\frac{\partial V}{\partial
u^{i}}+\frac{1}{2}\epsilon^{2}\lambda^{i}\lambda^{j}\frac{\partial^{2}V}{\partial
u^{i}\partial u^{j}}+\dots\,,$ (62)

while the various Jacobian factors can also be expanded

$\frac{\partial f^{i}}{\partial
x^{\mu}}=\epsilon\frac{\partial\lambda^{i}}{\partial
x^{\mu}},\qquad\frac{\partial f^{i}}{\partial
u^{j}}=\delta^{i}_{j}+\epsilon\frac{\partial\lambda^{i}}{\partial u^{j}}\,,$
(63)

before substituting into (60).

One might wonder whether such a base-point preserving bundle morphism of this
kind can ever capture the effects of a derivative field redefinition, which is
a change of section $\phi\to\psi$ such that $u\circ\psi$ is a function of
$u\circ\phi$ and its spacetime derivatives. The first point to note is that,
starting from the 2-derivative Lagrangian ${\cal L}[\phi,g_{E}]$ above, the
transformation (60) cannot generate terms with more than 2-derivatives. So, we
straightaway learn that this kind of bundle morphism cannot generically
capture a derivative field redefinition; at best, one might be able to
reproduce the effects of a derivative field redefinition if we truncate the
transformed Lagrangian also at two derivatives. We next consider a concrete
example, for which such a diffeomorphism can capture the effects of a
derivative field redefinitions – but we stress that this example is far from
generic.

####  3.3.4 Example: $\phi^{4}$ theory

Consider a 4d theory of a single real scalar field. In the bundle formulation,
we take a trivial bundle
$E=\mathbb{R}^{1,3}\times\mathbb{R}_{u}\xrightarrow{\pi}\mathbb{R}^{1,3}$, on
which we specify an initial fibred coordinate system $(x^{\mu},u)$. We take
the metric

$g_{E}=-\frac{1}{2}\frac{y\,u^{4}}{4!}\eta_{\mu\nu}dx^{\mu}\,dx^{\nu}+du\,du,\qquad
y\in\mathbb{R}$ (64)

which is flat in the field space coordinate $u$, corresponding to a canonical
2-derivative kinetic term. The real parameter $y$ is a coupling constant.
Pulling back along a section $\phi:x^{\mu}\to(x^{\mu},\phi(x))$, our initial
Lagrangian is simply

${\cal
L}[\phi]=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{y}{4!}\phi^{4}\,,$
(65)

that of ‘$\phi^{4}$ theory’ in 4d, with potential $V(\phi)=(y/4!)\phi^{4}$.

Now, consider a derivative field redefinition that is defined by a change of
section $\phi\to\psi$ where, in our coordinate system,141414We choose a field
redefinition that is consistent with a $\text{Poinc}\times
O(1)_{u}\cong\text{Poinc}\times\mathbb{Z}_{2}$ symmetry. The field
redefinition is consequently second-order in derivatives and ‘preserves the
number of fields’. This kind of field redefinition, which shifts $\phi$ by
‘$\Box\phi$’, can be used to eliminate redundancies due to the equations of
motion (EOMs), and will play an important role in the examples of higher-
derivative Lagrangians that we discuss in §6.

$\displaystyle x^{\mu}\circ\psi=x^{\mu}\,,\qquad
u\circ\psi=\left(1-\epsilon\partial_{\mu}\partial^{\mu}\right)(u\circ\phi)\,,$
(66)

where $1\gg\epsilon\in\mathbb{R}$. The first condition ensures the change of
section only acts ‘internally’ on the fibres, as is our assumption throughout.
Abusing notation as before and letting $\psi$ ($\phi$) also denote the
component values of the section $\psi$ ($\phi$) in the local fibre coordinate,
and letting $\Box:=\partial_{\mu}\partial^{\mu}$ as usual, we would write this
(in notation more familiar to the physicist) as

$\phi\mapsto\psi=\phi-\epsilon\Box\phi\,.$ (67)

Pulling back the metric $g$ along $\psi$ rather than $\phi$, we obtain the
transformed Lagrangian,

$\displaystyle{\cal L}[\psi,g]$
$\displaystyle=\frac{1}{2}\eta^{\rho\sigma}\langle\partial_{\rho}\otimes\partial_{\sigma},\,dx^{\mu}\otimes
dx^{\nu}\rangle\partial_{\mu}(u\circ\psi)\partial_{\nu}(u\circ\psi)-V(u\circ\psi)$
$\displaystyle=\frac{1}{2}\partial_{\mu}(\phi-\epsilon\Box\phi)\partial^{\mu}(\phi-\epsilon\Box\phi)-V(\phi-\epsilon\Box\phi)$
(68)
$\displaystyle=\frac{1}{2}(\partial\phi)^{2}-\epsilon\partial_{\mu}\Box\phi\partial^{\mu}\phi+\frac{1}{2}\epsilon^{2}\partial_{\mu}\Box\phi\partial^{\mu}\Box\phi-V+\epsilon\Box\phi\frac{dV}{d\phi}+\frac{1}{2}\epsilon^{2}(\Box\phi)^{2}\frac{d^{2}V}{d\phi^{2}}+\mathcal{O}(\epsilon^{3})$

exactly as one would expect from a derivative field redefinition, where we
have here kept the potential $V(\phi)$ general. We stress that there is no
obstacle to considering derivative field redefinitions in the geometric
picture, once we consider a field redefinition to be a change of section
(along which the unchanged metric is pulled back to form the Lagrangian).

Knowing that our bundle geometry construction can only capture Lagrangians
with up to 2-derivatives, it is instructive to also expand this only up to
2-derivative terms:

$\displaystyle{\cal L}[\phi,g_{E}]\mapsto{\cal L}[\psi,g_{E}]$
$\displaystyle=\frac{1}{2}(\partial\phi)^{2}\left(1-2\epsilon
V^{\prime\prime}(\phi)\right)-V(\phi)+\mathcal{O}(4\partial)\,$
$\displaystyle=\frac{1}{2}(\partial\phi)^{2}\left(1-\epsilon
y\phi^{2}\right)-\frac{y}{4!}\phi^{4}+\mathcal{O}(4\partial)\,,$ (69)

where we have also used IBPs, and the relation

$\partial_{\mu}F(\phi)=\partial_{\mu}\phi\frac{dF}{d\phi}\,.$ (70)

The result, truncating to 2-derivative order, is a $\phi$-dependent shift of
the kinetic term.

Truncated to this order, the effect of doing this derivative field
redefinition can be realised instead via a diffeomorphism on the bundle $E$.
Because the field redefinition we seek to replicate is a perturbation around
doing nothing, we assume the bundle morphism is also a perturbation around the
trivial morphism, of the form (61). By Poincaré invariance we also assume the
bundle morphism has no explicit dependence on $x^{\mu}$, so we take

$(x^{\mu},u)\mapsto(x^{\mu},u+\epsilon\lambda(u))\,.$ (71)

Subbing into (60), we get

$\displaystyle{\cal L}[\phi,g_{E}]\mapsto{\cal
L}[\phi,f_{E}^{\ast}g_{E}]=\frac{1}{2}(\partial\phi)^{2}\left(1+2\epsilon\lambda^{\prime}\right)-V-\epsilon\lambda
V^{\prime}+\mathcal{O}(\epsilon^{2})\,.$ (72)

Thus, in this example, we can reproduce the effects (3.3.4) of the derivative
field redefinition, truncated to operators with 2 derivatives and up to 4
field insertions, by choosing the function $\lambda=-y\phi^{3}/6$.

This is far from a generic example; for more general (2-derivative)
Lagrangians, derivative field redefinitions cannot be described by base-point
preserving bundle morphisms of the kind we have described – but they can
always be described as changes of section of the bundle. A relevant example
here was discussed in Appendix E of Cohen:2020xca ; there, starting from a
theory with zero Ricci scalar on the field space, a derivative field
redefinition in the EFT was performed (motivated by a simple non-derivative
rotation in the UV), that was found to introduce a singularity in the Ricci
scalar; this could not occur if the metric is pushed forward under a
diffeomorphism (since a diffeomorphism must act invertibly on tensors).

####  3.3.5 A limiting lemma

It is likely not even possible to remedy this by allowing for general bundle
morphisms $(f,\bar{f})$, as described by (47), which have a non-trivial
morphism on the base manifold $\Sigma$. That is, allowing totally general
bundle morphisms (even, in desperation, allowing for generic Poincaré
violation) is still not enough to fully capture any old ‘derivative change of
section’. A big clue to this is the following lemma, which is admittedly
proven for the case of vector bundles (saunders1989geometry , Lemma 2.2.9):

###### Lemma 3.1

If $(E,\pi,\Sigma)$ and $(H,\rho,\Sigma^{\prime})$ are vector bundles and
$(f,\bar{f})$: $(E,\pi,\Sigma)\to(H,\rho,\Sigma^{\prime})$ is a vector bundle
morphism such that $\bar{f}$ is a diffeomorphism, then
$\tilde{f}:\Gamma((E,\pi,\Sigma))\to\Gamma((H,\rho,\Sigma^{\prime}))$ defined
by $\tilde{f}(\phi)=f\circ\phi\circ\bar{f}^{-1}$ is a module homomorphism over
the ring isomorphism $\bar{f}^{-1*}:C^{\infty}(\Sigma)\to
C^{\infty}(\Sigma^{\prime})$

Crucially, a ‘derivative change of section’ such as
$\phi(x)\mapsto\psi=\partial_{\mu}\phi(x)$ (abusing notation in the now-
familiar way), or indeed the specific change of section (66) considered in the
previous example, is not a module homomorphism over the ring of smooth
functions, because differentiation doesn’t commute with multiplication. The
lemma then implies it is not possible to find any bundle morphism
$(f,\bar{f})$, where $\bar{f}$ is a diffeomorphism, such that
$\phi(x)\mapsto\partial_{\mu}\phi(x)$, since that would require the map to be
a homomorphism.

In this and the previous Section we have carefully reviewed and, we hope,
refined the geometric picture of the 2-derivative term in scalar EFTs, and how
one can think about field redefinitions geometrically as simply a change of
section. In doing so we have laid the groundwork for extending this geometric
approach to higher derivative terms. The key objects, mathematically, will be
jet bundles of the original field space bundle $E$, and metrics that we will
define thereon. Since jet bundles might be unfamiliar territory to most
effective field theorists, in the next Section we review the concept and
necessary definitions. Our main reference is the textbook of Saunders
saunders1989geometry .

##  4 Jet Bundles

Starting from the ‘field space bundle’ $(E,\Sigma,\pi)$, one can construct a
sequence of associated manifolds of higher dimension, all themselves fibres
bundles, called jet bundles. These jet bundles are indexed by an integer;
there is a ‘1-jet bundle’ denoted $J^{1}E$, a ‘2-jet bundle’ $J^{2}E$, and so
on; the ‘0-jet bundle’ is just $E$ itself. For our humble physics purposes,
$r$-jet bundles with higher $r$ will be needed to describe EFTs with more and
more derivatives; the 1-jet bundle shall suffice to describe all terms (at
least in SMEFT and HEFT) with up to four derivatives.151515The 1-jet bundle
also has well-known mathematical physics applications in formal treatments of
Lagrangian mechanics. See e.g. saunders1989geometry ; de1998lie ;
Sardanashvily:2009br .

Jets are built up from equivalence classes of sections, as follows. Starting
from a fibre bundle $(E,\Sigma,\pi)$, with a local fibred coordinate system
$(x^{\mu},u^{i})$ as introduced in §3, we say that two local sections
$\phi,\psi\in\Gamma_{x}(\pi)$ are $1$-equivalent at the base point $x$ if both
their values and the values of their first derivatives agree there – but
$\phi$ and $\psi$ can disagree in their higher derivatives. Recalling our
notation $\phi^{i}(x):=(u^{i}\circ\phi)(x)$ for the ‘components’ of a section
(given local fibre coordinates $u^{i}$), then $\phi\sim\psi$ iff

$\displaystyle\phi^{i}(x)=\psi^{i}(x)\,,\qquad\text{and}\qquad\frac{\partial\phi^{i}(x)}{\partial
x^{\mu}}=\frac{\partial\psi^{i}(x)}{\partial x^{\mu}}\,.$ (73)

It is easy to show that this equivalence relation is independent of the choice
of coordinate chart on our patch of $E$. The 1-equivalence class containing a
representative section $\phi$ is called the 1-jet of $\phi$ at $x$, denoted
$j_{x}^{1}\phi$. In a similar way, two local sections are $r$-equivalent at
$x\in\Sigma$, for some $r\in\mathbb{Z}_{\geq 0}$, if their first $r$
derivatives agree at $x$; the $r$-equivalence class containing $\phi$ is the
$r$-jet of $\phi$ at $x$, denoted $j_{x}^{r}\phi$.

The 1-jet bundle $J^{1}E$ is then defined to be the set of 1-equivalence
classes of sections,

$J^{1}E=\\{j^{1}_{x}\phi\,|\,x\in\Sigma,\phi\in\Gamma_{x}(\pi)\\}\,,$ (74)

which moreover has a natural structure as a differentiable manifold.
Continuing in a similar fashion, the $r$-jet bundle $J^{r}E$ is the set of
$r$-equivalence classes of sections – or, more prosaically, is the space of
field configurations that agree in their first $r$ derivatives. The $0$-jet
bundle is defined to be $J^{0}E=E$ itself.

Focussing our attention on the 1-jet bundle $J^{1}E$, from its definition one
can define two (ordinary) fibre bundles with total space $J^{1}E$. One is a
bundle over the original base space $\Sigma$, and the other is a bundle over
the original bundle $E$:161616More generally, the $J^{r}E$ bundle can always
be modelled as a fibre bundle over base space $J^{r-1}E$, or indeed as a fibre
bundle over any $J^{q<r}E$. (There is natural inductive limit of the sequence
$\\{J^{r}E\\}$ of jet bundles, that defines a space $J^{\infty}E$
Sardanashvily:2009br . One can model $J^{\infty}E$ by sections that agree as
$C^{\infty}$ maps – we will not discuss this infinite order jet bundle here.)

$\displaystyle\pi_{1}$
$\displaystyle:J^{1}E\to\Sigma\,:\,\,\,j^{1}_{x}\phi\mapsto x\,,$ (75)
$\displaystyle\pi_{1,0}$ $\displaystyle:J^{1}E\to
E:\,\,\,j^{1}_{x}\phi\mapsto\phi(x)\,.$ (76)

These projections are sometimes referred to as the ‘source’ and ‘target’
bundles respectively, for obvious reasons. These projections are compatible
with the original bundle structure of $E$, in that they clearly fit into a
commutative diagram

${{J^{1}E}}$${E}$${\Sigma}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{1,0}}$

involving the original projection $\pi:E\to\Sigma$. Thus,
$\pi_{1}=\pi\circ\pi_{1,0}$.

### Local coordinates on the 1-jet bundle

Recall that we start from local coordinates $(x^{\mu},u^{i})$ on the fibre
bundle $E$, or more precisely on an open set $\pi^{-1}(\mathcal{U})\subset E$,
and that a particular section $\phi:\Sigma\to E$ maps a point $x\in\Sigma$ to
the point in $E$ with coordinates $(x^{\mu},\phi^{i}(x^{\mu}))$. The 1-jet
bundle $J^{1}E$ then admits a set of induced local coordinates

$(x^{\mu},u^{i},u^{i}_{\mu})\,.$ (77)

The value of these coordinates at a point $j^{1}_{x}\phi\in J^{1}E$ are

$\displaystyle x^{\mu}\circ j^{1}_{x}\phi$ $\displaystyle=x^{\mu}\,,$
(spacetime points) (78) $\displaystyle u^{i}\circ j^{1}_{x}\phi$
$\displaystyle=\phi^{i}(x)\,,$ (field values) (79) $\displaystyle
u^{i}_{\mu}\circ j^{1}_{x}\phi$
$\displaystyle=\left.\frac{\partial\phi^{i}}{\partial x^{\mu}}\right|_{x}\,$
$\displaystyle\text{(first derivatives)}.$ (80)

Thus, the extra fibre coordinates in the 1-jet bundle
$(J^{1}E,\Sigma,\pi_{1})$ encode the first derivatives of sections passing
through that base point $x$ – which is intuitive given the definition of a
point $j^{1}_{x}\phi\in J^{1}E$ as the collection of all sections passing
through $x$ with the same value and first derivative there.

One can model the fibres of the bundle $(J^{1}E,E,\pi_{1,0})$, on which
$u^{i}_{\mu}$ provide fibre coordinates, as vector spaces Sardanashvily:2009br

$\pi_{1,0}^{-1}(x\in\Sigma,m\in M_{x})\cong T^{\ast}_{x}(\Sigma)\otimes
T_{m}(M)\,.$ (81)

This structure is encoded in the placement of indices of the derivative
coordinate $u^{i}_{\mu}$; it transforms as a vector with respect to its $i$
index, and a co-vector with respect to its $\mu$ index. We will often call
these extra coordinates $u^{i}_{\mu}$ ‘derivative coordinates’.

Summarising, the induced fibre coordinates on $(J^{1}E,\Sigma,\pi_{1})$, which
evaluate on a section to $(\phi^{i},\partial_{\mu}\phi^{i})$, provide the
familiar ingredients that we use to write down Lagrangians in field theory.
The 1-jet bundle provides a coordinate-free description of this data. The
geometry of $\Sigma$, $J^{0}E=E$ and $J^{1}E$, all of which play a big role in
the EFT formalism developed in this paper, is summarized schematically in
Figure 1.

Lastly, it is easy to find the dimension of the 1-jet manifold. Starting with
a general fibre bundle $E\to\Sigma$, with $\text{Dim}\Sigma=d$ and
$\text{Dim}E-\text{Dim}\Sigma=n$ (the $\mathbb{R}$-dimension of the fibre,
equivalently the number of real scalar fields), the $\\{u^{i}_{\mu}\\}$
provide $dn$ extra coordinates. Then

$\text{Dim}(J^{1}E)=\text{Dim}E+dn=d+n+dn\,.$ (82)

Let’s count the dimensions of $J^{1}E$ for some examples:

  * •

Real scalar field in $d$ dimensions: $\text{Dim}(J^{1}E)=2d+1$ (see §6.2);

  * •

HEFT or SMEFT: $\text{Dim}(J^{1}E)=24$ (see §7).

Going further, the dimensions of the jet manifolds $J^{r}E$ quickly blow up
when $d=\text{Dim}\Sigma$ is large; in general, it is easy to see that

$\text{Dim}(J^{r}E)=d+n\frac{(d+r)!}{d!r!}\,.$ (83)

For SMEFT or HEFT, we have $(J^{2}E)=64$ for the 2-jet bundle.

##  5 Higher-derivative EFTs from Jet Bundle Geometry

The 1-jet bundle, introduced in §4, defines a bundle over spacetime with local
fibre coordinates that we identify with scalar fields and their first
derivatives. While these degrees of freedom are treated independently on the
jet bundle, we always eventually pull back objects (like metrics) along
sections, whereupon the relation between the field and its derivative is
enforced (i.e. upon pullback $u^{i}_{\mu}$ will ‘remember’ that it is a
derivative of $u^{i}$).

The key idea, from the EFT perspective, is that we can consider more general
geometric structures involving higher-derivatives of fields in this larger
space $J^{1}E$. In particular, we will see how defining different metrics on
$J^{1}E$, equivalently ‘measures of distance’, encodes different scalar EFT
Lagrangian with up to 4 spacetime derivatives acting on the fields. In this
Section we outline how this works, for a general scalar EFT. In Sections 6 and
7, we will apply this to specific examples including the Higgs EFTs, which is
our primary interest.

###  5.1 Defining geometry on the 1-jet bundle

The subject of geometry on jet bundles is not so widely explored in the
mathematical literature. There are nonetheless ‘natural metrics’ that are
induced on jet bundles $J^{r}E$, given a metric on the original bundle
$E\xrightarrow{\pi}\Sigma$ that we take to be, locally, a product metric
$\eta\oplus g$, with $\eta$ flat (ignoring any potential contribution $V(u)$
for now) and $g(u)$ possibly curved, as described in §§2.1 and 3. To give the
reader a flavour for geometry on jet bundles, we begin with a brief sketch of
such a natural geometry on $J^{1}E$,171717Contact forms provide another source
of natural geometries on jet bundles, which can also be used to define a class
of 4-derivative EFT Lagrangians using the ideas in this paper. and see how it
offers a starting point for constructing Lagrangians with 4-derivatives.

We want to define a Riemannian metric on $J^{1}E$ considered as a manifold. We
already saw that $J^{1}E$ is itself a vector bundle $(J^{1}E,E,\pi_{1,0})$
with fibre $T^{\ast}_{x}(\Sigma)\otimes T_{m}(M)$. Defining a metric on
$J^{1}E$ means equipping the tangent bundle $T(J^{1}E)$ with an inner product

$g^{(1)}:T(J^{1}E)\times_{J^{1}E}T(J^{1}E)\to\mathbb{R}$ (84)

that is symmetric and non-degenerate, and that varies smoothly with the base
coordinate $j_{x}^{1}\phi$ (in other words, it is a tensor field on $J^{1}E$).
We use a notation whereby $g^{(r)}$ denotes a metric on the $r$-jet bundle
$J^{r}E$. Now, the tangent bundle $T(J^{1}E)$ is isomorphic to the pullback
bundle181818To show (85), we need some theorems concerning the structure of
the tangent bundle of such a vector bundle. There is a short exact sequence of
smooth vector bundles on $E$, namely $0\to VE\to TE\to HE\to 0$, where the
sub-bundle $VE:=\text{ker~{}}d\pi$ is called the ‘vertical bundle’, and the
quotient bundle $HE=TE/VE$ is called the ‘horizontal bundle’. Both these
bundles are isomorphic to pullback bundles along $\pi^{\ast}$; we have
$VE\cong\pi^{\ast}E$, and $HE\cong\pi^{\ast}TY$, which gives rise to the
isomorphism (85).

$T(J^{1}E)\cong\pi^{\ast}(J^{1}E\oplus TE)\,.$ (85)

We can therefore build a (fibre-wise) inner product on $T(J^{1}E)$ (i.e. a
metric on $J^{1}E$) out of (fibre-wise) inner products on the bundles $J^{1}E$
and $TE$. The latter is just the metric on $E$, which we already stipulated is
(locally) the product metric $\eta\oplus g$. The former is not much more
complicated, given the fibre is $T^{\ast}_{x}(\Sigma)\otimes T_{\phi}(X)$; we
define the inner product via $\eta^{-1}$ and $g$. Note the inverse spacetime
metric appears here, because we are contracting co-vectors with respect to
spacetime, but vectors with respect to the target space.

Putting things together, we can express this ‘natural’ choice of metric, which
recall is the one built from the metrics $\eta$ and $g$ that we already have
at our disposal, in our local coordinate system:

$g^{(1)}=\Lambda^{4}\,\eta_{\mu\nu}\,dx^{\mu}\otimes
dx^{\nu}+g_{ij}(u)\,du^{i}\otimes
du^{j}+\frac{1}{\Lambda^{2}}\eta^{\mu\nu}g_{ij}(u)\,du^{i}_{\mu}\otimes
du^{j}_{\nu}\,.$ (86)

We have introduced appropriate factors of $\Lambda$ to give everything
consistent dimensions.

###  5.2 Our first four-derivative Lagrangian

Now, starting from this metric on the jet bundle $J^{1}E$, which we intrepret
as a ‘derivative-generalisation of field space’, one can construct a
Lagrangian using exactly the same recipe that we described for the
2-derivative Lagrangian in §2.1. The only difference is that we now need to
pull the metric $g^{(1)}$ all the way back to spacetime, and so we should use
a section $j^{1}\phi$ of the fibre bundle $(J^{1}E,\Sigma,\pi_{1})$ introduced
above.

In particular, given an original section $\phi$ of $E$ (i.e. a scalar field
configuration), let

$j^{1}\phi\in\Gamma_{x}(\pi_{1})$ (87)

be its ‘prolongation’ to the 1-jet bundle, defined as the section of $\pi_{1}$
that passes through the 1-jet $j^{1}_{x}\phi$.191919Note that not every
section of the 1-jet bundle $\pi_{1}:J^{1}E\to\Sigma$ is a prolongation of a
section $\phi$ of $\pi:E\to\Sigma$, where the latter is for us identified with
a scalar field configuration. We are only interested in sections obtained in
this way.  Evaluating on the coordinate chart, its components are simply

$\displaystyle x^{\mu}\circ j^{1}\phi=x^{\mu}\,,\qquad u^{i}\circ
j^{1}\phi=\phi^{i}(x)\,,\qquad u^{i}_{\mu}\circ j^{1}\phi$
$\displaystyle=\left.\frac{\partial\phi^{i}}{\partial x^{\mu}}\right|_{x}\,,$
(88)

from (80).

We define the Lagrangian

${\cal
L}_{4\partial}[\phi,g^{(1)}]:=\frac{1}{2}\,\left\langle\eta^{-1},\,(j^{1}\phi)^{\ast}g^{(1)}\right\rangle\,,$
(89)

which we regard as a functional of the original section $\phi$. Assuming the
system of local coordinates set out in Eqs. (77)–(80), this gives the
Lagrangian

${\cal
L}_{4\partial}[\phi,g^{(1)}]=\frac{1}{2}\eta^{\mu\nu}g_{ij}(\phi(x))\partial_{\mu}\phi^{i}\partial_{\nu}\phi^{j}+\frac{1}{2\Lambda^{2}}\eta^{\rho\sigma}\eta^{\mu\nu}g_{ij}(\phi(x))\partial_{\rho}\partial_{\mu}\phi^{i}\partial_{\sigma}\partial_{\nu}\phi^{j}\,,$
(90)

up to a cosmological constant term coming from the first term in (86) that we
can neglect when doing flat-space quantum field theory (although its natural
appearance seems intriguing nonetheless).

Let us pause to make some further comments. First and foremost, we have shown
that defining a geometry on the 1-jet bundle naturally ‘refines’ our original
2-derivative Lagrangian, as constructed from field space geometry in §2.1, by
a particular 4-derivative ‘correction’ term that is also geometric in origin.
This is already, to our knowledge, a novel result in the context of EFT.

We can try to unpack the particular Lagrangian that we have obtained, which is
clearly not yet general, in the case of the Higgs EFTs. Let us therefore take
$\Sigma$ to be flat 4d spacetime and the target space $M$ is a real 4-manifold
equipped with the $SO(4)$ group action $\sigma_{\text{S}}$ for SMEFT, say. We
can, for illustration, keep only terms in the Lagrangian (90) that have two
field insertions. This is equivalent to taking only the leading term in the
expansion of the original $\sigma_{\text{S}}$-invariant metric,

$g_{ij}(u^{i})=\delta_{ij}+\dots\,$ (91)

In this limit, and contracting the various indices, the Lagrangian (90)
becomes

${\cal
L}=\frac{1}{2}\partial_{\mu}\phi\cdot\partial^{\mu}\phi+\frac{1}{2\Lambda^{2}}\partial_{\mu}\partial_{\nu}\phi\cdot\partial^{\mu}\partial^{\nu}\phi+\mathcal{O}(\phi^{4})\,.$
(92)

Interestingly, we recover the most general SMEFT Lagrangian with up to two
fields and up to four derivatives. The SM can, as for the purely 2-derivative
case, be obtained by taking the cut-off scale $\Lambda\to\infty$, which
decouples all the non-renormalisable operators from the EFT.

###  5.3 General geometries for general Lagrangians

In the previous two Subsections, we considered a metric on $J^{1}E$ that is
‘natural’, mathematically, given the geometry we started with on the field
space bundle $E$. We found that this metric delivered a 4-derivative EFT
Lagrangian, via the ‘obvious map’ to Lagrangians specified by Eq. (89).
Unsurprisingly, if we start from such a naturally induced metric on $J^{1}E$,
this will only ever get us a 4-derivative term whose form is determined by the
2-derivative terms already present in the action. For a general EFT
description, we of course want to consider 4-derivative terms that are
independent of the 2-derivative terms. This suggests that our geometry on the
1-jet bundle $J^{1}E$ should introduce new structures not already present in
the geometry of the field space bundle $E$.

This poses no problem, and in a sense simplifies the picture. One can shortcut
the mathematical niceties of §5.1, and instead take a more pedestrian approach
that we are used to as physicists; to wit, we should consider the most general
geometry on the 1-jet bundle that is consistent with symmetry. This is, after
all, the most natural approach to take in the spirit of EFT!

#### General Poincaré invariant 1-jet metrics

We will, however, make rather stringent assumptions about the structure and
symmetry of spacetime, to simplify our discussion (and specialise to those
EFTs we are most interested in as high-energy physicists). As previously, we
fix spacetime to be $\Sigma=\mathbb{R}^{1,3}$, flat Minkowski in $d$ spacetime
dimensions with metric $\eta=\eta_{\mu\nu}\,dx^{\mu}\otimes
dx^{\nu}=dt^{2}-d\vec{x}\cdot d\vec{x}$.202020Lest there is any confusion, the
scalar field theory Lagrangian will be defined in terms of a geometry on the
1-jet bundle $\pi_{1}:J^{1}E\to\Sigma$, and we equip $J^{1}E$ with a geometry
(i.e. a metric $g^{(1)}$) that is independent of the geometry on spacetime. In
particular, the metric components in the ‘base space directions’ of $J^{1}E$
need not agree with those components of $\eta$, the metric on $\Sigma$. We
moreover enforce Poincaré symmetry on the jet bundle geometry. In practice,
this means we must contract all $\mu$ indices (that derive from both the base
space coordinates $x^{\mu}$ and the derivative coordinates $u^{i}_{\mu}$) to
make Lorentz invariants, but we do not form contractions with $x^{\mu}$
vectors since that would break translations (nor can any explicit $x$
dependence appear in the metric components). We will discuss symmetries on the
jet bundle again in §5.5, using the language of bundle morphisms (building on
§3.3).

So, let us write down the most general such metric, in our local coordinates.
We keep all terms, including ‘cross-terms’ that mix different types of jet
bundle coordinates:

$\displaystyle g^{(1)}=$
$\displaystyle\begin{pmatrix}dx^{\mu}&du^{i}&du^{i}_{\mu}\end{pmatrix}[g^{(1)}]\begin{pmatrix}dx^{\nu}\\\
du^{j}\\\ du^{j}_{\nu}\end{pmatrix},$ $\displaystyle[g^{(1)}]$
$\displaystyle=\begin{pmatrix}\Lambda^{4}g_{\mu\nu}(\dots)\;&\Lambda^{2}g_{\mu
j}(\dots)&g^{\nu}_{\mu j}(\dots)\\\\[5.69054pt] \Lambda^{2}g_{\nu
i}(\dots)&g_{ij}(\dots)&\dfrac{1}{\Lambda}g_{ij}^{\nu}(\dots)\\\\[5.69054pt]
g^{\mu}_{\nu
i}(\dots)&\dfrac{1}{\Lambda}g_{ij}^{\mu}(\dots)&\dfrac{1}{\Lambda^{2}}g_{ij}^{\mu\nu}(\dots)\end{pmatrix}\,.$
(93)

Because the derivative coordinates themselves have pairs of indices, one
upstairs and one downstairs, it is convenient to use the same index notation
for the components of the metric; thus, a component such as $g_{ij}^{\mu}$ or
$g_{ij}^{\mu\nu}$ still has two downstairs jet bundle indices (i.e. all metric
components are of course components of a $(0,2)$ tensor, never a 3- or
4-tensor), but taken with respect to one (or two) derivative coordinates
respectively. The $(\dots)$ record the fact that, in general and before
imposing any symmetry, each metric component is a functions of all the jet
bundle coordinates – including the ‘derivative coordinates’. Symmetry will of
course restrict the form of these components dramatically.

Upon pulling back a general metric of the form (93) along a section
$j^{1}\phi$ of the jet bundle $(J^{1}E,\Sigma,\pi_{1})$ and forming the
Lagrangian (89), one obtains the following Lagrangian terms:

$\displaystyle{\cal L}[\phi,g^{(1)}]\,=\,\,$
$\displaystyle\eta^{\rho\sigma}\bigg{[}\frac{1}{2}\Lambda^{4}\delta^{\mu}_{\rho}\delta^{\nu}_{\sigma}\,g_{\mu\nu}(\phi,\partial\phi)+\Lambda^{2}\delta^{\mu}_{\rho}g_{\mu
j}(\phi,\partial\phi)\,\partial_{\sigma}\phi^{j}+\Lambda\delta^{\mu}_{\rho}g_{\mu
j}^{\nu}(\phi,\partial\phi)\,\partial_{\rho}\partial_{\nu}\phi^{i}$
$\displaystyle+\,$
$\displaystyle{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\frac{1}{2}g_{ij}(\phi,}\partial\phi{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0})\,\partial_{\rho}\phi^{i}\partial_{\sigma}\phi^{j}}+\frac{1}{2\Lambda}g_{ij}^{\mu}(\phi,\partial\phi)\,\partial_{\rho}\partial_{\mu}\phi^{i}\partial_{\sigma}\phi^{j}+\frac{1}{2\Lambda}g_{ij}^{\nu}(\phi,\partial\phi)\,\partial_{\rho}\phi^{i}\partial_{\sigma}\partial_{\nu}\phi^{j}$
$\displaystyle\,+$
$\displaystyle\frac{1}{2\Lambda^{2}}g_{ij}^{\mu\nu}(\phi,\partial\phi)\,\partial_{\rho}\partial_{\mu}\phi^{i}\partial_{\sigma}\partial_{\nu}\phi^{j}\,\bigg{]}\,.$
(94)

Let us unpack this general form of the Lagrangian a little. First, we
highlight in magenta the piece of this Lagrangian that is captured by the
usual field space geometry picture, as in Eq. (9). In fact, this $g_{ij}$ term
is already more general than the 2-derivative Lagrangian (9), because the
component function $g_{ij}(\phi,\partial\phi)$ is itself a function of first
derivatives, and so this term already includes a subset of 4-derivative EFT
operators (and indeed operators with $>4$ derivatives), not just 2-derivative
operators.

Another important remark to make is that we also capture terms with fewer than
2-derivatives from this geometry, coming from the first two terms on the RHS
of (5.3) – we already saw this feature in §3 when describing the 2-derivative
EFT using the field space bundle picture (which is equivalent to formulating
the theory geometrically on the ‘0-jet bundle’ $J^{0}E=E$). In particular, the
first term

${\cal L}\supset\eta^{\mu\nu}\Lambda^{4}g_{\mu\nu}(\phi,\partial\phi)$ (95)

can encode an arbitrary scalar potential contribution; taking
$g_{\mu\nu}=\frac{1}{d}\eta_{\mu\nu}V(\phi/\Lambda)+\dots$, and Taylor
expanding, the preceding equation becomes

${\cal
L}\supset\Lambda^{4}\sum_{n=0}^{\infty}V_{n}\left(\frac{\phi}{\Lambda}\right)^{n}\,.$
(96)

In the case of SMEFT or HEFT, when we impose $O(4)$ symmetry and so forbid
terms with odd powers of $\phi$, this contains the (super-)renormalisable
terms $c+\mu^{2}\phi^{2}-\lambda\phi^{4}$ appearing in the SM, where
$c=\Lambda^{4}V_{0}$, $\mu^{2}=\Lambda^{2}V_{2}$ and $\lambda=-V_{4}$.212121Of
course, tiny values are required for $V_{0}$ and $V_{2}$, given any viable EFT
cutoff scale $\Lambda$; this manifests the (tree-level) cosmological constant
and electroweak hierarchy problems. We emphasize that, in the usual geometric
approaches, the scalar potential does not have a geometric origin but rather
is added as an extra ingredient in defining the EFT – see our discussion in
§2.2.

Let us return to non-renormalisable operators, with more derivatives. Because
all of the metric components in (5.3) are arbitrary functions of $\phi$ and
$\partial\phi$, the Lagrangian (5.3) contains operators with arbitrary numbers
of derivatives – but not all such terms – in a similar way to the usual
geometric Lagrangian (9) containing operators with arbitrary numbers of
fields. The more precise statement here is that the 1-jet bundle geometric
Lagrangian (5.3) yields operators with

$D\leq N+2$ (97)

where $D$ is the number of derivatives in the operator, and $N$ is the number
of fields. The reason for the $+2$ on the RHS of (97) is the same reason that
the ordinary field space geometry (equivalently, the $J^{0}E$ case) describes
2-derivative terms, namely because there are already two derivatives due to
$g$ being a $(0,2)$ tensor.

We do not claim, in this fully general EFT without any symmetries, that we
capture all operators satisfying (97); to answer this would require careful
counting of non-redundant operators with unbounded mass dimension (since $N$
can be taken arbitrarily large), which is clearly difficult. Adapting Hilbert
series techniques (see e.g. Henning:2015daa ; Henning:2017fpj ) to jet bundle
‘field space’ might provide helpful tools for tackling such completeness
questions (should they be interesting). A more practical approach is to
truncate the general Lagrangian (5.3) and count e.g. all operators with a
maximum number of derivatives. For the 1-jet bundle, we are able to count
operators in the examples considered in §6 and §7, including for HEFT and
SMEFT, accounting for all integration by parts (IBP) redundancies; in all
cases, we find the complete basis of operators with up to 4-derivatives is
covered by the Lagrangian (5.3). In fact, we prove in general that by pulling
back a 1-jet bundle metric one obtains a complete non-redundant basis of 0-,
2-, and 4-derivative operators for any scalar EFT. See Appendix B.

Passing to higher jet bundles, one captures more and more operators (with more
and more derivatives, relative to the number of fields); specifically, for an
$r$-jet bundle geometry, one can obtain operators with

$D\leq rN+2$ (98)

number of derivatives; those operators with $(r-1)N<D-2<rN$ would not have
already been captured by the $(r-1)$-jet bundle geometry. We thus see how
going to higher jet bundles systematically captures more and more operators in
the EFT, and so provides a natural organising principle for the EFT expansion.
Indeed, one can prove that the $r$-jet bundle geometry can capture a complete
basis of operators in a general scalar EFT with up to $2(r+1)$ derivatives –
again, see Appendix B.

We will use 1-jet bundles to study several examples of scalar field theories
in §6, culminating in a geometric formulation of the Higgs EFTs, including all
operators containing up to 4 derivatives, in §7. Before we get into the nitty
gritty of these examples, in the remainder of this Section we want to dig a
bit deeper into how field redefinitions might be described in this context,
and how the various notions of ‘bundle morphism’ described in §3.3 lift to the
1-jet bundle description. This will also allow a more general formulation of
symmetries on the jet bundles.

###  5.4 Field redefinitions are (prolonged) changes of section

To recap, in the ‘bundle formulation’ of a scalar field theory that we
described in §3, we identified scalar fields with local sections
$\phi\in\Gamma_{x}(\pi)$ of bundles $(E,\Sigma,\pi)$ over spacetime $\Sigma$
(40). A field redefinition, in general, was identified as a change of section
$\phi\mapsto\psi\in\Gamma_{x}(\pi)$. Furthermore, for the special class of
field redefinitions that are ‘non-derivative’, this is equivalent to doing a
base-point preserving bundle morphism $(f,\bar{f})=(f_{E}:E\to
E,\text{id}_{\Sigma}:\Sigma\to\Sigma)$.

Upon passing to the 1-jet bundle $(J^{1},\Sigma,\pi_{1})$, which we suggest
provides the right geometric ingredients for constructing EFT Lagrangians with
more derivatives, the section $\phi$ is promoted to its prolongation
$j^{1}\phi\in\Gamma_{x}(\pi_{1})$, a section of the 1-jet bundle. This is
uniquely determined by the original section $\phi$, simply by
$j^{1}\phi(x)=j^{1}_{x}\phi$, the 1-jet of $\phi$ at $x$. The natural notion
of a ‘field redefinition’ in this picture is therefore sending

$j^{1}\phi\mapsto j^{1}\psi\,,$ (99)

the result of which is also a section of the 1-jet bundle $\pi_{1}$. We can
think of this as a ‘prolongation’ of the field redefinition $\phi\to\psi$.
Since the Lagrangian will be defined by pulling back objects (a metric) from
$(J^{1}E,\Sigma,\pi_{1})$ along a section, this tells us how we can implement
the field redefinition at the level of the Lagrangian – it is straightforward
to check that the Lagrangian transforms in the ‘right way’ (see §5.5.4 for
some more-or-less trivial examples). We believe this can accommodate any
(possibly derivatively-dependent, but sufficiently smooth) field redefinition.

###  5.5 Morphisms on the jet bundle

The notion of bundle morphisms that we gave in §3.3, and used to describe non-
derivative field redefinitions and (internal and spacetime) symmetries, can be
promoted to a notion of morphism on the 1-jet bundle (which extends suitably
to higher $r$-jet bundles), which we describe in this Section.

####  5.5.1 Prolongation of bundle morphisms

We start with the notion of bundle morphism $(f,\bar{f})$ introduced in Eq.
(45), where we straightaway fix both bundles to have the same base and total
space. Going to the 1-jet bundle $J^{1}E$, the prolongated morphism $j^{1}f$
is defined by its action on points in the jet bundle $J^{1}E$, which recall
are jets $j^{1}_{x}\phi$, by saunders1989geometry

$j^{1}(f,\bar{f})\,(j^{1}_{x}\phi)=j^{1}_{\bar{f}(x)}(f\circ\phi\circ\bar{f}^{-1})\,$
(100)

where $\bar{f}^{-1}$ is the inverse of $\bar{f}$, which we assume to be a
(local) diffeomorphism. The commutative diagram (45) is extended to

${{J^{1}E}}$${{J^{1}E}}$${E}$${E}$${\Sigma}$${\Sigma}$$\scriptstyle{\bar{f}}$$\scriptstyle{f}$$\scriptstyle{j^{1}f}$$\scriptstyle{\pi_{1,0}}$$\scriptstyle{\rho_{1,0}}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$
(101)

Specialising to the case of bundle morphisms that act trivially on the base
(47), i.e. taking $\bar{f}=\text{id}_{\Sigma}$, Eq. (100) simplifies to

$j^{1}(f)\,(j^{1}_{x}\phi)=j^{1}_{x}(f\circ\phi)\,.$ (102)

The commutative diagram (101) then simplifies to

${{J^{1}E}}$${{J^{1}E}}$${E}$${E}$${\Sigma}$$\scriptstyle{\pi}$$\scriptstyle{\rho}$$\scriptstyle{f}$$\scriptstyle{\pi_{1,0}}$$\scriptstyle{\rho_{1,0}}$$\scriptstyle{j^{1}f}$
(103)

Recall $\pi_{1}=\pi_{1,0}\circ\pi$ and likewise
$\rho_{1}=\rho_{1,0}\circ\rho$. For two sections
$j^{1}\phi\in\Gamma_{x}(\pi_{1})$ and $j^{1}\psi\in\Gamma_{x}(\rho_{1})$, the
‘prolonged’ commutative diagram implies the relation

$j^{1}\psi=j^{1}f\circ j^{1}\phi\,$ (104)

between sections, analogous to (50).

We can obtain the coordinate representation of the prolongated diffeomorphism
$j^{1}f$ by taking the composition with the various coordinate functions. For
the original base and ‘field space coordinates’ $x^{\mu}$ and $u^{i}$, we get

$\displaystyle x^{\mu}\circ j^{1}f$ $\displaystyle=x^{\mu}\circ\rho_{1}\circ
j^{1}f=x^{\mu},$ (105) $\displaystyle u^{i}\circ j^{1}f$
$\displaystyle=u^{i}\circ\rho_{1,0}\circ j^{1}f=u^{i}\circ f\,.$ (106)

That is, the prolongated morphism $j^{1}f$ acts exactly as $f$ on the
$x^{\mu}$ and $u^{i}$ components (remembering our restriction that the bundle
morphism $f$ acts trivially on the base space). Lastly, for the derivative
coordinates $u_{\mu}^{i}$, we can deduce their transformation under $j^{1}f$
by considering the action on a particular jet $j^{1}_{x}\phi$:222222We
reiterate that all formulae here are for the specific subset of bundle
morphisms that act trivially on the base.

$\displaystyle(u_{\mu}^{i}\circ j^{1}f)(j^{1}_{x}\phi)$
$\displaystyle=\partial_{\mu}\left(u^{i}\circ(f\circ\phi)\right)\big{|}_{x}$
(107) $\displaystyle=\partial_{\mu}(u^{i}\circ
f)\big{|}_{x}+\partial_{\mu}(u^{j}\circ\phi)\big{|}_{x}\frac{\partial(u^{i}\circ
f)}{\partial u^{j}}\bigg{|}_{x}$ $\displaystyle=\partial_{\mu}(u^{i}\circ
f)\big{|}_{x}+u_{\mu}^{j}\,\,\frac{\partial(u^{i}\circ f)}{\partial
u^{j}}\bigg{|}_{x}\,.$

One can think of this as a ‘total derivative’ of the transformed coordinate
functions $u^{i}\circ f$; it follows essentially from applying the chain rule.
To summarize, we can write the prolongated bundle morphism as:

$j^{1}f:(x^{\mu},u^{i},u_{\mu}^{i})\mapsto(x^{\mu},f^{i},\partial_{\mu}f^{i}+u_{\mu}^{j}\partial_{j}f^{i})\,,$
(108)

in our local fibred coordinates, where $f^{i}(x^{\mu},u^{j})=u^{i}\circ f$ as
in §3.3.

The construction we have described for prolonging a bundle morphism from
$\pi:E\to\Sigma$ to its 1-jet bundle $\pi_{1}:J^{1}\to\Sigma$ can be extended
straightforwardly to the higher-jet bundles. Since on the physics side we
focus on $J^{1}E$ in this paper, we omit details of this construction here.

####  5.5.2 Prolonging internal and spacetime symmetries to the 1-jet bundle

In §3.3.1 we defined notions of internal and spacetime symmetries as bundle
morphisms satisfying particular conditions. Namely, all symmetries leave the
Lagrangian invariant; internal symmetries are base-point preserving bundle
morphisms, which in local coordinates can be expressed as
$f_{g}:(x^{\mu},u^{i})\mapsto(x^{\mu},f^{i}_{g}(u^{i},x^{\mu}))$; spacetime
symmetries have a non-trivial action on the base space $\Sigma$ of the bundle.
We discussed the particular cases of the Higgs EFTs in §3.3.1.

How do these symmetries act on the 1-jet bundle? Having phrased symmetries in
the above language of bundle morphisms, there is an obvious way to ‘lift’ them
to the 1- and higher-jet bundles using the notion of prolongation, that we
have introduced in this Section.

#### Prolonging an internal symmetry

For an internal symmetry, we can directly apply the formula (108) for the
prolongation of a base-point preserving bundle morphism, which we have for
each group element $g\in G$, our symmetry Lie group. The set of prolonged
symmetry morphisms are232323As usual, all morphisms and prolongations thereof
are understood to be only local here; the formulae we write are valid only in
certain open submanifolds coordinatized by certain charts; in other words, all
manifolds appearing in commutative diagrams such as (103) should be understood
as open submanifolds of the objects written.

$j^{1}f_{g}:(x^{\mu},u^{i},u_{\mu}^{i})\mapsto(x^{\mu},f_{g}^{i},\partial_{\mu}f_{g}^{i}+u_{\mu}^{j}\partial_{j}f_{g}^{i})\,,\quad\forall
g\in G$ (109)

For the SMEFT/HEFT symmetries, we can, at least in a coordinate patch, write
$f_{O\in O(4)}^{i}=\sigma(O,u^{i})$, for $\sigma=\sigma_{\text{S}}$ or
$\sigma_{\text{H}}$ in the case of SMEFT or HEFT respectively. Let us unpack
this further in the case of SMEFT, for illustration, in which the group action
in our coordinate chart of choice is linear:

$f^{i}=O^{i}_{j}u^{j}\,,$ (110)

where $O^{i}_{j}$ is a constant $O(4)$ matrix in the fundamental
representation. Notice that, in addition to being an internal symmetry, it is
also a global symmetry, in the usual physicist’s sense, which is a restricted
kind of base-point preserving morphism. (It is the same restriction we have
often imposed when viewing field redefinitions as morphisms, where we appealed
to Poincaré invariance as our justification.) Thus, $\partial_{\mu}f^{i}=0$
while $\partial_{j}f^{i}=O^{i}_{j}$. The prolongation of the internal $O(4)$
symmetries to the 1-jet bundle is thus given by the set of morphisms:

$\\{j^{1}f_{O}:(x^{\mu},u^{i},u_{\mu}^{i})\mapsto(x^{\mu},O^{i}_{j}u^{j},O^{i}_{j}u_{\mu}^{j})\qquad\forall
O\in O(4)\,\\}\,.$ (111)

exactly as one would naïvely expect. In practice, this means that one forms
$O(4)$-invariants (say, when constructing invariant metrics on the 1-jet
bundle, and thence invariant Lagrangians) by contracting the upper case $i$
indices wherever they appear both in the $u^{i}$ coordinates and in the
derivative coordinates $u^{i}_{\mu}$. The simplicity of this group action,
which could hardly have been otherwise, will be frequently invoked in §7 when
we finally turn to constructing the 4-derivative Higgs EFT Lagrangians.

#### Prolonging a spacetime symmetry

Starting from the field space bundle $(E,\Sigma,\pi)$, a spacetime symmetry is
a collection of bundle morphisms $(f,\bar{f})$ for which $\bar{f}$ is a non-
trivial diffeomorphism on the base space. The Poincaré group action for our
Lorentz-invariant EFTs is still specified by Eq. (57). The prolonging of a
spacetime symmetry requires the more general version shown in Eqs. (100, 101).
This requires the generalisation of Eqs. (105–107) to the case where $\bar{f}$
is non-trivial. (We refer the interested reader to Section 4.2 of the book
saunders1989geometry  for these more general formulae, which we only use
explicitly here). The prolongation of the Poincaré symmetry morphisms (57)
gives:

$\\{j^{1}f_{a,L}:(x^{\mu},u^{i},u_{\mu}^{i})\mapsto(L^{\mu}_{\nu}x^{\nu}+a^{\mu},u^{i},(L^{-1})^{\nu}_{\mu}u_{\nu}^{i})\qquad\forall(a,L)\in
O(1,3)\,\\}\,.$ (112)

Notice that while the Poincaré action is non-linear on the $x^{\mu}$
coordinates, including the effects of translations, this does not appear in
the group action on the derivative coordinates upon prolonging the symmetry.
Moreover, notice that it is the inverse matrix $L^{-1}$ that appears
multiplying the derivative coordinates. Again, as for the $O(4)$ internal
indices, the straightforward action of the prolonged spacetime symmetry on the
derivative coordinates means that we can treat the $\mu$ index exactly as we
would any other Lorentz index when forming Lorentz-invariant contractions;
namely, we should contract with the Minkowski metric $\eta^{\mu\nu}$ on
spacetime, as we do everywhere in the examples that follow in §6 and §7.

####  5.5.3 Non-derivative field redefinitions (last time!)

Again, analogous to the situation we described in §3.3, not any change of
prolonged section $j^{1}\phi\to j^{1}\psi$ can be obtained by prolonging a
bundle morphism, as in (103). Indeed, the same considerations of §3.3, noting
in particular the lemma discussed in §3.3.5, suggest that a morphism $j^{1}f$
on the 1-jet bundle cannot reproduce the effects of a derivative field
redefinition (except under special circumstances, analogous to those
considered in Example 3.3.4),242424Later in §6.2, we will perform an analogous
exercise to that in §3.3.4 ; for a scalar field theory at the 4-derivative
level, which is described by geometry on the 1-jet bundle, we ask when (if
ever) a prolonged bundle morphism can reproduce the effects of a derivative
field redefinition.  but can reproduce the effects of non-derivative field
redefinitions. It is important to check, as we do in this Subsection, that
non-derivative field redefinitions are still implemented consistently by
prolonging the bundle morphism $f$ to $j^{1}f$.

Recall that a general Lagrangian (with up to 4 derivatives) is, in this
formulation, obtained by pulling back a metric $g^{(1)}$ from the 1-jet bundle
to $\Sigma$ along a section $j^{1}\phi\in\Gamma_{x}(\pi_{1})$, given an
initial section $\phi\in\Gamma_{x}(\pi)$, before contracting with the inverse
metric on spacetime:

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

Doing a prolongated bundle morphism $j^{1}f:J^{1}E\to J^{1}E$ on the 1-jet
bundle, obtained from a morphism $f:E\to E$ via (103), which is itself a
diffeomorphism on $J^{1}E$ considered as a manifold, the metric changes via
the usual transformation law

$g^{(1)}=g^{(1)}_{IJ}({\bf x})d{\bf x}^{I}\otimes d{\bf
x}^{J}\mapsto(j^{1}f)^{\ast}g^{(1)}=g^{(1)}_{KL}({\bf x}\circ
j^{1}f)\frac{\partial{\bf j^{1}f}^{K}}{\partial{\bf x}^{I}}\frac{\partial{\bf