# On the road(s) to the Standard Model

Rodrigo Alonso  Mia West  Institute for Particle Physics Phenomenology, Durham
University, South Road, Durham, DH1 3LE

###### Abstract

Experimental measurements point at the Standard Model (SM) as the theory of
electroweak symmetry breaking, but as we close in on our characterization the
question arises of what limits in theory space lead to the SM. The class of
theories with this property cannot be ruled out, only constrained to an ever
smaller neighbourhood of the SM. In contrast, which class of theories do not
posses this limit and can therefore potentially be ruled out experimentally?
In this work we study both classes and find evidence to support the Standard
Model Effective field theory being the single road to the Standard Model,
theories that fall outside this class keeping a ‘minimum distance’ from the SM
characterized by a cut-off of at most $4\pi v/g_{\rm SM}$.

††preprint: IPPP/21/29

##  I Introduction

Nowadays, particle physics finds itself in the midst of Electroweak Symmetry
Breaking (EWSB) exploration, the outcome of this endeavour will chart Nature’s
theory of elementary particles. Experimental data collated and compared with
predictions of theories of EWSB has narrowed down the range possibilities;
many a casualties lie indeed now discarded having been disproven by the
progress in our measurements. The Higgs boson discovery, coming up on a decade
old, was the main stroke in our map, subsequent data giving a profile that
resembles the one heralded by the Standard Model (SM). Theory considerations
have long pointed out the SM case for EWSB to be unstable under higher scale
corrections and indicated that new physics should lie in wait at the
electroweak scale. Whether these considerations should be revisited and our
theory perspective profoundly changed, or if instead patience is all that is
needed, the pressing question at present posed by experimental data is to
characterize the theory ‘neighbourhood’ of the SM. The claim that one observes
nothing but the SM at the LHC is indeed only as good our characterization of
what else we could observe; it is here we find value in the aforementioned
casualties. The aim in this work is to explore the consistent theory
neighbourhood of the Standard Model.

A long known and studied approach, or ‘trajectory’, to the SM is a linearly
realized Effective Field Theory (SMEFT), see [1] for a review, this road being
pointed at by the decoupling theorem [2]. The integration of any heavy
particle whose mass can be arbitrarily larger than the EWSB vev ($M>v$) in a
perturbative linear realization will yield the SMEFT; supersymmetry or
composite Higgs models fall into this category. Is this the only road to the
Standard Model, i.e. are there other consistent limits to obtain the SM
couplings for the known spectrum of elementary particles? As fundamental as
this topic is, on its present formulation the candidate preceded the question;
Higgs Effective Theory [3, 4] is an EFT that encompasses the SMEFT but extends
beyond it and might offer new roads. In HEFT, a linear realization is not
assumed (though admissible in certain limit) and is indeed the most general
Lorentz and gauge invariant theory with the known spectrum of particles (which
suggests it should be possible to formulate it in terms of amplitudes). The
theories that this EFT describes but fall out of SMEFT, which will be called
here theories of the quotient space HEFT/SMEFT or simply quotient EFTs 111In
other works these are called, with a slight abuse of notation, HEFT., could
contain a path to the SM other than via SMEFT. This quotient space is
characterized as missing a point in field space which is left invariant under
an $O(4)$ transformation [5, 6], be it because it is not present or because
the would be invariant point is singular [7]. A geometric formalism was used
to derive this result and also aids in exploring the properties of theories
without field redundancies, as introduced in [5, 6], and followed up in [7, 8,
9] \- it is also adopted here. Some theories in HEFT/SMEFT quotient space have
been formulated while having a perturbative expansion [7]; they have been
found to have a cut-off of $\sim 4\pi v$ and no limit can be taken within them
that yields the SM. It has been suggested that all of this this quotient space
shares this property of a finite $v$-bound cut-off [10] with further evidence
provided in [9], which means in turn that they all could be casualties of our
exploration with present and future machines. This question has been explored
so far with perturbative unitarity bounds, while here it is looked at with
semi-classical arguments.

This letter is structured as follows. Section II introduces geometry from
amplitudes, and sec. II.1 presents the basis in Riemann normal coordinates.
This first part has been rendered review, rather than new results, by virtue
of [9] although all results here are derived independently. Section. II.2
presents theory and experimental bounds on the curvature plane while III
characterizes SMEFT on this plane. In sec. IV, example models of SMEFT and
quotient space are presented and characterized in the curvature plane. Sec. V
presents theories in quotient space arising from geometry rather than explicit
models and finds candidate quotient theories that seem to approach the SM. A
semi-classical argument for the finite cut-off of theories in quotient space
is given in VI.

##  II Geometry and Amplitudes

For simplicity, $O(4)\supset SU(2)\times U(1)$ invariance in the EWSB sector
is assumed. We take the high energy limit and make use of the equivalence
theorem. The Higgs singlet field is denoted $h$, and the Goldstones swallowed
by the $W$ and $Z$ bosons as $\varphi^{a}$, $a=1,2,3$.

Let us start by defining our geometry from the scattering-matrix $S$ in order
to depart from a common-place, basis-invariant magnitude in particle physics.
Following the line-integral definition for general amplitudes valid also in
the UV, we have ($S=1-i\mathcal{A}$):

$\displaystyle-R_{h+h-}$ $\displaystyle=\frac{1}{2\pi
i}\oint\frac{1}{s_{12}^{2}}\mathcal{A}_{W^{+}_{1}W^{-}_{2}\to hh}$ (1)
$\displaystyle-R_{+-+-}$ $\displaystyle=\frac{1}{2\pi
i}\oint\frac{1}{s_{12}^{2}}\mathcal{A}_{W^{+}_{1}W^{+}_{2}\to W^{+}W^{+}}$ (2)
$\displaystyle-\nabla_{h}R_{+h-h}$ $\displaystyle=\frac{1}{2\pi
i}\oint\frac{1}{s_{12}^{2}}\mathcal{A}_{W_{1}^{+}W_{2}^{-}\to hhh}$ (3)
$\displaystyle-\nabla_{h}R_{+-+-}$ $\displaystyle=\frac{1}{\pi
i}\oint\frac{1}{s_{12}^{2}}\mathcal{A}_{W_{1}^{+}W_{2}^{+}\to
W_{3}^{+}W_{4}^{+}h}$ (4) $\displaystyle=\frac{1}{\pi
i}\oint\frac{1}{s_{34}^{2}}\mathcal{A}_{W_{1}^{+}W_{2}^{+}\to
W_{3}^{+}W_{4}^{+}h}$ (5)

where $s_{ij}=(p_{i}+p_{j})^{2}$. Indices in the Riemann tensor run through
$h,a=1,2,3$ and the $\pm$ entries are given by contracting an $a$-index with
the projector $(\delta^{a}_{\,1}\pm i\delta^{a}_{\,2})/\sqrt{2}$, for example

$\displaystyle R_{h+h-}$
$\displaystyle=R_{hahb}\frac{(\delta^{a}_{\,1}+i\delta^{a}_{\,2})}{\sqrt{2}}\frac{(\delta^{b}_{\,1}-i\delta^{b}_{\,2})}{\sqrt{2}}$
(6)

While the above definition is useful to include UV models and derive
positivity bounds [11], in practice we will work with the low energy EFT. In
which case the correspondence is taking our geometry from the order
$\mathcal{O}(s)$ coefficients in a Taylor expansion. What’s more is they
capture all terms to this order. Being explicit,

$\displaystyle\mathcal{A}_{W_{1}^{+}W_{2}^{-}\to hh}=$ $\displaystyle-
s_{12}R_{+h-h}$ (7) $\displaystyle\mathcal{A}_{W^{+}_{1}W^{+}_{2}\to WW}=$
$\displaystyle-s_{12}R_{+-+-}$ (8)
$\displaystyle\mathcal{A}_{W_{1}^{+}W_{2}^{-}\to hhh}=$ $\displaystyle-
s_{12}\nabla_{h}R_{+h-h}$ (9) $\displaystyle\mathcal{A}_{W^{+}_{1}W^{+}_{2}\to
W_{3}^{+}W_{4}^{+}h}=$
$\displaystyle-\frac{s_{12}+s_{34}}{2}\nabla_{h}R_{+-+-}$ (10)

where we neglected masses assuming $s\gg M_{W}^{2},M_{Z}^{2},m_{h}^{2}$.

This starting point makes evident that our tensor, $R$, and its derivatives
are physical and field redefinition (coordinate) invariant. Even if intuitive,
this last statement should be qualified. On the geometry side, having defined
tensor entries rather than invariants, one has that these change under
coordinate transformations - albeit with well defined properties. They are
nonetheless the same for local (defined around the vacuum) transformations of
our fields which leave the amplitudes the same [12]:

$\displaystyle\hat{\phi}^{i}=$
$\displaystyle\left(\delta^{i}_{j}+\sum_{k=1}c^{k}_{j}\phi^{k}\right)\phi^{j}$
(11)

so that after quantization both fields produce a particle out of the vacuum,

$\displaystyle\langle p|\phi^{i}|0\rangle=\langle p|\hat{\phi}^{i}|0\rangle$
(12)

with $|p\rangle$ the state associated with the field. It is for this type of
transformation that the $S$ matrix will be left invariant, and tensors
evaluated at the vacuum transform trivially, since:

$\displaystyle\left.\frac{\partial\phi^{i}}{\partial\hat{\phi}^{j}}\,\right|_{\phi=0}=\delta^{i}_{j}$
(13)

Still, from where we stand the definition of Riemann tensor components in
terms of amplitudes seems arbitrary and potentially inconsistent. So let us
now turn to the Lagrangian theory which yields such relations.

###  II.1 Riemann Normal Coordinates

Take the metric that the Riemann tensor derives from in eqs. (1-5) as
$G_{ij}(\phi)$, with $i,j=h,1,2,3$, $\phi=(h,\varphi^{a})$ $a=1,2,3$. The
amplitudes in eqs. (1-5) follow from the action

$\displaystyle S=$ $\displaystyle\frac{1}{2}\int
d^{4}x\partial_{\mu}\phi^{i}G_{ij}\partial^{\mu}\phi^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
d^{4}x\left(\partial_{\mu}h\partial^{\mu}h+F(h)^{2}g_{ab}\partial^{\mu}\varphi^{a}\partial_{\mu}\varphi^{b}\right)$
(14)

In matrix notation, our parametrization of the metric reads

$\displaystyle G_{ij}=\left(\begin{array}[]{cc}1&\\\
&F^{2}g_{ab}\end{array}\right)$ (17)

where off-diagonal entries are forbidden by symmetry and $g_{ab}$ is the
metric on the 3-sphere which we find useful to represent via the unit vector
$u(\varphi)$:

$\displaystyle g_{ab}=$ $\displaystyle\frac{\partial
u(\varphi)}{\partial\varphi^{a}}\frac{\partial
u(\varphi)}{\partial\varphi^{b}}$ $\displaystyle u\cdot u$ $\displaystyle=1$
(18)

with $u$ transforming as a vector under $O(4)$. It follows that the non-
vanishing elements of the Riemann tensor and its first covariant derivative
are

$\displaystyle R_{abcd}$
$\displaystyle=\left(\frac{1}{v^{2}}-(F^{\prime})^{2}\right)F^{2}g_{a[c}g_{bd]}$
(19) $\displaystyle R_{ahbh}$ $\displaystyle=-F^{\prime\prime}F\tilde{g}_{ab}$
(20) $\displaystyle\nabla_{h}R_{ahbh}$
$\displaystyle=F^{2}\left(-\frac{F^{\prime\prime}}{F}\right)^{\prime}g_{ab}$
(21) $\displaystyle\nabla_{h}R_{abcd}$
$\displaystyle=F^{4}\left(\frac{1}{v^{2}F^{2}}-\frac{(F^{\prime})^{2}}{F^{2}}\right)^{\prime}g_{a[c}g_{bd]}$
(22) $\displaystyle\nabla_{a}R_{hbcd}$
$\displaystyle=\frac{F^{4}}{2}\left(\frac{1}{v^{2}F^{2}}-\frac{(F^{\prime})^{2}}{F^{2}}\right)^{\prime}g_{a[c}g_{bd]}$
(23)

where prime denotes differentiation with respect to $h$ and it is useful to
define

$\displaystyle R_{h}$ $\displaystyle\equiv-\frac{F^{\prime\prime}}{F}$
$\displaystyle R_{\varphi}$
$\displaystyle\equiv\frac{1}{v^{2}F^{2}}-\frac{(F^{\prime})^{2}}{F^{2}}$ (24)

Verifying that these tensor entries appear as coefficients in the 4- and
5-point amplitudes is a matter of computing amplitudes: expanding our metric
around the vacuum and adding over the various diagrams, e.g. see fig. 1 for
those contributing to $WW\to hhh$, relations (1-5) are recovered. The $O(4)$
symmetry in our system reduces the number of independent components and
amplitudes to $R_{h}$, $R_{\varphi}$ and its derivatives.

Figure 1:  Diagrams for the $\mathcal{O}(s)$ contribution to the $WWhhh$
amplitude in the basis of eq. (14).

Geometry does tell us however, that there is a frame where this computation is
particularly simple: the frame where our coordinates follow geodesics, i.e.
Riemann normal coordinates (RNC).

Let us then go into a brief outline of RNC. One can solve iteratively the
Geodesic equation:

$\displaystyle\frac{d^{2}\phi^{i}}{d^{2}\sigma}+\Gamma^{i}_{jk}(\phi)\frac{d\phi^{j}}{d\sigma}\frac{d\phi^{k}}{d\sigma}=0$
(25)

in an expansion which assumes the dependence on $\phi$ of $\Gamma$ admits a
Taylor expansion and introduces new coordinates $\phi^{\prime}$ defined to
second order as

$\phi^{\prime
i}=\phi^{i}+\frac{1}{2}\Gamma^{i}_{jk}(0)\phi^{j}\phi^{k}+\mathcal{O}(\phi^{3})$

Together with a metric in the new coordinates and to $\phi^{\prime 3}$ order
[13]:

$G(\phi^{\prime})_{ij}=G(0)_{ij}+\phi^{\prime k}\phi^{\prime
l}\frac{1}{3}R_{iklj}+\frac{1}{6}\phi^{\prime k}\phi^{\prime l}\phi^{\prime
m}\nabla_{m}R_{iklj}$

For concreteness, one can work out this transformation for our metric to find:

$\displaystyle\left(\begin{array}[]{c}h^{\prime}\\\
\varphi^{\prime}\end{array}\right)=\left(\begin{array}[]{c}h-FF^{\prime}\varphi^{2}/2\\\
\varphi^{a}+F^{\prime}h\varphi^{a}/F+\Gamma^{a}_{bc}\varphi^{b}\varphi^{c}/2\end{array}\right)+\mathcal{O}(\phi^{3})$
(30)

The use of RNC is the reduction to parametrization independent magnitudes,
i.e. Riemann tensor and its derivatives with the Christoffel symbols absent in
our frame. In an analogy with general relativity, this is the free-falling
frame where tidal effects reveal the geometry of the space-time manifold. In
practice, there are no 3-point amplitudes 222They are reinstated however once
we account for massive states. and the interacting Lagrangian for 4-point
reads:

$\displaystyle\mathcal{L}_{4}^{\rm RNC}=$
$\displaystyle\frac{1}{6}R_{hahb}\left(2h\partial
h\varphi^{a}\partial\varphi^{b}-(\partial
h)^{2}\varphi^{a}\varphi^{b}-h^{2}\partial\varphi^{a}\partial\varphi^{b}\right)$
$\displaystyle+\frac{1}{6}R_{abcd}\partial\varphi^{a}\varphi^{b}\varphi^{c}\partial\varphi^{d}$
(31)

The first line gives the Feynman rule

$\varphi^{a}(p_{1})$$\varphi^{b}(p_{2})$$h(p_{3})$$h(p_{4})$
$\displaystyle\frac{iR_{ahbh}}{3}$
$\displaystyle\left(\begin{array}[]{c}(p_{1}+p_{2})(p_{3}+p_{4})\\\
+2p_{1}p_{2}+2p_{3}p_{4}\end{array}\right)$ (34)

which evaluated on-shell is the sole diagram needed to compute $A_{WW\to hh}$
in this frame. For 5-point vertexes, we have

$\displaystyle\mathcal{L}_{5}^{\rm RNC}=$
$\displaystyle\frac{1}{12}(\nabla_{h}R_{\partial\varphi
hh\partial\varphi}+\nabla_{h}R_{\partial h\varphi\varphi\partial
h}+2\nabla_{h}R_{\partial h\varphi h\partial\varphi})$
$\displaystyle+\frac{1}{12}(\nabla_{h}R_{\partial\varphi\varphi\varphi\partial\varphi}+2\nabla_{\varphi}R_{\partial\varphi
h\varphi\partial\varphi})$ (35)

where the term $\nabla_{\varphi}R_{dh\varphi\varphi d\varphi}$ cancels due to
the Riemann tensor asymmetry; and with abuse of notation
$V_{\varphi}=V_{a}\varphi^{a}$, similarly for $h$. For the 5-point amplitude,
again due to the absence of 3-point vertexes, evaluating the Feynman rule that
follows from the 5-point action yields the result (i.e. in this frame there is
only the last diagram in fig. 1 to compute). Amplitudes for six or more
particles in total do require a sum over diagrams and contain, in addition,
poles which nevertheless can be derived from lower-point amplitudes, see [9].

###  II.2 Experimental and theory constraints on curvature

Unitarity constrains the magnitude of curvature, and its derivatives, for a
given c.m. energy $s$, to the 4-point level. Symbolically

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where the first partial wave for $W^{+}W^{-}$ gives

$\displaystyle\left(\frac{R_{\varphi}s}{16\pi}\right)^{2}+\frac{1}{2}\left(\frac{R_{h}s}{8\pi}\right)^{2}\leq
1$ (36)

where we have accounted for the amplitude being real. One can also select the
$W^{+}W^{+}$ channel, but the emphasis in here is on bounds which are
sensitive to both curvatures simultaneously which helps to better close some
corners in the curvature plane.

One can use these constraints to determine the theory cut-off in terms of
curvature; however, here we turn this around to note that given that we have
explored energies up to $s\sim v^{2}$ and no new states have showed up, we can
set an upper limit on curvature.

This limit is super-seeded by experimental bounds from LHC which bound Higgs
couplings. In the conventional parametrization, one has:

$\displaystyle
F(h)^{2}=1+2a\frac{h}{v}+b\frac{h^{2}}{v^{2}}+\mathcal{O}(h^{3})$ (37)

which gives a curvature around the origin

$\displaystyle
v^{2}\left(R_{\varphi}(0),R_{h}(0)\right)=\left(1-a^{2},-(b-a^{2})\right)$
(38)

itself related to amplitudes as, substituting (24,19,20) on (7,8),

$\displaystyle\mathcal{A}_{W^{+}_{1}W^{+}_{2}\to WW}=$ $\displaystyle
s_{12}R_{\varphi}$ (39) $\displaystyle\mathcal{A}_{W_{1}^{+}W_{2}^{-}\to hh}=$
$\displaystyle-s_{12}R_{h}$ (40)

Translating bounds on the coefficients from present and future measurements
into curvature, we present the plot in fig. 2. The value in both sets of
constraints is to put into context how much of the theory-consistent curvature
space have we explored experimentally.

From the outer-most to inner-most region of fig. 2: the (outer-most) grey
region is excluded due to unitarity; up to the blue region is excluded by
current LHC bounds (the region is translated from bounds on $a$ in [14], and
$b$ in [15]); finally, up to the green and orange (inner-most) regions we
present expected exclusion limits for HL-LHC and FCC respectively. The
projected bounds on $R_{\varphi},R_{h}$ are derived using sensitivity
predictions of $a$ (HL-LHC, [16]; FCC-ee, [17]); and $b$ ([18] for both HL-LHC
and FCC-hh), around their SM values. All uncertainties and projected
sensitivities are displayed at the $95\%$ confidence level; where multiple
sensitivity estimates are given, the most conservative is selected. Note that
HL-LHC bounds used here predate the LHC ones so that the seemingly marginal
improvement is likely an underestimation.

Figure 2: Theoretically (grey), and experimentally (up to blue) excluded (up
to 95% confidence level) regions of the curvatures $R_{h},R_{\varphi}$ which
are related to electroweak amplitudes as in eqs (40,39); and sensitivity
limits of future colliders (HL-LHC, up to green; FCC, up to orange), also up
to 95% confidence level. See text for detail. The plot scales linearly within
the dashed box, and logarithmically outside.

##  III Correlation of curvature in SMEFT

In the linear realization and to first order (with our assumption of $O(4)$
invariance) we have:

$\displaystyle R_{\varphi}=R_{h}$ (41)

Which is to say the coefficients of $s$ in the 4-point amplitudes for
$W^{+}W^{+}$ scattering and $W^{+}W^{-}\to hh$ in eqs (40,39) are anti-
correlated. Correlations do appear in the linear parametrization of SMEFT in
HEFT [19] in line with what we find here, nonetheless in this section we go
into some length of how this can be derived to display the utility of a
geometric language.

A simple argument to show there is a correlation, if a bit more abstract, is
to use Riemann normal coordinates and custodial symmetry around the
$O(4)$-symmetric point - which admits Cartesian coordinates. In this frame,
the metric reads

$\displaystyle
G_{ij}(\phi)=\delta_{ij}+\frac{1}{3}R_{iklj}\phi^{k}\phi^{l}+\mathcal{O}(\phi^{3})$
(42)

and a linear realization of $O(4)$ symmetry dictates that the Riemann tensor
be of the form $R(\delta_{il}\delta_{kj}-\delta_{kl}\delta_{ij})$, with a
single unknown $R$. A transformation from Cartesian to polar coordinates then
reveals $R_{h}=R_{\varphi}$.

The collapse of the two curvatures into a single one can also be derived
matching the two EFTs:

$\displaystyle\frac{\left(\partial h^{2}+F^{2}\partial\varphi^{2}\right)}{2}=$
$\displaystyle K\left(\frac{H^{\dagger}H}{M^{2}}\right)(\partial
H^{\dagger}H)^{2}$
$\displaystyle+G\left(\frac{H^{\dagger}H}{M^{2}}\right)D_{\mu}H^{\dagger}D^{\mu}H$
(43)

where it should be understood from a general SMEFT action, we transformed to a
basis where the Higgs singlet is canonically normalized.

This exercise yields, to order $M^{-4}$

$\displaystyle R_{\varphi}$
$\displaystyle=-3\frac{G^{\prime}(0)}{M^{2}}+\frac{H^{\dagger}H}{M^{4}}\left(2(G^{\prime}(0))^{2}-\frac{5}{2}G^{\prime\prime}(0)\right)$
(44) $\displaystyle R_{h}$
$\displaystyle=-3\frac{G^{\prime}(0)}{M^{2}}+\frac{H^{\dagger}H}{M^{4}}\left(4(G^{\prime}(0))^{2}-5G^{\prime\prime}(0)\right)$
(45)

which also reveals the correlation is lost at order $M^{-4}$.

Finally, and in a direct connection with observables, one can compute the
amplitude which has been used to define our curvature, the computation itself
getting rid of any field redundancy. Take the non-canonically normalized
action

$\displaystyle\mathcal{L}=\frac{1}{2}\frac{c_{H\Box}}{M^{2}}(\partial_{\mu}H^{\dagger}H)^{2}+\frac{c_{HDD}}{M^{2}}H^{\dagger}HD_{\mu}H^{\dagger}D^{\mu}H$
(46)

After normalization of the theory, computation of diagrams such as those shown
in fig. 3, where we note that in this frame there is a $h^{3}$ coupling that
scales with $s$ and must be accounted for, yields

$\displaystyle{\mathcal{A}}_{W^{+}W^{+}\to
W^{+}W^{+}}=\frac{s}{M^{2}}\left(c_{H\Box}-c_{HDD}\right)$ (47)
$\displaystyle{\mathcal{A}}_{W^{+}W^{-}\to
hh}=-\frac{s}{M^{2}}\left(c_{H\Box}-c_{HDD}\right)$ (48)

and hence the direct connection with SMEFT geometry as

$\displaystyle\left(R_{\varphi}\,,\,R_{h}\right)=\frac{1}{M^{2}}\left(c_{H\Box}-c_{HDD},c_{H\Box}-c_{HDD}\right)\,.$
(49)

Figure 3:  A selection of diagrams for the $WWhh$ and $WWWW$ amplitudes with
the action in eq. (46)

##  IV Models as probes into HEFT

Recent study of EFT has shown that UV completion might impose extra
constraints on an otherwise seemingly valid EFT, as is the case of positivity
constraints [20]. It should be said that these constraints on the curvatures
themselves $R_{h}$ and $R_{\varphi}$ do not restrict their sign, but reveal
the need for doubly-charged states if the curvature is negative [11]. It is
for these reasons that this section looks at models and introduces two new
representations under $O(4)$ as

$\displaystyle{\bf h}$ $\displaystyle:\quad 4\quad{\rm of}\quad O(4)$ (50)
$\displaystyle\Phi$ $\displaystyle:\quad 9\quad{\rm of}\quad O(4)\,\,({\rm
traceless\,\,symmetric})$ (51) $\displaystyle S$ $\displaystyle:\quad
1\quad{\rm of}\quad O(4)$ (52)

with the results of positivity constraints suggesting $S$ and $\Phi$ will
produce positive and negative curvature respectively. Note that ${\bf h}$ is
the Higgs doublet $H$ in a real representation as

$\displaystyle\left(\tilde{H},H\right)$
$\displaystyle=\hat{\sigma}_{I}\frac{{\bf h}^{I}}{\sqrt{2}}$ (53)

with $\tilde{H}=\epsilon H^{*}$ and $\sigma^{I}=(\sigma^{i},1)$ with
$\sigma^{i}$ the Pauli matrices. We consider the addition of a 9 and a 1
separately with respective actions

$\displaystyle\mathcal{L}_{S}=\frac{1}{2}D_{\mu}{\bf h}^{T}D_{\mu}{\bf
h}+\frac{1}{2}(\partial S)^{2}-V({\bf h},S^{2})$ (54)
$\displaystyle\mathcal{L}_{\Phi}=\frac{1}{2}D_{\mu}{\bf h}^{T}D_{\mu}{\bf
h}+\frac{1}{2}{\rm Tr}\left(D_{\mu}\Phi D^{\mu}\Phi\right)-V({\bf h},\Phi)$
(55)

The key distinction is whether ${\langle{\Phi}\rangle}=0$ or not, which
depends on the sign of its mass term and its mixing as induced by the
potential.

###  IV.1  Only $h$ acquires a vev, SMEFT case

In this subsection we momentarily restrict the $O(4)$ symmetry to $SO(4)$ to
allow for tri-linear couplings. First for the singlet $S$ case, we take a
potential as

$\displaystyle V=-\frac{g_{*}m_{S}}{2}S\,{\bf
h}^{2}+\frac{m_{S}^{2}}{2}S^{2}+\frac{m_{\bf h}^{2}}{2}{\bf h}^{2}$ (56)

extra terms allowed by the symmetry will give controlled corrections to the
result and we neglect them. Integrating the field $S$ at tree level returns

$\displaystyle\mathcal{L}_{\rm eff}$
$\displaystyle=\frac{1}{2}\frac{g_{*}m_{S}}{2}{\bf
h}^{2}\frac{1}{\partial^{2}+m_{S}^{2}}\frac{g_{*}m_{S}}{2}{\bf h}^{2}$ (57)
$\displaystyle=\frac{g_{*}^{2}}{2}(H^{\dagger}H)^{2}+\frac{g_{*}^{2}}{2m_{S}^{2}}(\partial(H^{\dagger}H))^{2}+\mathcal{O}\left(\partial^{4}\right)$
(58)

then via eq. (49)

$\displaystyle\left(R_{\varphi},R_{h}\right)=\left(\frac{g_{*}^{2}}{m_{S}^{2}},\frac{g_{*}^{2}}{m_{S}^{2}}\right)$
(59)

i.e. positive curvature for the singlet case, as expected.

Along the same lines, the potential for the symmetric representation is

$\displaystyle V=-\frac{g_{*}m_{\Phi}}{2}{\bf h}^{T}\Phi{\bf
h}+\frac{m_{\Phi}^{2}}{2}\Phi^{2}+\frac{m_{\bf h}^{2}}{2}{\bf h}^{2}$ (60)

The integration now returns, to dimension six:

$\displaystyle\mathcal{L}_{\rm eff}=$ $\displaystyle\frac{g_{*}^{2}}{8}{\rm
Tr}\left[\left({\bf h}{\bf h}^{T}-\frac{{\bf
h}^{2}}{4}\right)\frac{m_{\Phi}^{2}}{\Box+m_{\Phi}^{2}}\left({\bf h}{\bf
h}^{T}-\frac{{\bf h}^{2}}{4}\right)\right]$ (61) $\displaystyle=$
$\displaystyle\frac{3g_{*}^{2}}{8}(H^{\dagger}H)^{2}+\frac{g_{*}^{2}}{m^{2}_{\Phi}}\left(H^{\dagger}HDH^{\dagger}DH+\frac{(\partial
H^{\dagger}H)^{2}}{8}\right)$

where $\Box=D_{\mu}D^{\mu}$ and one has that the operator does yield negative
curvature:

$\displaystyle\left(R_{\varphi},R_{h}\right)=\left(-\frac{3g_{*}^{2}}{4m_{\Phi}^{2}},-\frac{3g_{*}^{2}}{4m_{\Phi}^{2}}\right)\,.$
(62)

###  IV.2 Both $\Phi$ and h break the symmetry, HEFT/SMEFT quotient space

As we will show, this case does not belong in SMEFT and stands as a
representative of quotient space. We take the extension of a mexican hat
potential for two fields as:

$\displaystyle V(\Phi)=$
$\displaystyle-\frac{\vec{m}^{2}}{2}\cdot\left(\begin{array}[]{c}{\bf
h}^{2}\\\ \Phi^{2}\end{array}\right)+\left(\begin{array}[]{c}{\bf h}^{2}\\\
\Phi^{2}\end{array}\right)^{T}\frac{\lambda}{8}\,\left(\begin{array}[]{c}{\bf
h}^{2}\\\ \Phi^{2}\end{array}\right)$ (69)
$\displaystyle-\frac{\tilde{\lambda}}{8}{\bf h}^{T}\Phi\Phi\,{\bf
h}+\frac{\tilde{\lambda}_{\Phi}}{8}{\rm Tr}\left(\Phi\Phi\Phi\Phi\right)$ (70)

with ${\vec{m}}^{2}$ a 2-vector and $\lambda$ a $2\times 2$ symmetric matrix.
Since $\Phi$ acquires a vev, we take $\tilde{\lambda}>0$ which triggers
$O(4)\to O(3)$ and preserves custodial symmetry. Linear terms in the fields
are absent, contrary to the previous case which since we restore $O(4)$ in
place of $SO(4)$. The key question as will be shown is to consistently compute
particle couplings and masses from an explicit potential.

The Goldstone boson Lagrangian and couplings to the radial singlet modes
$\delta h$, $\delta\Phi$ read:

$\displaystyle\mathcal{L}=$ $\displaystyle\frac{1}{2}\left((v_{\bf h}+\delta
h)^{2}+C_{9}(v_{\Phi}+\delta\Phi)^{2}\right)\frac{g_{ab}}{v^{2}}D^{\mu}\varphi^{a}D_{\mu}\varphi^{b}$
(71)

where

$\displaystyle C_{9}$ $\displaystyle=\frac{2\times 4}{4-1}\,,$ $\displaystyle
v^{2}$ $\displaystyle=v_{{\bf h}}^{2}+C_{9}v_{\Phi}^{2},$
$\displaystyle\sin\beta$ $\displaystyle=\sqrt{C_{9}}\frac{v_{\Phi}}{v},$ (72)

and

$\displaystyle{\langle{{\bf h}}\rangle}$
$\displaystyle=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ v_{\bf
h}\end{array}\right)$ $\displaystyle{\langle{\Phi}\rangle}$
$\displaystyle=\frac{v_{\Phi}}{2\sqrt{3}}\left(\begin{array}[]{cccc}1&&&\\\
&1&&\\\ &&1&\\\ &&&-3\end{array}\right)$ (81)

the generalization of $C_{9}$ to $SO(N)$ being $C_{N(N+1)/2-1}=2N/(N-1)$. Take
the mixing for the singlet radial modes $\delta{\bf h}$ and $\delta\Phi$ as
(note that no other field in $\Phi$ or ${\bf h}$ is a singlet of $SO(3)$ so we
know these two only mix among each other):

$\displaystyle\left(\begin{array}[]{c}\delta{\bf h}\\\
\delta\Phi\end{array}\right)=\left(\begin{array}[]{cc}\cos\omega&-\sin\omega\\\
\sin\omega&\cos\omega\end{array}\right)\left(\begin{array}[]{c}h\\\
\tilde{h}\end{array}\right)$ (88)

Putting the above back in the Lagrangian for the Goldstones and taking $h$ to
be the lightest singlet, one obtains in our basis of eq. (14,37)

$\displaystyle a$
$\displaystyle=c_{\omega}c_{\beta}+\sqrt{C_{9}}s_{\beta}s_{\omega}$
$\displaystyle b=$ $\displaystyle c_{\omega}^{2}+C_{9}s_{\omega}^{2}$ (89)

Note that the limit of no mixing gives $b=1$ and a parametrization of the
curvature $R_{h}=-R_{\varphi}$ orthogonal to the SMEFT with a potential new
road to the SM. The question to be answered is then: can one take
$\omega=\beta=0$ while keeping $m_{\tilde{h}}\gg m_{h}$ and maintaining
perturbativity?

To answer this question we should express $\omega$ and $\beta$ in terms of
physical masses and couplings, then use eq. (38) to substitute and find
curvature as a function of physical masses and couplings. In practice we have
to solve for the potential. The value of the fields that minimize $V$ can be
read off after rearranging as

$\displaystyle V(v_{\bf
h},v_{\Phi})=\left({\vec{v}}^{2}-2\hat{\lambda}^{-1}{\vec{m}}^{2}\right)^{T}\frac{\hat{\lambda}}{8}\left(\vec{v}^{2}-2\hat{\lambda}^{-1}{\vec{m}}^{2}\right)$
(90)

with

$\displaystyle{\vec{v}}^{2}$ $\displaystyle=2\hat{\lambda}^{-1}{\vec{m}}^{2}$
$\displaystyle\hat{\lambda}$
$\displaystyle=\lambda+\left(\begin{array}[]{cc}&-3\tilde{\lambda}/8\\\
-3\tilde{\lambda}/8&7\tilde{\lambda}_{\Phi}/12\end{array}\right)$ (93)

Next, expanding around the vevs we find the mass matrix for the singlets
$\delta{\bf h},\delta\Phi$ as

$\displaystyle M^{2}=$ $\displaystyle{\rm Diag}(v)\,\hat{\lambda}\,{\rm
Diag}(v)=U\,{\rm Diag}(m_{h}^{2},m_{\tilde{h}}^{2})\,U^{T}$ (94)

with ${\rm Diag}(v)=\delta^{ij}v_{j}$. The aim is to express $\omega,\beta$ as
$\omega(m_{h},m_{\tilde{h}},\hat{\lambda},v),\beta(m_{h},m_{\tilde{h}},\hat{\lambda},v)$,
which can be done by taking the determinant of the mass matrix

$\displaystyle{\rm det}(M^{2})$ $\displaystyle=v_{\bf h}^{2}v_{\Phi}^{2}{\rm
det}(\hat{\lambda})=m_{h}^{2}m_{\tilde{h}}^{2}$ (95)

and combining the eigenvector equations into

$\displaystyle\sin(2\omega)=\frac{2v_{\bf
h}v_{\Phi}}{m_{h}^{2}-m_{\tilde{h}}^{2}}\hat{\lambda}_{{\bf h}\Phi}$ (96)

to obtain

$\displaystyle\sin(2\omega)$
$\displaystyle=\frac{2m_{h}m_{\tilde{h}}}{m_{h}^{2}-m_{\tilde{h}}^{2}}\frac{\hat{\lambda}_{{\bf
h}\Phi}}{\sqrt{\det(\hat{\lambda})}}$ (97) $\displaystyle\sin(2\beta)$
$\displaystyle=\sqrt{C_{9}}\frac{2m_{h}m_{\tilde{h}}}{v^{2}\sqrt{\det\hat{\lambda}}}$
(98)

No obstacle prevents taking $\omega\to 0$ with $\hat{\lambda}_{{\bf h}\Phi}\to
0$, but it is evident that $\beta$ cannot be arbitrarily close to zero while
keeping $\tilde{h}$ massive and respecting unitarity. Qualitatively then, we
have a minimum attainable curvature as:

$\displaystyle\left(v^{2}R_{\varphi}\geq\frac{3m_{h}^{2}m_{\tilde{h}}^{2}}{8\pi^{2}v^{4}}\,,\,\,v^{2}R_{h}\leq-\frac{3m_{h}^{2}m_{\tilde{h}}^{2}}{8\pi^{2}v^{4}}\right)$
(99)

where we took the unitarity bound on $\hat{\lambda}$ that follows from the
4-pt amplitude for $\delta h$ and $\delta\Phi$, see e.g. [21]. This result,
being proportional to the extra state mass, yields a naive cut-off
$R=\frac{4\pi}{\Lambda^{2}}$ with inverse dependence on the new physics scale:

$\displaystyle\frac{\Lambda^{2}}{v^{2}}\sim\frac{(4\pi)^{3}}{\lambda_{\rm
SM}}\frac{v^{2}}{m_{\tilde{h}}^{2}}$ (100)

so that the largest cut-off, or the closest to the SM couplings one can get,
is attained for the lowest new physics scale. How low this scale can be while
still being able to assume an EFT applies can be estimated from the amplitude
for $W$ scattering, mediated by the singlets in the full theory

$\displaystyle-\mathcal{A}=\frac{s}{v^{2}}\left(1-c_{\beta}^{2}\frac{s}{s-m_{h}^{2}}-s_{\beta}^{2}\frac{s}{s-m_{\tilde{h}}^{2}}\right)+(s\to
t)$ (101)

The plot in fig. 4 shows the region in the curvature plane that the models
discussed in this section cover. In particular for the minimum mass of the
extra singlet we take the limit of $m_{\tilde{h}}\gtrsim 350$ GeV from [22] as
reference.

Figure 4:  Range of curvature for SMEFT and quotient theories, on the same
background as Fig. 2. Two quotient theories are plotted: the yellow region
shows curvature for the symmetric representation with
${\langle{\Phi}\rangle}\neq 0$, and the dark-grey region shows a hyperbolic
manifold (see sec. V). The black line shows SMEFT curvature; on which the
purple and red dots represent the singlet and the symmetric representation
with ${\langle{\Phi}\rangle}=0$ examples from sec. IV respectively. The outer-
most to inner-most dots are evaluated with coupling $g_{*}=1$ and heavy
singlet mass: 500 GeV, 1 TeV, 1.5 TeV, 2 TeV and 4 TeV.

##  V Manifolds

The above HEFT cases fall into the category of manifolds with a singularity,
as one can see by integrating out heavy states [7]. In contrast, one can also
have that no $O(4)$-symmetric point is present and the manifold is smooth at
every point. This section visualizes both types of manifolds, together with
those that admit a SMEFT description. Consider (higher dimensional)
cylindrical coordinates, the gauge symmetry acts rotating along the axis and
orthogonal to this rotation we have a cylindrical radial coordinate $\rho$ and
a ‘height’ $z$. Our manifolds are hypersurfaces within this 5d space
parametrized by $h$ and $\varphi^{a}$

$\displaystyle(\rho(h)u(\varphi),z(h))$ (102)

With a line element:

$\displaystyle
d\ell^{2}=\left(\left(\frac{d\rho}{dh}\right)^{2}\pm\left(\frac{dz}{dh}\right)^{2}\right)dh^{2}+\rho(h)^{2}du^{2}$
(103)

which defines the 4-d metric, where the plus sign is for Euclidean 5d space
and the minus for the metric with a $(-1,1,1,1,1)$ signature. In our basis,
eq. (14), $dh^{2}$ has unit coefficient which can always be attained by a
field redefinition. In terms of geometry, the singlet Higgs field $h$ equals
distance in field space for fixed $u$. From the equation above and our basis
it also follows that $F(h)=\rho(h)/v$ with $F(0)=1$ giving $\rho(0)=v$. For
convenience let us define $\theta=(h+h_{0})/f$ with $f$ a new physics scale.

The most symmetric manifolds are $S^{4}$, $R^{4}$ & $\mathcal{H}^{4}$ which
are parametrized in our basis as

$\displaystyle S^{4}$ $\displaystyle(f\sin(\theta)u,f\cos(\theta))$ (104)
$\displaystyle R^{4}$ $\displaystyle((h+v)u,0)$ (105)
$\displaystyle\mathcal{H}^{4}$ $\displaystyle(f\sinh(\theta)u,f\cosh(\theta))$
(106)

and yield constant (field-independent) curvature:

$\displaystyle R_{\varphi},$ $\displaystyle R_{h}$ $\displaystyle
S^{4},\mathcal{H}^{4}$ $\displaystyle\pm\frac{1}{f^{2}},$
$\displaystyle\pm\frac{1}{f^{2}}$ (107)

while the $f\to\infty$ limit yields $R^{4}$ which corresponds to the SM.
Indeed these manifolds can be described in SMEFT and correspond to Composite
Higgs Models [23] or negative curvature models [24].

###  V.1 quotient space theories with a singularity

A one-parameter deformation of the manifolds above takes us into quotient
space with a singularity at the origin:

$\displaystyle{\rm deformed\,}S^{4}$
$\displaystyle\left(fs_{\gamma\theta}u,\int
dh\sqrt{1-\gamma^{2}c^{2}_{\gamma\theta}}\right)$ (108) $\displaystyle{\rm
deformed\,}\mathcal{H}^{4}$ $\displaystyle\left(fsh_{\gamma\theta}u,\int
dh\sqrt{\gamma^{2}ch^{2}_{\gamma\theta}-1}\right)$ (109)

where $s_{\gamma\theta}=\sin(\gamma\theta)$ and the singularity is made
evident by the curvature

$\displaystyle R_{\varphi}$ $\displaystyle R_{h}$ $\displaystyle{\rm
deformed\,}S^{4}$
$\displaystyle\frac{1-\gamma^{2}}{f^{2}s^{2}_{\gamma\theta}}+\frac{\gamma^{2}}{f^{2}},$
$\displaystyle\frac{\gamma^{2}}{f^{2}}$ (110) $\displaystyle{\rm
deformed\,}\mathcal{H}^{4}$
$\displaystyle\frac{1-\gamma^{2}}{f^{2}sh^{2}_{\gamma\theta}}-\frac{\gamma^{2}}{f^{2}},$
$\displaystyle-\frac{\gamma^{2}}{f^{2}}$ (111)

since the origin, and would-be-$O(4)$ invariant point, $\theta=0$, returns
$R_{\varphi}=\infty$. This singularity is present for any $\gamma\neq\pm 1$
which seemingly presents a way to approximate the SM by sending first
$f\to\infty$ while keeping $fs_{\gamma\theta_{0}}(fsh_{\gamma\theta_{0}})=v$
constant, then $\gamma\to 1$. Indeed in this limit,
$\partial^{n}R\propto(1-\gamma^{2})$ and contributions to amplitudes of an
arbitrary number of particles cancel. Nonetheless and quite relevantly in this
limit, the singularity is just a field distance $v/\gamma$ from the vacuum
$h=0$. The model in the section above with a symmetric representation taking a
vev also belongs to the quotient theories with singularities, yet it showed
that the SM point cannot be reached. So it could be that the deformed
manifolds have no UV completion, yet from low energy we see no indication for
it. This highlights the need for a bound based purely in the EFT perspective
to comprise all possibilities.

Figure 5:  Examples of manifolds which belong in SMEFT (a), or in quotient
space (b,c,d) with the gauge symmetry action being rotation around the $z$
axis. SMEFT manifolds in (a) correspond to: Composite models (yellow), the SM
(green), and negative curvature models (blue). quotient manifolds (b,d) are
smooth, while (c) present a singularity and both (c,d) are in a class which
resembles the SM around the vacuum. For (d), part of the manifolds has been
cut out for better visualization.

###  V.2 Smooth quotient theories

On the other hand, one could have smooth manifolds in quotient space,
$\rho\neq 0\,\forall\,h$; we take here as examples a torus and a hyperbola (in
Euclidean space)

$\displaystyle{\rm torus}$
$\displaystyle((\rho_{0}+fc_{\theta})u,fs_{\theta})$ (112) $\displaystyle{\rm
hyperbola}$ $\displaystyle((\rho_{0}+fch_{\hat{\theta}})u,fsh_{\hat{\theta}})$
(113)

where $\hat{\theta}=(\hat{h}(h)+\hat{h}_{0})/f$ with
$(dh/d\hat{h})^{2}=sh_{\hat{\theta}}^{2}+ch_{\hat{\theta}}^{2}$ as follows
from our normalization in Euclidean 5d. In terms of curvature, these manifolds
give:

$\displaystyle R_{\varphi}$ $\displaystyle R_{h}$ $\displaystyle{\rm Torus}$
$\displaystyle\frac{\cos(\theta)^{2}}{v^{2}},$
$\displaystyle\frac{\cos(\theta)}{fv}$ (114) $\displaystyle{\rm Hyperbola}$
$\displaystyle\frac{ch_{\hat{\theta}}^{2}}{(ch_{\hat{\theta}}^{2}+sh_{\hat{\theta}}^{2})v^{2}},$
$\displaystyle\frac{-ch_{\hat{\theta}}}{(ch_{\hat{\theta}}^{2}+sh_{\hat{\theta}}^{2})^{2}fv}$
(115)

We see that the hyperbola does not go through the zero curvature point for any
value of $f,\theta$, always keeping a distance as the explicit model in the
previous section did. The torus however for $\theta=\pi/2$ does have both
curvatures vanish, yet by construction the manifold is not $R^{4}$. Visually,
for this point we are sitting atop of the torus and for its first two
derivatives it does resemble a plane, but its third derivative is non-
vanishing and indeed $R^{\prime}_{h}=1/f^{2}v$ which is bounded from below
given $\rho_{0}>f$ and for $\theta=\pi/2$, $v=\rho_{0}$.

This nonetheless illustrates the possibility of manifolds that do look locally
like the SM to the $n$th derivative, yet do not go through the origin. Let us
take on such set of manifolds labelled by $n$

$\displaystyle F_{(n)}(h)$
$\displaystyle=1+\frac{h}{v}+c_{n}\left(\frac{h}{v}\right)^{n}$
$\displaystyle|c_{n}|$ $\displaystyle>\frac{(n-1)^{n-1}}{n^{n}}$ (116)

The manifolds associated with these $F_{n}$ for $n=3,4,5$ are plotted in fig.
5 and they resemble a plane and hence the SM ever more accurately for
increasing $n$ around $h=0$.

##  VI Obstacles in the road to the SM

We have encountered HEFT/SMEFT quotient theories which either come from smooth
manifolds with no $O(4)$-invariant point, or manifolds which get arbitrarily
close to the would be $O(4)$-invariant point, but the point itself is
singular.

A number of UV complete theories yield quotient theories with singularities at
the origin. From working out an explicit example, we have seen that these can
only get within a finite distance of the SM point. This explicit computation
relied on knowledge of the full theory, but here we attempt to give an
argument as to why quotient theories are not a road to the SM model in purely
low energy grounds.

Let us turn to semi-classical arguments. Consider the Higgs field as sourced
by a probe particle $i$ localized in a region $\sigma_{x}$ and with a mass
$m_{i}>m_{h}$. This configuration is, of course, short lived yet for times
smaller than the decay rates one might consider such system. The
renormalizable linear realization gives an equation of motion 333The spin 0, 1
case has an extra $h/v$ times the source which we dropped,

$\displaystyle(-\Box-m^{2}_{h})h(x)=\frac{m_{i}}{v}J_{i}(x)$ (117)

where

$\displaystyle{\rm Spin\,1/2}$ $\displaystyle J_{i}$ $\displaystyle=\langle
i|\bar{\psi}\psi|i\rangle$ (118) $\displaystyle{\rm Spin\,1}$ $\displaystyle
J_{i}$ $\displaystyle=-\langle i|m_{i}V_{\mu}V^{\mu}|i\rangle$ (119)

and the particle state is

$\displaystyle|i\rangle=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\Psi(p)\frac{a_{i,p}^{\dagger}}{\sqrt{2E_{p}}}|0\rangle$
(120)

Away from the localized source the field is

$\displaystyle h(r>\sigma_{x})$
$\displaystyle=\frac{m_{i}}{v}\int\frac{d^{4}xd^{4}q}{(2\pi)^{4}}\frac{e^{iq(x-y)}\hat{J}_{i}(\vec{x})}{q^{2}-m^{2}}$
(121) $\displaystyle\simeq-\frac{m_{i}}{v}\frac{e^{-m_{h}r}}{4\pi r}$ (122)

where in the second line we assumed that the current $J_{i}$ is the same as
the probability density, as we shall see justified in the non-relativistic
limit.

Consider now the candidate quotient theories that resemble the Standard Model
to a high degree, examples given in the previous section are the functions
$F_{(n)}$ as given in (116) or the deformed $S^{4},\mathcal{H}^{4}$ theories
(110,111). The solution above should be a good first approximation certainly
for large distances $r>1/m_{h}$ where the field value is exponentially close
to the vacuum value. However, at shorter distances if our candidate theories
truly present a limit in which the SM couplings are recovered, the solution
should still be a good approximation. The field value nonetheless increases
with decreasing distance and if there is a singularity, in this SM limit, it
is just a distance $v/\gamma\simeq v$ away in field space. Conversely, for
smooth quotient theories, even if our series example $F_{(n)}$ resembles the
SM locally around the vacuum, the corrections in eq. (117) read
$1+nc_{n}(h/v)^{n}$ with $nc_{n}\sim 1$ for $n\gg 1$ and would dominate over
the SM for $h\sim v$. This is indeed the same condition for both types of
theories and yields a naive minimum distance or cut-off

$\displaystyle\frac{h(\sigma_{x}<r<m_{h}^{-1})}{v}\simeq\frac{m_{i}}{v}\frac{1}{4\pi
vr}$ (123) $\displaystyle\frac{h(r_{0})}{v}\sim 1\quad{\rm
for}\quad\frac{1}{r_{0}}\equiv\Lambda\sim 4\pi v\frac{v}{m_{i}}$ (124)

This points at a cut-off an inverse coupling factor higher than other
estimates based on pertubative unitarity. Nevertheless, quantum mechanics has
something to say about our implicit assumption $\sigma_{x}<r_{0}$. Indeed
$r_{0}\sim(m^{2}_{i}/4\pi v^{2})m_{i}^{-1}$ is smaller than the inverse mass
of a particle for perturbative couplings (which is the case for the SM) but in
order to localize the particle in a distance smaller than the inverse mass,
the uncertainty principle dictates a range of momenta that extends to the
relativistic regime. In this high energy limit, our current $J_{i}$ suffers a
relativist factor $m/E$ suppression as explicit evaluation of the matrix
elements shows when going beyond the non-relativistic approximation. For a
fermion, one has

$\displaystyle
J_{i}(x)=\int\frac{d^{3}pd^{3}k}{(2\pi)^{6}}\frac{\bar{u}(k)u(p)}{\sqrt{2E_{p}2E_{k}}}e^{i(p-k)x}\Psi^{*}(k)\Psi(p)$
(125)

which implies that the space-integral over the source $J_{i}$ is suppressed
and the field value at a distance $r>\sigma_{x}$ is

$\displaystyle\frac{h(\sigma_{x}<r)}{v}$
$\displaystyle=\frac{N(m_{i}\sigma_{x})}{4\pi
vr}\frac{m_{i}}{v^{2}}=\frac{N(\sigma_{x}m_{i})}{rm_{i}}\alpha_{i}$ (126)
$\displaystyle N(m_{i},\sigma_{x})$ $\displaystyle=\frac{\int
d^{3}k(m_{i}/E_{p})|\Psi(p)|^{2}}{\int
d^{3}k|\Psi(p)|^{2}}\quad\alpha_{i}=\frac{m_{i}^{2}}{4\pi v^{2}}$ (127)

which is the same result for spin $1/2$ and $1$. This suppression implies that
the pre-factor of $\alpha_{i}$ in the eq. (126) is at most order one, which
would then require an order one $\alpha_{i}$ to probe $(h/v)\sim 1$. Note that
this $\alpha_{i}$ will be at the edge of perturbative unitarity, although loop
corrections will be supressed by $\sim 1/(4\pi)$.

As an estimate, we take a Gaussian distribution $\Psi\sim
e^{-(p\sigma_{x})^{2}/2}$ and evaluate the potential at a distance
$r=2\sigma_{x}$ which encloses 95% of the probability density to find that
with $\alpha_{i}\sim 2$ the cut-off, or inverse distance, where we would probe
$h\sim v$ would be $r_{0}=0.6m_{i}^{-1}$,

$\displaystyle\Lambda\sim\sqrt{\frac{8\pi\sigma_{x}m_{i}}{N(\sigma_{x}m_{i})}}\Bigg{|}_{m_{i}\sigma\sim
0.3}v\simeq 2{\,\rm TeV}\,.$ (128)

The nature of EWSB and the question of whether a symmetric $O(4)$ point exits
should be independent of the introduction of our probe particle $i$, although
admittedly the fact that one would require couplings on the pertubative edge
makes the above a rough estimate.

The naive scaling from eq. (123) does, however, point towards the typical
scale for non-perturbative effects. This is indeed the natural scale for
answering non-local questions about our theory. While the detailed study of
this effect will be presented elsewhere [25] here we sketch the modifications
in a well known non-perturbative effect, sphalerons, whose energy is

$\displaystyle E_{\rm sph}\sim\frac{4\pi v}{g}$ (129)

In particular, the topological argument by Manton [26] has to do with a loop
(parametrized by $\mu$) of mappings from the sphere at spatial infinity to the
vacuum manifold, characterized by our unit vector $u$, i.e.
$u(\theta,\phi;\mu)$ and holds regardless of the Higgs singlet role.
Nonetheless the boundary conditions to find the energy of the potential
barrier have to be drastically changed in quotient theories. Indeed, the
proposed field at the top of the barrier $\mu=\pi/2$ in [26] is $({\bf
h}=h(r)u)$

$\displaystyle\bf h$
$\displaystyle=h(r)\left(\begin{array}[]{c}s_{\mu}s_{\theta}c_{\phi}\\\
s_{\mu}s_{\theta}s_{\phi}\\\ s_{\mu}c_{\mu}(c_{\theta}-1)\\\
s_{\mu}^{2}c_{\theta}+c_{\mu}^{2}\end{array}\right)$ $\displaystyle{\rm B.C.}$
$\displaystyle\left\\{\begin{array}[]{c}h(0)=0\,,\\\
h(\infty)=v\,.\end{array}\right.$ (136)

In particular, the condition at the origin, that the Higgs field go to its
symmetry preserving $O(4)$ symmetric point, is demanded to remove dependence
on angular variables of the Higgs doublet at the origin where $\theta,\phi$
are ill-defined. For quotient theories, it is clear that this does not apply
given that an $O(4)$ point is absent or singular. One can introduce a radial
dependent function on $u$ itself such that

$\displaystyle u(\theta,\phi,r\to\infty)$ $\displaystyle=u_{\infty}$
$\displaystyle u(\theta,\phi,r\to 0)$ $\displaystyle\to u_{0}$ (137)

The boundary conditions on $h$ would naively be $h^{\prime}(0)=0$. In either
case, the quotient theory effect is an order one modification which serves as
a handle to tell quotient theories apart from the Standard Model.

##  VII Summary

This work studied the quotient space HEFT/SMEFT, and the potential limits to
recover the SM other than via SMEFT with the use of a geometric formulation.
Explicit examples, which include perturbative UV complete models, can and will
be told apart from the SMEFT case by future experiments via projection of
measurements on the curvature plane defined from the $WW$ scattering and
$WW\to hh$ amplitudes (see fig. 4). These examples of quotient space
HEFT/SMEFT theories do not offer a limit to recover the SM and possess a
finite cut-off. In contrast to these, quotient theories were formulated in
sec. V which resemble the SM amplitudes for arbitrary precision and number of
particles. While these theories look like the SM model around the vacuum, at a
Higgs-singlet-distance of $\sim v$ they reveal their quotient space nature.
Making use of semi-classical arguments to displace the Higgs field by $\sim
v$, we find an argument for general theories in quotient space to be
distinguishable from the SM when probing the theory at an energy (inverse
distance) of at most $4\pi v/g_{\rm SM}$. Our discussion applies to quotient
theories both with and without singularities (non-analyticities). The most
pressing outstanding question is the characterization of experimental
signatures that follow from the semi-classical arguments given here.

##  VIII Acknowledgements

R. A. and M.W. are supported by the STFC under Grant No. ST/P001246/1.

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