(4.59)

and we require this relation in the following, for the consistency of seeking
theories constructible within the ALT shift. With this condition, we obtain
the contact term as

$\displaystyle\left[A_{WWh^{++}h^{--}}^{(\lambda_{1}\lambda_{2})}\right]_{\mathrm{contact}}$
$\displaystyle=-\frac{\left(g_{WWh}^{(2)}\right)^{2}}{4m_{W}^{4}}\langle\mathbf{12}\rangle[\mathbf{21}].$
(4.60)

Again, we learn the same lesson as the $WW\gamma h^{--}$ scattering; the on-
shell method automatically reproduces the relations between three- and four-
point amplitudes as we have seen throughout this paper, but if the gauge
symmetry relates three-point couplings that are seemingly different from the
bottom-up point of view, we need to impose them by hand in the on-shell
method.

####  $WWtt$ amplitude

We obtain similar relation as in Eq. (4.29) for this case since there is no
additional $tth^{++}$ or $tth^{--}$ three-point amplitude. We find the
relation

$\displaystyle{m_{t}}g_{tbW}^{2}=-g_{htt}^{(0)}g_{WWh}^{(0)}$ (4.61)

#### Coupling relations and $\rho$ parameter

By requiring the cancellations of the $\mathcal{O}(E^{2})$ term, we have
obtained

$\displaystyle 0$
$\displaystyle=4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}-\left(g_{WWh}^{(0)}\right)^{2}+\left(g_{WWh}^{(2)}\right)^{2},$
(4.62) $\displaystyle 0$
$\displaystyle=g_{WWh}^{(0)}g_{ZZh}^{(0)}-\frac{g_{WWZ}^{2}m_{Z}^{4}}{m_{W}^{2}},$
(4.63) $\displaystyle 0$
$\displaystyle=g_{ZZh}^{(0)}m_{W}^{2}-g_{WWh}^{(0)}m_{Z}^{2},$ (4.64)
$\displaystyle 0$
$\displaystyle=g_{WWZ}g_{WWh}^{(2)}\left(4-\frac{m_{Z}^{2}}{m_{W}^{2}}\right)+2g_{WWh}^{(2)}g_{hhZ}^{(2)}.$
(4.65)

Moreover, we have seen that we need to impose additional relations among
three-point couplings to satisfy the Ward identity of the four-point
amplitudes. These are given by

$\displaystyle 0$ $\displaystyle=g_{hh\gamma}^{(2)}+2g_{WW\gamma},$ (4.66)
$\displaystyle 0$
$\displaystyle=4\left(g_{WWZ}g_{hhZ}^{(2)}+g_{WW\gamma}g_{hh\gamma}^{(2)}\right)m_{W}^{2}+\left(g_{WWh}^{(2)}\right)^{2}.$
(4.67)

These relations fix the couplings between the Higgs and the gauge bosons as

$\displaystyle g_{WWh}^{(0)}=g_{WWZ}m_{Z},\quad
g_{ZZh}^{(0)}=\frac{g_{WWZ}m_{Z}^{3}}{m_{W}^{2}},\quad\left(g_{WWh}^{(2)}\right)^{2}=4m_{W}^{2}\left[g_{WWZ}^{2}\left(2-\frac{m_{Z}^{2}}{m_{W}^{2}}\right)+2g_{WW\gamma}^{2}\right],$
(4.68)

and

$\displaystyle g_{hh\gamma}^{(2)}=-2g_{WW\gamma},\quad
g_{hhZ}^{(2)}=-g_{WWZ}\left(2-\frac{m_{Z}^{2}}{m_{W}^{2}}\right).$ (4.69)

In addition, the $\rho$ parameter is fixed as

$\displaystyle\rho=\frac{g^{2}m_{W}^{2}}{g_{WWZ}^{2}m_{Z}^{2}}=\frac{1}{2}.$
(4.70)

Therefore, our calculation suggests that the doubly charged and neutral
scalars for this unitarization are in the SU(2) triplet.

###  4.3 Other UV completion

So far we have studied UV completion of the EW theory by adding scalar
particles. However, UV completion by scalar particles is not the only option
(forgetting about the experimental results) and we can add e.g. vector
particles to improve the high-energy behavior of the amplitudes. For instance,
we can add a vector particle, $Z^{\prime}$, which couples to the $W$-boson
with the coupling $g_{WWZ^{\prime}}$. Then, in the same way as Sec. 3.2, we
obtain the high-energy behavior of the $WWWW$ scattering as

$\displaystyle\left.A_{WWWW}^{(LLLL)}\right|_{Z,\gamma}$
$\displaystyle=-\frac{s+u}{4m_{W}^{4}}(4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}-3g_{WWZ^{\prime}}^{2}m_{Z^{\prime}}^{2})+\mathcal{O}(E^{0}),$
(4.71)

where $m_{Z^{\prime}}$ is the mass of the $Z^{\prime}$-boson, and we now have
$g^{2}=g_{WWZ}^{2}+g_{WW\gamma}^{2}+g_{WWZ^{\prime}}^{2}$. Therefore
$Z^{\prime}$ can unitarize this specific channel if

$\displaystyle 0$
$\displaystyle=4g^{2}m_{W}^{2}-3g_{WWZ}^{2}m_{Z}^{2}-3g_{WWZ^{\prime}}^{2}m_{Z^{\prime}}^{2}.$
(4.72)

In this case, we need to study amplitudes with external $Z^{\prime}$, and it
is known that in the end, we need an inifnite tower of spin-1 particles, as
discussed in [38, 39]. Instead, we can try UV completion by higher-spin
particles such as spin-2, 3, and so on. The on-shell method is suitable for
this purpose since it allows us to write down the amplitudes with arbitrary
spins, at least in principle. We leave a detailed study on possible UV
completions of the EW theory, including higher spin particles, in the on-shell
formalism as a future work.

##  5 Conclusion and discussion

Figure 2: Summary of recursive construction of EW theory.

On-shell recursion relations are a key component of the amplitude program.
They have proven to be very powerful and efficient in constructing $n$-point
amplitudes of massless particles, compared to the usual Feynman diagram
approach. With the progress of the little-group covariant massive spinor-
helicity formalism [17], a natural direction to pursue is to build massive
higher-point amplitudes from lower-point ones. However, without an appropriate
momentum shift, such construction suffers from contact term ambiguity because
of the non-vanishing boundary terms.

To address this issue, we have proposed the ALT shift, which is applicable in
the same way for massive and massless theories, in [20] that generates no
boundary term in massive QED and electroweak theories, or more generally
renormalizable theories with spin $\leq 1$. In addition, we have highlighted
in this paper another remarkable feature of the ALT shift, which is its
connection to Ward identity. Under the ALT shift and imposing the Ward
identity of the massless theory from the beginning, one not only pins down the
(spontaneously broken) gauge theory but also guarantees the on-shell
constructibility. This connection is closely related to the fact that the
shifted momentum is proportional to the transverse polarization vector, which
is a feature of the proposed ALT shift. As a consequence, the contact terms
coming from gauge symmetry naturally arise under the ALT shift.

Numerous examples of four-point amplitudes are presented in this work. We
schematically summarize them in Figure 2. First, without the introduction of
the Higgs boson, we have recovered all the contact terms in electroweak theory
that cancel the fastest energy growths of the factorizable part. The Standard
Model Higgs scalar can then be incorporated by the unitarity requirement. We
have also successfully reproduced the Higgs coupling to gauge bosons from the
bottom-up using the on-shell constructive method. Beyond the Standard Model,
our ALT shift can be applied to other renormalizable UV completions, and we
have showcased the one with a Higgs triplet. Here again, we have correctly
reproduced the relations between couplings by requiring unitarity and
occasionally imposing gauge symmetry via demanding Ward identity.

The method developed here and in our previous work [20] may lead to many
interesting applications. Apart from the Higgs doublet and triplet, other UV
completions can be studied with the ALT shift in the on-shell framework. It
may also be used to study e.g. EW radiations within the SM at high energy
relevant for future colliders (see e.g. [40] as a recent related work). Beyond
the context of the EW theory, one may also attempt to generalize ALT shift for
massive higher spin amplitude, which can be applied to analyze Compton
amplitude of general spinning compact objects [41, 42, 43, 44] as well as
study its generalization to higher dimensional massive spinors [45]. The ALT
shift may be useful for understanding the general properties of tree-level
massive scattering amplitudes. For instance, the color-kinematic duality and
double copy relations of massless gauge theory and gravity are proven at tree
level with an arbitrary number of external particles [46, 47, 48, 49, 50, 51,
52, 53]. Here the on-shell recursion relation plays a crucial role as it
allows us to study the general properties of the amplitudes based on
mathematical induction. Therefore, we may envision that the ALT shift will
enable us to extend our understanding of these properties to full massive
amplitudes.

## Acknowledgements

We thank N. Christensen, B. Feng, and Y. Shadmi for helpful discussion. This
work is supported in part by U.S. Department of Energy Grant No. DESC0011842.
Y.E. and Z.L. acknowledge the Aspen Center for Physics, where the part of the
work was performed, which is supported by National Science Foundation grant
PHY-2210452. The (Feynman-like) diagrams in this paper are drawn by TikZ-
Feynman [54].

##  Appendix A Massive spinor formalism

In this appendix, we summarize the massive spinor formalism and conventions
used in this paper. We express the four-momentum $p^{\mu}$ in the spinor space
as

$\displaystyle
p_{a\dot{a}}=p_{\mu}\left[\sigma^{\mu}\right]_{a\dot{a}}=\begin{pmatrix}p^{0}-p^{3}&-p^{1}+ip^{2}\\\
-p^{1}-ip^{2}&p^{0}+p^{3}\end{pmatrix},\quad
p^{\dot{a}a}=p_{\mu}\left[\bar{\sigma}^{\mu}\right]^{\dot{a}a}=\begin{pmatrix}p^{0}+p^{3}&p^{1}-ip^{2}\\\
p^{1}+ip^{2}&p^{0}-p^{3}\end{pmatrix}.$ (A.1)

The determinant is given by

$\displaystyle\mathrm{det}\left[p_{a\dot{a}}\right]=\mathrm{det}\left[p^{\dot{a}a}\right]=m^{2},$
(A.2)

where $m$ is the mass of the particle. We also note that

$\displaystyle p_{i}^{\dot{a}a}(p_{j})_{a\dot{a}}=2p_{i}\cdot
p_{j},\quad\epsilon^{ab}\epsilon^{\dot{a}\dot{b}}p_{b\dot{b}}=(p^{\dot{a}a})^{T}.$
(A.3)

#### Massive spinor

In general, if $m_{i}\neq 0$ for a particle $i$, we can write the momentum
$p_{i}$ as [17]

$\displaystyle[p_{i}]_{a\dot{a}}=|i\rangle_{a}^{I}[i|_{\dot{a}I},\quad I=1,2.$
(A.4)

Correspondingly we have

$\displaystyle[p_{i}]^{\dot{a}a}=|i]_{I}^{\dot{a}}\langle i|^{aI}.$ (A.5)

The little group is the transformation that keeps the momentum unchanged.
Therefore in this expression, the little group acts on the indices $I$ and is
SU(2) for real momentum. By taking its determinant we obtain

$\displaystyle\mathrm{det}[|i\rangle_{a}^{I}]\times\mathrm{det}[[i|_{\dot{a}I}]=m_{i}^{2}.$
(A.6)

We fix the relative normalization as

$\displaystyle\mathrm{det}[|i\rangle_{a}^{I}]=\mathrm{det}[[i|_{\dot{a}I}]=m_{i}.$
(A.7)

We raise and lower the indices $I$, $J$ by the anti-symmetric tensor
$\epsilon^{IJ}$ and $\epsilon_{IJ}$ with $\epsilon^{12}=-\epsilon_{12}=1$ as

$\displaystyle\lambda^{I}=\epsilon^{IJ}\lambda_{J},\quad\lambda_{I}=\epsilon_{IJ}\lambda^{J}.$
(A.8)

Since $\epsilon_{IJ}|i\rangle^{I}_{a}|i\rangle^{J}_{b}$ is anti-symmetric with
respect to $a$ and $b$, it should be proportional to $\epsilon_{ab}$, and the
same holds for $\epsilon^{IJ}[i|_{\dot{a}I}[i|_{\dot{b}J}$. The determinant
then fixes the overall normalization as

$\displaystyle\epsilon_{IJ}|i\rangle^{I}_{a}|i\rangle^{J}_{b}=m_{i}\epsilon_{ab},\quad\epsilon^{IJ}[i|_{\dot{a}I}[i|_{\dot{b}J}=-m_{i}\epsilon_{\dot{a}\dot{b}},$
(A.9)

and similarly

$\displaystyle\langle
i^{I}i^{J}\rangle=-m_{i}\epsilon^{IJ},\quad[i_{I}i_{J}]=-m_{i}\epsilon_{IJ},$
(A.10)

where we denote

$\displaystyle\langle ij\rangle=\langle
i|^{a}|j\rangle_{a}=\epsilon^{ab}|i\rangle_{b}|j\rangle_{a},\quad[ij]=[i|_{\dot{a}}|j]^{\dot{a}}=\epsilon_{\dot{a}\dot{b}}|j]^{\dot{a}}|i]^{\dot{b}}.$
(A.11)

From these relations, we obtain

$\displaystyle\langle
i|^{aI}[p_{i}]_{a\dot{a}}=-m_{i}[i|_{\dot{a}}^{I},\quad[p_{i}]_{a\dot{a}}|i]^{\dot{a}I}=m_{i}|i\rangle_{a}^{I},$
(A.12)

which is nothing but the Dirac equation. In the same way, we obtain

$\displaystyle
p^{\dot{a}a}|i\rangle^{I}_{a}=m_{i}|i]^{\dot{a}I},\quad[i|_{\dot{a}}^{I}p^{\dot{a}a}=-m_{i}\langle
i|^{aI}.$ (A.13)

We use the bold notation introduced in [17] to express spinors with floating
little-group indices, when no ambiguity would arise with the abbreviation.

#### Massless spinor

If $m=0$, we need only one pair of spinors to express the momentum as

$\displaystyle p_{a\dot{a}}=|p\rangle_{a}[p|_{\dot{a}}.$ (A.14)

In this case the little group is U(1) for real momentum. For complex momentum,
one can extend this to the little-group scaling.

#### Dirac wavefunctions

The Dirac wavefunctions, $u$ and $v$, satisfy the Dirac equation

$\displaystyle\left[\not{p}-m\right]u(p)=0,\quad\left[\not{p}+m\right]v(p)=0.$
(A.15)

One can easily see that this is solved as

$\displaystyle u^{I}(p_{i})=\begin{pmatrix}|i\rangle_{a}^{I}\\\
|i]^{\dot{a}I}\end{pmatrix},\quad
v^{I}(p_{i})=\begin{pmatrix}|i\rangle_{a}^{I}\\\
-|i]^{\dot{a}I}\end{pmatrix}.$ (A.16)

Note that $u(-p)$ and $v(p)$ satisfy the same equation, and hence

$\displaystyle u^{I}(-p_{i})=\begin{pmatrix}|i\rangle_{a}^{I}\\\
-|i]^{\dot{a}I}\end{pmatrix},\quad
v^{I}(-p_{i})=\begin{pmatrix}|i\rangle_{a}^{I}\\\
|i]^{\dot{a}I}\end{pmatrix}.$ (A.17)

In other words, we have the relations

$\displaystyle|-i\rangle^{I}=|i\rangle^{I},\quad|-i]^{I}=-|i]^{I}.$ (A.18)

In the same way we can show that

$\displaystyle\bar{u}^{I}(p)=\begin{pmatrix}-\langle
i|^{aI}&[i|_{\dot{a}}^{I}\end{pmatrix},\quad\bar{v}^{I}(p)=\begin{pmatrix}\langle
i|^{aI}&[i|_{\dot{a}}^{I}\end{pmatrix}.$ (A.19)

They satisfy

$\displaystyle\epsilon_{IJ}u^{I}(p)\bar{u}^{J}(p)=\not{p}+m,\quad\epsilon_{IJ}v^{I}(p)\bar{v}^{J}(p)=\not{p}-m.$
(A.20)

#### Helicity basis and polarization vector

We work in the helicity basis to define the ALT shift in the main text, and
here we summarize its basic properties. The solution of the Dirac equation is
given by

$\displaystyle u^{I}(p)=\begin{pmatrix}\sqrt{p\cdot\sigma}\xi^{I}\\\
\sqrt{p\cdot\bar{\sigma}}\xi^{I}\end{pmatrix}.$ (A.21)

We can choose $\xi^{I}$ as the eigenstates of the helicity operator $\hat{h}$,
given by

$\displaystyle\hat{h}=\frac{\hat{p}\cdot\vec{\sigma}}{2}=\frac{1}{2}\begin{pmatrix}\cos\theta&\sin\theta
e^{-i\phi}\\\ \sin\theta e^{i\phi}&-\cos\theta\end{pmatrix},$ (A.22)

where $(\theta,\phi)$ denotes the solid angle of the momentum. The eigenvalues
are $\pm 1/2$ and the eigenstates are

$\displaystyle\xi^{+}=\begin{pmatrix}c\\\
s\end{pmatrix},\quad\xi^{-}=\begin{pmatrix}-s^{*}\\\
c\end{pmatrix},\quad\mathrm{where}~{}~{}c=\cos\frac{\theta}{2},~{}~{}s=e^{i\phi}\sin\frac{\theta}{2},$
(A.23)

which satisfy $\hat{h}\xi^{\pm}=\pm\xi^{\pm}/2$. As we saw before, the angle
and square brackets are given by

$\displaystyle|p\rangle_{a}^{I}=\sqrt{p\cdot\sigma}\xi^{I},\quad|{p}]^{\dot{a}I}=\sqrt{p\cdot\bar{\sigma}}\xi^{I}.$
(A.24)

With this identification, we obtain

$\displaystyle|{p}\rangle_{a}^{I}=\sqrt{E-p}\begin{pmatrix}c\\\
s\end{pmatrix}\delta^{I}_{+}+\sqrt{E+p}\begin{pmatrix}-s^{*}\\\
c\end{pmatrix}\delta^{I}_{-},$ (A.25)
$\displaystyle[{p}|_{\dot{a}}^{I}=\sqrt{E+p}\begin{pmatrix}-s&c\end{pmatrix}\delta^{I}_{+}-\sqrt{E-p}\begin{pmatrix}c&s^{*}\end{pmatrix}\delta^{I}_{-}.$
(A.26)

This indeed satisfies

$\displaystyle\epsilon_{IJ}|{p}\rangle_{a}^{I}[{p}|_{\dot{a}}^{J}=p_{a\dot{a}},$
(A.27)

with $\epsilon_{+-}=-1$ (or $``+"=1$ and $``-"=2$). We therefore define

$\displaystyle|i\rangle_{a}=\sqrt{E_{i}+p_{i}}\begin{pmatrix}-s_{i}^{*}\\\
c_{i}\end{pmatrix},\quad[i|_{\dot{a}}=\sqrt{E_{i}+p_{i}}\begin{pmatrix}-s_{i}&c_{i}\end{pmatrix},$
(A.28)
$\displaystyle|\eta_{i}\rangle_{a}=\sqrt{E_{i}-p_{i}}\begin{pmatrix}c_{i}\\\
s_{i}\end{pmatrix},\quad[\eta_{i}|_{\dot{a}}=-\sqrt{E_{i}-p_{i}}\begin{pmatrix}c_{i}&s_{i}^{*}\end{pmatrix},$
(A.29)

which satisfy

$\displaystyle\langle i\eta_{i}\rangle=[i\eta_{i}]=m_{i}.$ (A.30)

The momentum is written as

$\displaystyle(p_{i})_{a\dot{a}}=|i\rangle_{a}[i|_{\dot{a}}-|\eta_{i}\rangle_{a}[\eta_{i}|_{\dot{a}},$
(A.31)

where

$\displaystyle|{i}\rangle^{+}=|\eta_{i}\rangle,~{}~{}|{i}\rangle^{-}=|i\rangle,~{}~{}[{i}|^{+}=[i|,~{}~{}[{i}|^{-}=[\eta_{i}|.$
(A.32)

The polarization vectors of the gauge bosons are defined in the helicity basis
as

$\displaystyle\epsilon_{i}^{(+)}=\sqrt{2}\frac{|\eta_{i}\rangle[i|}{m_{i}},\quad\epsilon_{i}^{(-)}=-\sqrt{2}\frac{|i\rangle[\eta_{i}|}{m_{i}},\quad\epsilon_{i}^{(L)}=\frac{|i\rangle[i|+|\eta_{i}\rangle[\eta_{i}|}{m_{i}}.$
(A.33)

The norms are given by

$\displaystyle\epsilon^{(+)}_{i}\cdot\epsilon^{(-)}_{i}=\epsilon^{(L)}_{i}\cdot\epsilon^{(L)}_{i}=-1.$
(A.34)

Note that, in a general basis, the polarization vector has two little group
indices symmetrized as

$\displaystyle[\epsilon_{i}^{IJ}]_{a\dot{a}}=\frac{\sqrt{2}}{m_{i}}|{i}\rangle_{a}^{\\{I}[{i}|_{\dot{a}}^{J\\}}.$
(A.35)

The above three polarization vectors correspond to

$\displaystyle\epsilon_{i}^{++}=\epsilon_{i}^{(+)},\quad\epsilon_{i}^{--}=-\epsilon_{i}^{(-)},\quad\epsilon_{i}^{\\{+,-\\}}=\epsilon_{i}^{(L)},$
(A.36)

where the symmetrization, $\\{+,-\\}$, is understood to involve the factor
$1/\sqrt{2}$ in its normalization. For a massless particle such as the photon,
we can define the limit

$\displaystyle\lim_{m_{i}\to
0}\frac{|\eta_{i}\rangle}{m_{i}}=|\xi_{i}\rangle,\quad\lim_{m_{i}\to
0}\frac{|\eta_{i}]}{m_{i}}=|\xi_{i}],$ (A.37)

where $|\xi_{i}\rangle$ and $|\xi_{i}]$ are the arbitrary reference spinors.
With them, the polarization vector is given by

$\displaystyle\epsilon_{i}^{(+)}=\sqrt{2}\frac{|\xi_{i}\rangle[i|}{\langle
i\xi_{i}\rangle},\quad\epsilon_{i}^{(-)}=-\sqrt{2}\frac{|i\rangle[\xi_{i}|}{[i\xi_{i}]}.$
(A.38)

Note that the polarization vectors satisfy the completeness relation,

$\displaystyle\sum_{\lambda=\pm,L}[\epsilon_{i}^{(\lambda)}]_{a\dot{a}}[\epsilon_{i}^{(-\lambda)}]_{b\dot{b}}=-2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}+\frac{[p_{i}]_{a\dot{a}}[p_{i}]_{b\dot{b}}}{m_{i}^{2}},$
(A.39)

for the massive case, and

$\displaystyle\sum_{\lambda=\pm}[\epsilon_{i}^{(\lambda)}]_{a\dot{a}}[\epsilon_{i}^{(-\lambda)}]_{b\dot{b}}=-2\epsilon_{ab}\epsilon_{\dot{a}\dot{b}}+\frac{[p_{i}]_{a\dot{a}}[\bar{p}_{i}]_{b\dot{b}}+[\bar{p}_{i}]_{a\dot{a}}[{p}_{i}]_{b\dot{b}}}{p_{i}\cdot\bar{p}_{i}},\quad\bar{p}_{i}=|\xi_{i}\rangle[\xi_{i}|,$
(A.40)

for the massless case, respectively.

Going back to the vector notation, these are given by

$\displaystyle\sum_{\lambda=\pm,L}[\epsilon_{i}^{(\lambda)}]_{\mu}[\epsilon_{i}^{(-\lambda)}]_{\nu}=-\eta_{\mu\nu}+\frac{[p_{i}]_{\mu}[p_{i}]_{\nu}}{m_{i}^{2}},$
(A.41)

for the massive case, and

$\displaystyle\sum_{\lambda=\pm}[\epsilon_{i}^{(\lambda)}]_{\mu}[\epsilon_{i}^{(-\lambda)}]_{\nu}=-\eta_{\mu\nu}+\frac{[p_{i}]_{\mu}[\bar{p}_{i}]_{\nu}+[\bar{p}_{i}]_{\mu}[{p}_{i}]_{\nu}}{p_{i}\cdot\bar{p}_{i}},$
(A.42)

for the massless case.

Polarization vectors and fermion wavefunctions are eigenstates of spin
operator $J_{k}^{0}$ where spin-axis is aligned along momentum $k^{\mu}$, and
they are related by spin-raising and lowering operator $J^{\pm}_{k}$. These
operators are the three generators of the little group SU(2) for real
momentum,

$\displaystyle J_{k}^{0}$ $\displaystyle=\frac{1}{2}\left(-\lvert
k\rangle_{\alpha}\dfrac{\partial}{\partial\lvert
k\rangle_{\alpha}}+\lvert\eta_{k}\rangle_{\alpha}\dfrac{\partial}{\partial\lvert\eta_{k}\rangle_{\alpha}}+[k\lvert_{\dot{\alpha}}\dfrac{\partial}{\partial[k\lvert_{\dot{\alpha}}}-[\eta_{k}\lvert_{\dot{\alpha}}\dfrac{\partial}{\partial[\eta_{k}\lvert_{\dot{\alpha}}}\right),$
(A.43) $\displaystyle J_{k}^{-}$ $\displaystyle=-\left(\lvert
k\rangle_{\alpha}\dfrac{\partial}{\partial\lvert\eta_{k}\rangle_{\alpha}}+[\eta_{k}\lvert_{\dot{\alpha}}\dfrac{\partial}{\partial[k\lvert_{\dot{\alpha}}}\right),\quad
J_{k}^{+}=\left(\lvert\eta_{k}\rangle_{\alpha}\dfrac{\partial}{\partial\lvert
k\rangle_{\alpha}}+[k\lvert_{\dot{\alpha}}\dfrac{\partial}{\partial[\eta_{k}\lvert_{\dot{\alpha}}}\right).$
(A.44)

The commutation relations among these differential operators are

$\displaystyle[J_{k}^{0},J_{k}^{\pm}]=\pm
J_{k}^{\pm},\qquad[J_{k}^{+},J_{k}^{-}]=-2J_{k}^{0}$ (A.45)

and so we observe that $J^{\pm}_{k}$ serve as the spin raising and lowering
operators for the eigenstate of $J_{k}^{0}$. We can apply spin operators to
the polarization tensors as

$\displaystyle J_{k}^{0}\epsilon_{k}^{(\pm)}$
$\displaystyle=\pm\epsilon_{k}^{(\pm)},\quad
J_{k}^{0}\epsilon_{k}^{(L)}=0,\quad
J^{\mp}_{k}\epsilon_{k}^{(\pm)}=-\frac{1}{\sqrt{2}}\epsilon_{k}^{(L)},\quad
J_{k}^{\pm}\epsilon_{k}^{(L)}=\sqrt{2}\epsilon_{k}^{(\pm)},\quad
J^{\pm}_{k}p_{k}=0.$ (A.46)

##  Appendix B On-shell construction of electroweak theory in vector
representation

For readers’ convenience, in this appendix, we express the on-shell
computation of the EW gauge boson scatterings in Sec. 3 in the vector basis.

###  B.1 Three-point amplitude

The three-point amplitudes involving only the gauge bosons, defined in Sec.
3.1, are expressed in the vector basis as

$\displaystyle A_{WWZ}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=g_{WWZ}\epsilon_{\alpha}^{(\lambda_{1})}(p_{1})\epsilon_{\beta}^{(\lambda_{2})}(p_{2})V^{\alpha\beta\mu}(p_{1},p_{2},p_{3})\epsilon_{\mu}^{(\lambda_{3})}(p_{3}),$
(B.1) $\displaystyle A_{WW\gamma}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=g_{WW\gamma}\epsilon_{\alpha}^{(\lambda_{1})}(p_{1})\epsilon_{\beta}^{(\lambda_{2})}(p_{2})V^{\alpha\beta\mu}(p_{1},p_{2},p_{3})\epsilon_{\mu}^{(\lambda_{3})}(p_{3}),$
(B.2)

where $p_{3}=-p_{1}-p_{2}$ and

$\displaystyle
V_{\alpha\beta\mu}(p_{1},p_{2},p_{3})=\eta_{\alpha\beta}(p_{1}-p_{2})_{\mu}+\eta_{\beta\mu}(p_{2}-p_{3})_{\alpha}+\eta_{\mu\alpha}(p_{3}-p_{1})_{\beta}.$
(B.3)

The Ward identity (3.3) is given by

$\displaystyle\left.A_{WWZ}\right|_{\epsilon_{k}\to
p_{k}}=-g_{WWZ}(m_{i}^{2}-m_{j}^{2})\epsilon_{i}\cdot\epsilon_{j},\quad\left.A_{WW\gamma}\right|_{\epsilon_{3}\to
p_{3}}=0.$ (B.4)

###  B.2 $WWWW$ amplitude

By using the completeness relations (A.41) and (A.42), Eq. (3.8) is given by

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=-\sum_{I=Z,\gamma}\sum_{i=2,4}\frac{g_{WWI}^{2}}{p_{1i}^{2}-m_{I}^{2}}\frac{1}{{z^{+}_{1i}-z^{-}_{1i}}}$
$\displaystyle\times\epsilon_{1}^{\alpha}\epsilon_{i}^{\beta}\left[{z^{+}_{1i}}\left[{V_{\alpha\beta}}^{\mu}(\hat{p}_{1},\hat{p}_{i},-\hat{p}_{1i})\times
V_{\gamma\delta\mu}(\hat{p}_{3},\hat{p}_{j},-\hat{p}_{3j})\right]_{z=z^{-}_{1i}}-(z_{1i}^{+}\leftrightarrow
z_{1i}^{-})\right]\epsilon_{3}^{\gamma}\epsilon_{j}^{\delta}.$ (B.5)

The product of $V$ contains up to the quadratic terms in $z$, and we obtain

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{I=Z,\gamma}\sum_{i=2,4}\frac{g_{WWI}^{2}}{p_{1i}^{2}-m_{I}^{2}}\epsilon_{1}^{\alpha}\epsilon_{i}^{\beta}{V_{\alpha\beta}}^{\mu}({p}_{1},{p}_{i},-{p}_{1i})\times
V_{\gamma\delta\mu}({p}_{3},{p}_{j},{p}_{1i})\epsilon_{3}^{\gamma}\epsilon_{j}^{\delta}$
$\displaystyle+\sum_{i=2,4}\frac{g^{2}}{2q_{1}\cdot
q_{i}}\epsilon_{1}^{\alpha}\epsilon_{i}^{\beta}{V_{\alpha\beta}}^{\mu}(q_{1},q_{i},-q_{1i})\times
V_{\gamma\delta\mu}(q_{3},q_{j},-q_{3j})\epsilon_{3}^{\gamma}\epsilon_{j}^{\delta},$
(B.6)

which is equivalent to Eq. (3.14), where the first term is from the
$z^{0}$-term while the second term is from the $z^{2}$-term. To simplify the
second term, we remind ourselves that we take the particle 1 as transverse,
$\lambda_{1}=\pm$, and hence
$\epsilon_{1}^{\alpha}={q_{1}^{\alpha}}/{c_{1}m_{1}}$. We then obtain

$\displaystyle\sum_{i=2,4}\frac{1}{2q_{1}\cdot
q_{i}}\epsilon_{1}^{\alpha}\epsilon_{i}^{\beta}{V_{\alpha\beta}}^{\mu}(q_{1},q_{i},-q_{1i})\times
V_{\gamma\delta\mu}(q_{3},q_{j},-q_{3j})\epsilon_{3}^{\gamma}\epsilon_{j}^{\delta}=-\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[2\eta^{\alpha\gamma}\eta^{\beta\delta}-\eta^{\alpha\beta}\eta^{\gamma\delta}-\eta^{\alpha\delta}\eta^{\beta\gamma}\right]\epsilon_{3}^{\gamma}\epsilon_{4}^{\delta}.$
(B.7)

Therefore, combining all the terms, we obtain

$\displaystyle A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-\sum_{i=2,4}\left[\frac{g_{WWZ}^{2}}{p_{1i}^{2}-m_{Z}^{2}}+\frac{g_{WW\gamma}^{2}}{p_{1i}^{2}}\right]\epsilon_{1}^{\alpha}\epsilon_{i}^{\beta}{V_{\alpha\beta}}^{\mu}({p}_{1},{p}_{i},-{p}_{1i})V_{\gamma\delta\mu}({p}_{3},{p}_{j},{p}_{1i})\epsilon_{3}^{\gamma}\epsilon_{j}^{\delta}$
$\displaystyle-g^{2}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[2\eta^{\alpha\gamma}\eta^{\beta\delta}-\eta^{\alpha\beta}\eta^{\gamma\delta}-\eta^{\alpha\delta}\eta^{\beta\gamma}\right]\epsilon_{3}^{\gamma}\epsilon_{4}^{\delta},$
(B.8)

which is equivalent to Eq. (3.20), expressed in the vector basis.

###  B.3 $WWZZ$ amplitude

We use the completeness relation (A.41) to write Eq. (3.25) as

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=-g_{WWZ}^{2}\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\frac{1}{{z^{+}_{1i}-z^{-}_{1i}}}$
$\displaystyle\times\epsilon_{1}^{\alpha}\epsilon_{i}^{\gamma}\left[{z^{+}_{1i}}\left[V_{\alpha\mu\gamma}(\hat{p}_{1},-\hat{p}_{1i},\hat{p}_{i})\left(\eta^{\mu\nu}-\frac{\hat{p}_{1i}^{\mu}\hat{p}_{1i}^{\nu}}{m_{W}^{2}}\right)V_{\nu\beta\delta}(-\hat{p}_{2j},\hat{p}_{2},\hat{p}_{j})\right]_{z=z^{-}_{1i}}-(z_{1i}^{+}\leftrightarrow
z_{1i}^{-})\right]\epsilon_{2}^{\beta}\epsilon_{j}^{\delta}.$ (B.9)

With the help of the Ward identity of the three-point amplitudes, we rewrite
it as

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-
g_{WWZ}^{2}\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\epsilon_{1}^{\alpha}\epsilon_{i}^{\gamma}\left[V_{\alpha\mu\gamma}({p}_{1},-{p}_{1i},{p}_{i})\times\left(\eta^{\mu\nu}-\frac{{p}_{1i}^{\mu}{p}_{1i}^{\nu}}{m_{W}^{2}}\right)V_{\nu\beta\delta}(-{p}_{2j},{p}_{2},{p}_{j})\right]\epsilon_{2}^{\beta}\epsilon_{j}^{\delta}$
$\displaystyle+\sum_{i=3,4}\frac{g_{WWZ}^{2}}{2q_{1}\cdot
q_{i}}\epsilon_{1}^{\alpha}\epsilon_{i}^{\gamma}V_{\alpha\mu\gamma}(q_{1},-q_{1i},q_{i}){V^{\mu}}_{\beta\delta}(-q_{2j},q_{2},q_{j})\epsilon_{2}^{\beta}\epsilon_{j}^{\delta},$
(B.10)

which corresponds to Eq. (3.27), where again the first term comes from $z^{0}$
while the second term from $z^{2}$. In the same way as the $WWWW$ scattering,
we can simplify the second term by assuming that $\lambda_{1}=\pm$, and we
obtain

$\displaystyle A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=$
$\displaystyle-
g_{WWZ}^{2}\sum_{i=3,4}\frac{1}{p_{1i}^{2}-m_{W}^{2}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[V_{\alpha\mu\gamma}({p}_{1},-{p}_{1i},{p}_{i})\times\left(\eta^{\mu\nu}-\frac{{p}_{1i}^{\mu}{p}_{1i}^{\nu}}{m_{W}^{2}}\right)V_{\nu\beta\delta}(-{p}_{2j},{p}_{2},{p}_{j})\right]\epsilon_{i}^{\gamma}\epsilon_{j}^{\delta}$
$\displaystyle+g_{WWZ}^{2}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[2\eta_{\alpha\beta}\eta_{\gamma\delta}-\eta_{\alpha\gamma}\eta_{\beta\delta}-\eta_{\alpha\delta}\eta_{\beta\gamma}\right]\epsilon_{3}^{\gamma}\epsilon_{4}^{\delta}.$
(B.11)

This corresponds to Eq. (3.28), expressed in the vector basis.

###  B.4 $WWtt$ amplitude

For the $WWtt$ amplitude, additional fermions in the spectrum introduce the
following three-point interactions

$\displaystyle A_{tbW}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=g_{tbW}\bar{v}(p_{2})\gamma_{\mu}\epsilon^{\mu}(p_{3})P_{L}{u}(p_{1}),$
(B.12) $\displaystyle A_{ttZ}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=\bar{v}(p_{2})\gamma_{\mu}\epsilon^{\mu}(p_{3})(g_{ttZ}^{L}P_{L}+g_{ttZ}^{R}P_{R}){u}(p_{1}),$
(B.13) $\displaystyle A_{tt\gamma}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=g_{tt\gamma}\bar{v}(p_{2})\gamma_{\mu}\epsilon^{\mu}(p_{3}){u}(p_{1}).$
(B.14)

Then we can write Eq. (3.35) as

$\displaystyle A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=-\sum_{I=Z,\gamma}\frac{1}{p_{12}^{2}-m_{I}^{2}}\frac{1}{{z^{+}_{12}-z^{-}_{12}}}$
$\displaystyle\times\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\bar{v}_{4}\left(z_{12}^{+}\left[{g_{WWI}V_{\alpha\beta}}^{\mu}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})\gamma_{\mu}(g_{ttI}^{L}P_{L}+g_{ttI}^{R}P_{R})\right]_{z=z^{-}_{12}}-(z_{12}^{+}\leftrightarrow
z_{12}^{-})\right)u_{3}$
$\displaystyle-\frac{1}{p_{14}^{2}-m_{b}^{2}}\frac{1}{{z^{+}_{14}-z^{-}_{14}}}$
$\displaystyle\times\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\bar{v}_{4}\left(z_{14}^{+}\left[g_{tbW}^{2}\gamma_{\alpha}P_{L}\hat{p}^{\mu}_{14}\gamma_{\mu}\gamma_{\beta}P_{L}\right]_{z=z^{-}_{14}}-(z_{14}^{+}\leftrightarrow
z_{14}^{-})\right)u_{3}.$ (B.15)

We notice that each term contains one insertion of momentum in the numerator
either from the function
${V_{\alpha\beta}}^{\mu}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})$ or from
internal fermion. These in turn will generate terms linear in $z$ or terms
independent in $z$. Summing over all poles, linear terms in $z$ cancels and we
find the total amplitude is of the form

$\displaystyle A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}$
$\displaystyle=-\sum_{I=Z,\gamma}\frac{1}{p_{12}^{2}-m_{I}^{2}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\bar{v}_{4}\left({g_{WWI}V_{\alpha\beta}}^{\mu}({p}_{1},{p}_{2},-{p}_{12})\gamma_{\mu}\left[g_{ttI}^{L}P_{L}+g_{ttI}^{R}P_{R}\right]\right)u_{3}$
$\displaystyle\qquad-\frac{1}{p_{14}^{2}-m_{b}^{2}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\bar{v}_{4}\left(g_{tbW}^{2}\gamma_{\alpha}P_{L}{p}^{\mu}_{14}\gamma_{\mu}\gamma_{\beta}P_{L}\right)u_{3}.$
(B.16)

This is the vector representation of Eq. (3.38).

##  Appendix C UV completion of electroweak theory in vector representation

Again for readers’ convenience, in this appendix, we express the on-shell
computation of the UV completion of the EW theory in Sec. 4 in the vector
basis.

###  C.1 A neutral scalar: Higgs doublet

We begin with including one neutral real scalar field $h$, which we call the
Higgs, discussed in Sec. 4.1. The three-point amplitudes are expressed as

$\displaystyle
A_{WWh}^{(\lambda_{1},\lambda_{2})}=g_{WWh}\epsilon_{1}\cdot\epsilon_{2},\quad
A_{ZZh}^{(\lambda_{1},\lambda_{2})}=g_{ZZh}\epsilon_{1}\cdot\epsilon_{2},\quad
A_{ffh}^{(\lambda_{1}\lambda_{2})}=g_{ffh}\bar{v}_{2}u_{1},$ (C.1)

Therefore the amplitudes involving only the gauge bosons are given by

$\displaystyle\left.A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=g_{WWh}^{2}\sum_{i=2,4}\frac{(\epsilon_{1}\cdot\epsilon_{i})(\epsilon_{3}\cdot\epsilon_{j})}{p_{1i}^{2}-m_{h}^{2}},$
(C.2)
$\displaystyle\left.A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=g_{WWh}g_{ZZh}\frac{(\epsilon_{1}\cdot\epsilon_{2})(\epsilon_{3}\cdot\epsilon_{4})}{p_{12}^{2}-m_{h}^{2}},$
(C.3)

where $j\neq i$ and $j=2,4$ in the above sum. These correspond to Eqs. (4.3)
and (4.6). The $WWZh$ scattering contains the term only up to linear in $z$ in
the numerator at the pole, and therefore it is given by a simple product of
the three-point amplitudes without the momentum shift, as we have explained in
the main text. Therefore we obtain

$\displaystyle A_{WWZh}^{(\lambda_{1}\lambda_{2}\lambda_{3})}=$
$\displaystyle-\frac{g_{WWZ}g_{ZZh}}{p_{12}^{2}-m_{Z}^{2}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}V_{\alpha\beta\gamma}(p_{1},p_{2},-p_{12})\epsilon_{3}^{\gamma}$
$\displaystyle+\sum_{i=1,2}(-1)^{i}\frac{g_{WWZ}g_{WWh}}{p_{i3}^{2}-m_{W}^{2}}\epsilon_{i}^{\alpha}\epsilon_{3}^{\beta}V_{\alpha\mu\beta}(p_{i},-p_{i3},p_{3})\left(\eta^{\mu\nu}-\frac{p_{i3}^{\mu}p_{i3}^{\nu}}{m_{W}^{2}}\right)\epsilon_{j\nu},$
(C.4)

in the vector notation where $j\neq i,3,4$. This corresponds to Eq. (4.14).
Finally, after using the completeness relation (A.41), the $WWhh$ amplitude
(4.19) is written as

$\displaystyle A_{WWhh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=-\sum_{i=3,4}\epsilon_{\mu}^{(\lambda_{1})}\left[\frac{g_{WWh}^{2}m_{W}^{2}}{p_{1i}^{2}-m_{W}^{2}}\left(\eta^{\mu\nu}-\frac{{p}_{1i}^{\mu}{p}_{1i}^{\nu}}{m_{W}^{2}}\right)+g_{WWh}^{2}\frac{q_{1i}^{\mu}q_{1i}^{\nu}}{2q_{1}\cdot
q_{i}}\right]\epsilon_{\nu}^{(\lambda_{2})},$ (C.5)

which corresponds to Eq. (4.21). By taking $\epsilon_{1}=q_{1}/c_{1}m_{W}$, we
obtain

$\displaystyle A_{WWhh}^{(\lambda_{1}\lambda_{2})}$
$\displaystyle=-\epsilon_{\mu}^{(\lambda_{1})}\left[\sum_{i=3,4}\frac{g_{WWh}^{2}m_{W}^{2}}{p_{1i}^{2}-m_{W}^{2}}\left(\eta^{\mu\nu}-\frac{{p}_{1i}^{\mu}{p}_{1i}^{\nu}}{m_{W}^{2}}\right)+\frac{g_{WWh}^{2}}{2}\eta^{\mu\nu}\right]\epsilon_{\nu}^{(\lambda_{2})}.$
(C.6)

This corresponds to Eq. (4.22). For $WWtt$ calculations, in addition to Eq.
(B.16), we must add the Higgs contribution which is of the form

$\displaystyle
A_{WWtt}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}=g_{tth}g_{WWh}\frac{(\epsilon_{1}\cdot\epsilon_{2})\bar{v}_{4}u_{3}}{p_{12}^{2}-m_{h}^{2}}$
(C.7)

This corresponds to Eq. (4.26).

###  C.2 Neutral and doubly charged scalar: Higgs triplet

We next express the Higgs triplet case discussed in Sec. 4.2 in the vector
notation. The three-point amplitudes are given by

$\displaystyle
A_{WWh^{0}}^{(\lambda_{1},\lambda_{2})}=g_{WWh}^{(0)}\epsilon_{1}\cdot\epsilon_{2},\quad
A_{ZZh^{0}}^{(\lambda_{1},\lambda_{2})}=g_{ZZh}^{(0)}\epsilon_{1}\cdot\epsilon_{2},\quad
A_{WWh^{\mp\mp}}^{(\lambda_{1},\lambda_{2})}=g_{WWh}^{(2)}\epsilon_{1}\cdot\epsilon_{2},$
(C.8)

for those involving two gauge bosons, and

$\displaystyle
A_{h^{++}h^{--}Z}^{(\lambda_{3})}=g_{hhZ}^{(2)}(p_{1}-p_{2})\cdot\epsilon_{3},\quad
A_{h^{++}h^{--}\gamma}^{(\lambda_{3})}=g_{hh\gamma}^{(2)}(p_{1}-p_{2})\cdot\epsilon_{3},$
(C.9)

for those involving one gauge boson, respectively. Eqs. (4.37) and (4.40) are
then expressed in the vector basis as

$\displaystyle\left.A_{WWWW}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=\left(g_{WWh}^{(0)}\right)^{2}\left[\sum_{i=2,4}\frac{(\epsilon_{1}\cdot\epsilon_{i})(\epsilon_{3}\cdot\epsilon_{j})}{p_{1i}^{2}-m_{0}^{2}}\right]+\left(g_{WWh}^{(2)}\right)^{2}\frac{(\epsilon_{1}\cdot\epsilon_{3})(\epsilon_{2}\cdot\epsilon_{4})}{p_{13}^{2}-m_{2}^{2}},$
(C.10)
$\displaystyle\left.A_{WWZZ}^{(\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4})}\right|_{h}$
$\displaystyle=g_{WWh}^{(0)}g_{ZZh}^{(0)}\frac{(\epsilon_{1}\cdot\epsilon_{2})(\epsilon_{3}\cdot\epsilon_{4})}{p_{12}^{2}-m_{0}^{2}},$
(C.11)

where $j\neq 1,3,i$. For $W^{+}W^{+}Zh^{--}$ and $W^{+}W^{+}\gamma h^{--}$, by
using the completeness relation of the intermediate gauge bosons, we obtain

$\displaystyle A_{WWZh^{--}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=-\frac{2g_{WWh}^{(2)}g_{hhZ}^{(2)}}{p_{12}^{2}-m_{2}^{2}}(\epsilon_{1}\cdot\epsilon_{2})(\epsilon_{3}\cdot
p_{4})-\sum_{i=1,2}\left[\frac{g_{WWZ}g_{WWh}^{(2)}}{p_{i3}^{2}-m_{W}^{2}}\epsilon_{i}^{\alpha}\epsilon_{3}^{\beta}V_{\alpha\mu\beta}(p_{i},-p_{i3},p_{3})\left(\eta^{\mu\nu}-\frac{p_{i3}^{\mu}p_{i3}^{\nu}}{m_{W}^{2}}\right)\epsilon_{j\nu}\right],$
(C.12) $\displaystyle A_{WW\gamma
h^{--}}^{(\lambda_{1}\lambda_{2}\lambda_{3})}$
$\displaystyle=-\frac{2g_{WWh}^{(2)}g_{hh\gamma}^{(2)}}{p_{12}^{2}-m_{2}^{2}}(\epsilon_{1}\cdot\epsilon_{2})(\epsilon_{3}\cdot
p_{4})-\sum_{i=1,2}\left[\frac{g_{WW\gamma}g_{WWh}^{(2)}}{p_{i3}^{2}-m_{W}^{2}}\epsilon_{i}^{\alpha}\epsilon_{3}^{\beta}V_{\alpha\mu\beta}(p_{i},-p_{i3},p_{3})\left(\eta^{\mu\nu}-\frac{p_{i3}^{\mu}p_{i3}^{\nu}}{m_{W}^{2}}\right)\epsilon_{j\nu}\right],$
(C.13)

in the vector notation where $j\neq 3,4,i$. These correspond to Eqs. (4.47)
and (4.51). In particular, the Ward identity of the $W^{+}W^{+}\gamma h^{--}$
amplitude requires

$\displaystyle g_{hh\gamma}^{(2)}=-2g_{WW\gamma}.$ (C.14)

Finally, the $W^{+}W^{-}h_{a}^{++}h_{b}^{--}$ amplitude is expressed as

$\displaystyle A_{WWh^{++}h^{--}}^{(\lambda_{1}\lambda_{2})}=$
$\displaystyle-\frac{g_{WWZ}g_{hhZ}^{(2)}}{p_{12}^{2}-m_{Z}^{2}}\frac{1}{z_{12}^{+}-z_{12}^{-}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[z_{12}^{+}\left[V_{\alpha\beta\mu}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})(\hat{p}_{3}^{\mu}-\hat{p}_{4}^{\mu})\right]_{z_{12}^{-}}-(z_{12}^{+}\leftrightarrow
z_{12}^{-})\right]$
$\displaystyle+\frac{2g_{WW\gamma}^{2}}{p_{12}^{2}}\frac{1}{z_{12}^{+}-z_{12}^{-}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[z_{12}^{+}\left[V_{\alpha\beta\mu}(\hat{p}_{1},\hat{p}_{2},-\hat{p}_{12})(\hat{p}_{3}^{\mu}-\hat{p}_{4}^{\mu})\right]_{z_{12}^{-}}-(z_{12}^{+}\leftrightarrow
z_{12}^{-})\right]$
$\displaystyle+\frac{\left(g_{WWh}^{(2)}\right)^{2}}{p_{14}^{2}-m_{W}^{2}}\frac{1}{z_{14}^{+}-z_{14}^{-}}\epsilon_{1}^{\alpha}\epsilon_{2}^{\beta}\left[z_{12}^{+}\left[-\eta_{\alpha\beta}+\frac{\hat{p}_{14\alpha}\hat{p}_{14\beta}}{m_{W}^{2}}\right]_{z_{12}^{-}}-(z_{12}^{+}\leftrightarrow
z_{12}^{-})\right].$ (C.15)

Each term contain $z^{0}$ and $z^{2}$ terms, and by focusing on the $z^{2}$
term and taking $q_{1}=c_{1}m_{W}\epsilon_{1}$, we obtain

$\displaystyle\left[A_{WWh^{++}h^{--}}^{(\lambda_{1}\lambda_{2})}\right]_{\mathrm{contact}}$
$\displaystyle=\frac{1}{2c_{1}m_{W}^{3}}\left[2\left(g_{WWZ}g_{hhZ}^{(2)}+g_{WW\gamma}g_{hh\gamma}^{(2)}\right)m_{W}^{2}\epsilon_{2}\cdot(q_{3}-q_{4})+\left(g_{WWh}^{(2)}\right)^{2}\epsilon_{2}\cdot
q_{3}\right],$ (C.16)

which corresponds to Eq. (4.58). Therefore, the final result does not depend
on these variables only if

$\displaystyle 0$
$\displaystyle=4\left(g_{WWZ}g_{hhZ}^{(2)}+g_{WW\gamma}g_{hh\gamma}^{(2)}\right)m_{W}^{2}+\left(g_{WWh}^{(2)}\right)^{2},$
(C.17)

and with this condition we obtain the contact term as

$\displaystyle\left[A_{WWh^{++}h^{--}}^{(\lambda_{1}\lambda_{2})}\right]_{\mathrm{contact}}$
$\displaystyle=-\frac{\left(g_{WWh}^{(2)}\right)^{2}}{4m_{W}^{2}}\epsilon_{1}\cdot\epsilon_{2}.$
(C.18)

This corresponds to Eq. (4.60).

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# Van Hove Singularity Arising from Mexican-Hat-Shaped Inverted Bands in the
Topological Insulator Sn-doped Bi1.1Sb0.9Te2S

Wenchao Jiang  Bowen Li  Xiaomeng Wang  Guanyu Chen  Tong Chen  Ying Xiang
Wei Xie  Yaomin Dai<EMAIL_ADDRESS>Xiyu Zhu  Huan Yang  Jian Sun
<EMAIL_ADDRESS>Hai-Hu Wen<EMAIL_ADDRESS>National Laboratory of Solid
State Microstructures and Department of Physics, Collaborative Innovation
Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

(September 2, 2024)

###### Abstract

The optical properties of Sn-doped Bi1.1Sb0.9Te2S, the most bulk-insulating
topological insulator thus far, have been examined at different temperatures
over a broad frequency range. No Drude response is detected in the low-
frequency range down to 30 cm-1, corroborating the excellent bulk-insulating
property of this material. Intriguingly, we observe a sharp peak at about 2
200 cm-1 in the optical conductivity at 5 K. Further quantitative analyses of
the line shape and temperature dependence of this sharp peak, in combination
with first-principles calculations, suggest that it corresponds to a van Hove
singularity arising from Mexican-hat-shaped inverted bands. Such a van Hove
singularity is a pivotal ingredient of various strongly correlated phases.

Stimulated by the recent discovery of unconventional superconductivity (SC)
Cao et al. (2018a) and correlated insulating behavior Cao et al. (2018b) in
‘magic-angle’ twisted bilayer graphene, flat bands have quickly become one of
the hottest topics in condensed matter physics Marchenko et al. (2018);
Yankowitz et al. (2019); Yin et al. (2019); Li et al. (2019); Tarnopolsky et
al. (2019). Electronic systems having flat bands close to the Fermi level
($E_{F}$) are of great interest, because in such systems, with the quantum
kinetic energy being quenched, the ground state is achieved by minimizing
Coulomb interaction energy, which consequently results in a variety of
strongly correlated phases, such as unconventional SC Cao et al. (2018a),
Mott-like insulator Cao et al. (2018b), and Wigner crystallization Wu et al.
(2007). In addition, flat bands give rise to a van Hove singularity (VHS) in
the density of states (DOS). Theoretical studies have shown that such a VHS,
when situated near $E_{F}$, can induce exotic quantum phenomena including the
fractional quantum Hall effect Sun et al. (2011); Tang et al. (2011); Neupert
et al. (2011), charge density wave (CDW) Rice and Scott (1975), magnetism
Zhang et al. (2010); Tasaki (1998); Okada and Oshiyama (2001) and SC Hirsch
and Scalapino (1986); Newns et al. (1992); Tang and Fu (2014); Kopnin et al.
(2011).

Owing to the fascinating physics presented by flat-band systems, the search
for materials with flat bands near $E_{F}$ is highly motivated. To date, flat
bands have been realized in several systems including twisted bilayer graphene
Cao et al. (2018a, b); Marchenko et al. (2018); Li et al. (2010), high-$T_{c}$
cuprates Gofron et al. (1994); Dessau et al. (1993), kagome magnet Co3Sn2S2
Yin et al. (2019), heavy fermion compounds Hoshino and Kuramoto (2013); Pfau
et al. (2013), as well as some lattice models Wu et al. (2007); Sun et al.
(2011); Tang et al. (2011); Neupert et al. (2011); Zhang et al. (2010); the
exotic quantum phenomena found in these systems are believed to be closely
tied to the flat bands. For instance, in ‘magic-angle’ twisted bilayer
graphene, the electronic bands near $E_{F}$ are flat, and VHSs associated with
these flat bands have been experimentally detected Li et al. (2010). Mott-like
insulating states are discovered at half-filling of these flat bands Cao et
al. (2018b), while unconventional SC emerges upon doping the material away
from the correlated insulating states Cao et al. (2018a). In the high-$T_{c}$
cuprate YBa2Cu4O8 Gofron et al. (1994), an “extended” VHS has been observed at
about 19 meV below $E_{F}$. This VHS, which arises from a flat band along the
$\Gamma$-$Y$ direction, is believed to be responsible for the high SC
transition temperature. Recently, Yin _et al_. Yin et al. (2019) reveal a VHS
at 8 meV below $E_{F}$ stemming from a flat band with non-trivial Berry phase
in the kagome magnet Co3Sn2S2, which is likely to be the origin of the
negative magnetism they observed. Theoretical calculations based on some
lattice models Sun et al. (2011); Tang et al. (2011); Neupert et al. (2011)
have revealed flat bands with a nonzero Chern number; partial filling of these
flat bands can lead to fractional quantum Hall states without magnetic field.

In addition to these systems, topological insulators (TIs) are also expected
to possess flat bands in the bulk. Theoretical calculations have pointed out
that deeply inverted bands in TIs unavoidably develop a “Mexican-hat-shaped”
dispersion Yazyev et al. (2012); He et al. (2014), as shown by the schematic
in Fig. 1(a). Such Mexican-hat-shaped bands give rise to a VHS in the DOS at
the band edge, which may not only account for the symmetry breaking
instability of TIs He et al. (2014), such as magnetism Zhang et al. (2013) and
SC Hor et al. (2010); Liu et al. (2015); Zhong et al. (2014); An et al.
(2018); Zhang et al. (2011); Cai et al. (2018); Cui et al. (2019), but also
provide a platform for exploring exotic quantum phenomena, particularly the
topological SC Fu and Kane (2008); Fu and Berg (2010).

Optical spectroscopy is a powerful technique to reveal such a VHS in TIs if
$E_{F}$ lies in the bulk band gap, because electronic transitions between
Mexican-hat-shaped bands inevitably produce a divergently sharp peak in the
optical conductivity, which measures the joint density of states (JDOS)
Dressel and Grüner (2002); Basov and Timusk (2005). Optical studies of TIs
have mainly focused on the band gap and bulk carriers Akrap et al. (2012); Di
Pietro et al. (2012); Aleshchenko et al. (2014); Dubroka et al. (2017);
LaForge et al. (2010); Post et al. (2013); Xi et al. (2014), the phonon modes
Akrap et al. (2012); LaForge et al. (2010), as well as the topological surface
states Reijnders et al. (2014); Post et al. (2015). We notice that an
absorption peak near the bulk band gap appears in the optical conductivity of
Bi2Te2Se Akrap et al. (2012); Di Pietro et al. (2012); Dubroka et al. (2017);
Reijnders et al. (2014); Aleshchenko et al. (2014). However, no detailed
analysis of this peak has been done. To date, evidence for the VHS arising
from Mexican-hat-shaped inverted bands in TIs remains elusive.

Thus far, Sn-doped Bi1.1Sb0.9Te2S (Sn-BSTS) has been found to be the most
bulk-insulating TI whose $E_{F}$ sits in the bulk band gap Kushwaha et al.
(2016), providing a perfect system to trace the VHS arising from Mexican-hat-
shaped inverted bands through optical spectroscopy. The only optical study on
this material is in the THz range, focusing on the topological surface states
Cheng et al. (2016). Here, we present a detailed optical study on Sn-BSTS. Our
optical conductivity shows no Drude response down to 30 cm-1, confirming the
bulk-insulating nature of this material. A sharp peak is clearly observed at
about 2 200 $\textrm{cm}^{-1}$ in the optical conductivity at 5 K. By
comparing our experimental results with theoretically calculated optical
conductivity and band structure, we conclude that the sharp peak in the
optical conductivity represents a VHS associated with Mexican-hat-shaped
inverted bands.

High-quality Bi1.08Sn0.02Sb0.9Te2S single crystals with large and shiny
surfaces were synthesized using the Bridgeman method Kushwaha et al. (2016).
Figure 1(b) shows the resistivity $\rho$ of our sample as a function of
temperature ($T$), which agrees well with published data Kushwaha et al.
(2016). In the high-temperature region, $\rho$ increases exponentially with
decreasing $T$, suggesting that the transport properties are dominated by the
insulating bulk states. The decrease of $\rho$ at low $T$s is a signature of
surface-dominated transport Kushwaha et al. (2016); Ren et al. (2010); Taskin
et al. (2011): the bulk resistivity becomes so large at low $T$ that it is
short-circuited by the metallic topological surface states. The details about
our optical measurements and Kramers-Kronig analysis are given in the
supplementary material Sup  (see, also, ref. Homes et al. (1993)).

Figure 1: (a) Schematic plot of the Mexican-hat dispersion (solid lines)
formed by deeply inverted bands (dashed lines) in TIs. (b) $T$-dependent
resistivity of Sn-BSTS. (c) Reflectivity of Sn-BSTS up to 4 000
$\textrm{cm}^{-1}$ at different temperatures. The inset shows the far-infrared
reflectivity at 300 K.

Figure 1(c) displays the measured reflectivity $R(\omega)$ up to 4 000
$\textrm{cm}^{-1}$ at several selected temperatures; the inset shows
$R(\omega)$ in the far-infrared range at 300 K, which is dominated by strong
IR-active phonons LaForge et al. (2010); Akrap et al. (2012). In the presence
of free carriers, $R(\omega)$ would exhibit an upturn in the low-frequency
range Akrap et al. (2012); Reijnders et al. (2014). Such an upturn is not
observed in the low-frequency $R(\omega)$ of Sn-BSTS as shown in the inset of
Fig. 1(c), indicative of an insulating sample. The most striking feature in
$R(\omega)$ at 5 K is a step at 2 400 $\textrm{cm}^{-1}$. This step feature
diminishes and shifts to lower frequency as $T$ rises. Moreover, fringes
resulting from interference between light reflected from the top and bottom
surfaces appear below the step, and disappear at about 250 $\textrm{cm}^{-1}$
where phonon absorption emerges. The fringes can be seen more clearly in the
zoomed-in window in the inset of Fig. 1(c). The emergence of interference
fringes suggests that the sample becomes transparent in this frequency range.

The real part of the optical conductivity $\sigma_{1}(\omega)$ provides more
straightforward information, as it is directly related to the JDOS Sup .
Figure 2(a) shows $\sigma_{1}(\omega)$ of Sn-BSTS at different $T$s up to 4
500 $\textrm{cm}^{-1}$. The inset depicts $\sigma_{1}(\omega)$ measured at 300
K below 400 $\textrm{cm}^{-1}$, where IR-active phonons can be clearly
identified as discrete sharp peaks. The red solid circle at zero frequency
represents the dc conductivity at 300 K from transport measurements, which
agrees well with the low-frequency extrapolation of $\sigma_{1}(\omega)$. In
the low-frequency region, $\sigma_{1}(\omega)$ is vanishingly small, and no
Drude response is evidently resolved for all temperatures, consistent with the
insulating nature of our sample. For $\sigma_{1}(\omega)$ at 5 K, with
increasing frequency, an absorption edge at $\sim$1 500 $\textrm{cm}^{-1}$
develops, leading to a sharp peak at $\sim$2 200 $\textrm{cm}^{-1}$. As $T$ is
elevated, this peak becomes weaker and quickly shifts to lower frequency. The
noise-like signal setting in below the peak is interference fringes as
discussed for $R(\omega)$.

Figure 2: (a) Optical conductivity of Sn-BSTS up to 4 500 $\textrm{cm}^{-1}$
at several representative temperatures. Inset: Far-infrared optical
conductivity at 300 K, showing IR-active phonons. The red solid circle at zero
frequency denotes the transport value of the dc conductivity at 300 K. (b)
Optical conductivity with offset, highlighting the sharp peak between 1 600
and 2 200 $\textrm{cm}^{-1}$. Solid lines through the data are fitting results
at different temperatures.

In the following, we concentrate on the sharp peak at $\sim$2 200
$\textrm{cm}^{-1}$ in the $\sigma_{1}(\omega)$ spectra. Since IR-active
phonons are not expected at such a high frequency, we deduce that this peak
originates from electronic transitions. Furthermore, we can safely rule out
electronic transitions associated with impurity bands as the origin of such a
sharp peak in $\sigma_{1}(\omega)$ of Sn-BSTS Sup ; Gaymann et al. (1995). As
$\sigma_{1}(\omega)$ is proportional to the JDOS, a quasidivergent peak in
$\sigma_{1}(\omega)$ explicitly signifies a VHS in the JDOS of Sn-BSTS Van
Hove (1953).

Further information about the VHS in Sn-BSTS may be obtained through a
quantitative analysis of $\sigma_{1}(\omega)$. Van Hove has pointed out that
the DOS of electron bands has singularities at points where $|\nabla_{k}(E)|$
vanishes Van Hove (1953). These points in the band structure are known as
critical points and the corresponding singularities in the DOS are called
VHSs. The optical response of interband critical points can be described by a
standard analytic line shape for the complex dielectric function
Lautenschlager et al. (1987)

$\tilde{\varepsilon}(\omega)=C-Ae^{i\phi}(\omega-E_{t}+i\Gamma)^{n},$ (1)

where $A$, $E_{t}$, $\Gamma$ and $\phi$ stand for the amplitude, energy
threshold, broadening and excitonic phase angle of the critical point,
respectively. For three-dimensional (3D) critical points $n=1/2$. The real
part of the optical conductivity $\sigma_{1}(\omega)$ and the imaginary part
of the dielectric function $\varepsilon_{2}(\omega)$ have the following simple
relation:

$\sigma_{1}(\omega)=\frac{2\pi}{Z_{0}}\omega\varepsilon_{2}(\omega),$ (2)

where $Z_{0}$ = 377 $\Omega$ is the vacuum impedance. We fit this equation to
the measured $\sigma_{1}(\omega)$ at all $T$s and plot the fitting results as
solid lines through the data in Fig. 2(b). Note that Eq. (1) gives an
excellent description to the line shape of $\sigma_{1}(\omega)$ at all
measured temperatures. From the fitting, we can extract the $T$ dependence of
two key parameters: $E_{t}$ the energy threshold of the VHS, and $\phi$ which
contains information about the type of the VHS or the dispersion of the
electronic bands.

Figure 3: $T$ dependence of $E_{t}$ (a) and $\phi$ (b) extracted from the fit
at all measured temperatures. The inset of (b) displays $\varepsilon_{2}$ of
Sn-BSTS measured at 5 K. (c)-(f) Schematic diagrams of
$\varepsilon_{2}(\omega)$ for different types of VHSs. (g) Schematic plot of
$\varepsilon_{2}(\omega)$ for a VHS with $\phi=1$.

As portrayed by Fig. 3(a), $E_{t}$ increases with decreasing $T$. This implies
that the bulk band gap increases as $T$ is lowered. In TIs, inverted and non-
inverted band gaps exhibit opposite temperature dependence Strauss (1967);
Yazyev et al. (2012). Studies on TIs with a small band gap, such as Pb1-xSnxTe
Dimmock et al. (1966), Pb1-xSnxSe Strauss (1967); Dziawa et al. (2012) and
ZrTe5 Xu et al. (2018), have revealed that non-inverted gaps decrease upon
cooling, while gaps between inverted bands increase with decreasing $T$
Dimmock et al. (1966); Strauss (1967); Dziawa et al. (2012); Xu et al. (2018).
Based on these properties of TIs, it is tempting to associate the VHS in Sn-
BSTS with inverted bands. However, the increase of the band gap with
decreasing $T$ may not be taken as conclusive evidence for inverted bands,
since the band gap in conventional semiconductors, e.g. GaAs Lautenschlager et
al. (1987), Si and Ge Lautenschlager et al. (1985), also increase with
decreasing $T$. Further evidence for the VHS associated with inverted bands in
Sn-BSTS will be provided by theoretical calculations, and a comparison between
the calculations and experimental data.

Having examined the $T$ dependence of $E_{t}$, we next look into $\phi$ and
its evolution with $T$. Figure 3(b) shows that $\phi\simeq 1$ at low
temperatures, and it decreases with increasing $T$. For a 3D system, there are
four types of VHSs corresponding to four types of interband critical points
(labeled $M_{1}$, $M_{2}$, $M_{3}$, and $M_{0}$) in the band structure Van
Hove (1953); Lautenschlager et al. (1987); Yu and Cardona (2010); $\phi$ takes
the values of 0, $\pi/2$, $\pi$, and $3\pi/2$ for $M_{1}$, $M_{2}$, $M_{3}$,
and $M_{0}$, respectively. Figures 3(c)-3(f) schematically illustrate
$\varepsilon_{2}(\omega)$ calculated from Eq. (1) for all four types of VHSs.
While $M_{0}$ [Fig. 3(f)] and $M_{3}$ [Fig. 3(e)] refer to a minimum and a
maximum in the interband separation, respectively, $M_{1}$ and $M_{2}$
represent saddle points [Figs. 3(c) and 3(d)]. The experimentally determined
$\phi$ for Sn-BSTS does not have a value of integer multiples of $\pi/2$. This
implies that the VHS in Sn-BSTS does not simply fall into the basic
classification of VHSs $M_{0}$-$M_{3}$. In addition, $\phi$ varies with $T$,
from which we may deduce that the electronic band dispersion changes with $T$.

It’s worthwhile to notice that regular VHSs manifest themselves as step-like
features in $\varepsilon_{2}(\omega)$, whereas the one with $\phi=1$, as shown
in Fig. 3(g), produces a divergently sharp peak. Such a peak, which can also
be clearly identified in the measured $\varepsilon_{2}(\omega)$ [inset of Fig.
3(b)], unequivocally suggests that the VHS in Sn-BSTS reflects a more singular
JDOS than regular ones. This recalls the expected Mexican-hat-shaped
dispersion for deeply inverted bands in TIs He et al. (2014); Yazyev et al.
(2012). Theoretical calculations based on an analytical model He et al. (2014)
have shown that for non-inverted and shallowly inverted bands, a step feature,
representing a regular VHS, is obtained in the DOS around the band edge. In
stark contrast, for deeply inverted bands, a sharp peak, which corresponds to
a VHS arising from Mexican-hat-shaped bands, emerges in the DOS around the
band edge. These results point to that the VHS in Sn-BSTS is most likely
associated with Mexican-hat-shaped inverted bands.

In order to provide more insights into the VHS as well as the electronic band
dispersion in Sn-BSTS, we performed first-principles calculations Sup ; Abt et
al. (1994); Ambrosch-Draxl and Sofo (2006).

Figure 4: Calculated band structures of BiSbTe2S at $a/a_{0}=1.05$ (a) without
SOC and (b) with SOC. (c) Calculated band structure at different $a/a_{0}$,
simulating the effect of $T$ on the band structure. (d) Calculated
$\sigma_{1}(\omega)$ at various $a/a_{0}$. (e) Calculated $\sigma_{1}(\omega)$
at $a/a_{0}=1.05$ (blue curve) which is decomposed into contributions from
different band combinations.

Figures 4(a) and 4(b) compare the calculated band structures of BiSbTe2S at
$a/a_{0}=1.05$ without and with SOC. A conspicuous change in the band
structure induced by turning on SOC is an anti-crossing feature near the
$\Gamma$ point, which is a prototypical signature of inverted bands. The band
inversion can also be realized from the calculated orbit-resolved band
structure Sup . Figure 4(c) shows an enlarged view of the calculated band
structure near the $\Gamma$ point for various $a/a_{0}$ ratios. It is
noticeable that the inverted bands exhibit distinct Mexican-hat-shaped
dispersion, which is expected to yield a VHS in the JDOS He et al. (2014). The
separation between the Mexican-hat-shaped inverted bands increases as
$a/a_{0}$ is reduced, in accord with the $T$ dependence of $E_{t}$ determined
from the measured $\sigma_{1}(\omega)$. Furthermore, the dispersion of these
inverted bands varies slightly with $a/a_{0}$. Such an effect has been
inferred from the change of $\phi$ with $T$ extracted from
$\sigma_{1}(\omega)$. All these facts explicitly indicate that the VHS we
observed in $\sigma_{1}(\omega)$ arises from the Mexican-hat-shaped inverted
bands near the $\Gamma$ point.

To further support our conclusion, we calculated $\sigma_{1}(\omega)$ at
different $a/a_{0}$, as well as contributions from different band combinations
to $\sigma_{1}(\omega)$ for $a/a_{0}=1.05$. The calculated
$\sigma_{1}(\omega)$ at $a/a_{0}=1.05$, shown as the blue curve in Fig. 4(d),
agrees well with the measured $\sigma_{1}(\omega)$ at 5 K. A sharp peak can be
clearly seen at about 2 400 $\textrm{cm}^{-1}$. Such a sharp peak is absent in
$\sigma_{1}(\omega)$ calculated from the band structure without SOC Sup ,
indicating its close relation with inverted bands. With increasing $a/a_{0}$,
the peak in the calculated $\sigma_{1}(\omega)$ diminishes and shifts towards
lower frequency, qualitatively mimicking the $T$ dependence of the measured
$\sigma_{1}(\omega)$. In Fig. 4(e), the calculated $\sigma_{1}(\omega)$ at
$a/a_{0}=1.05$ (blue curve) is decomposed into contributions from different
band combinations. More specifically, the cyan, green, orange and red curves
in Fig. 4(e) denote the $\sigma_{1}(\omega)$ components contributed by the
electronic transitions V1-C1, V1-C2, V2-C1 and V2-C2, respectively. Here, the
electronic bands V1, V2, C1 and C2 are traced out as colored lines in Fig.
4(b). The black curves in Fig. 4(e) represent contributions from electronic
transitions between bands at higher energies [black lines in Fig. 4(b)]. It is
obvious from Fig. 4(e) that only electronic transitions between the inverted
bands near the $\Gamma$ point contribute to the sharp peak at about 2 400
$\textrm{cm}^{-1}$ in $\sigma_{1}(\omega)$. This offers further support for
our conclusion that the sharp peak we observed in the $\sigma_{1}(\omega)$ of
Sn-BSTS corresponds to a VHS induced by the deeply inverted bands with
Mexican-hat dispersions near the $\Gamma$ point.

Finally, we notice that the calculated $\sigma_{1}(\omega)$ [Fig. 4(e)] can
not describe the presence of absorption below the VHS, which gives rise to a
much broader absorption edge in the measured $\sigma_{1}(\omega)$ [Fig. 2(a)].
A close inspection of the band structures in Figs. 4(b) and 4(c) reveals that
the distortions of the valence and conduction bands towards a Mexican-hat
shape are asymmetric, resulting in an indirect band gap of $\sim$230 meV near
the $\Gamma$ point. The energy of this indirect band gap is much lower than
the VHS energy threshold $\sim$2 400 $\textrm{cm}^{-1}$ (297 meV) in the
calculated $\sigma_{1}(\omega)$. This means that indirect transitions can
occur below the VHS energy threshold. Since indirect electronic transitions
are not included in our first-principles calculations, it is natural that no
low-energy absorption is present in the calculated $\sigma_{1}(\omega)$. In
contrast, electronic transitions across the indirect band gap should have
considerable contributions to the experimental $\sigma_{1}(\omega)$, which may
account for the presence of absorption below the VHS, i.e. a much broader
absorption onset in the measured $\sigma_{1}(\omega)$.

The observation of a VHS arising from Mexican-hat-shaped inverted bands in a
TI deserves further attention, because it may not only elucidate the origin of
many observed symmetry breaking phases in TIs, but also pave the way for the
realization of more exotic quantum phenomena. Extensive studies have shown
that systems with a VHS in the DOS resulting from flat bands near $E_{F}$ are
generally unstable against symmetry breaking phases, such as CDW Rice and
Scott (1975), SC Hirsch and Scalapino (1986); Newns et al. (1992); Tang and Fu
(2014); Kopnin et al. (2011) and magnetism Zhang et al. (2010); Tasaki (1998);
Okada and Oshiyama (2001); Yin et al. (2019). In this sense, the magnetic
quantum phase transition in Cr-doped Bi2(SexTe1-x)3 that is driven by bulk
band topology Zhang et al. (2013), and the emergence of SC induced by doping
Hor et al. (2010); Liu et al. (2015); Zhong et al. (2014), pressure An et al.
(2018); Zhang et al. (2011); Cai et al. (2018) or protonation Cui et al.
(2019) in TIs are likely to be intimately tied to the VHS resulting from
Mexican-hat-shaped inverted bands. Moreover, driving a TI into the
superconducting state may lead to a topological superconductor Fu and Kane
(2008); Fu and Berg (2010), whose surface states host Majorana fermions which
have potential applications in topological quantum computing. It is
interesting that Sn-BSTS exhibits pressure-induced SC with the maximum
$T_{c}\simeq 12$ K (the highest among pressurized TIs), but structural phase
transitions have also been observed An et al. (2018); Cai et al. (2018), which
does not guarantee the topological states of this material. Nevertheless, with
such an unusual VHS, Sn-BSTS may be a promising system for the realization of
topological SC, if the Mexican-hat-shaped inverted bands are tuned to $E_{F}$
through doping Hor et al. (2010); Liu et al. (2015); Zhong et al. (2014),
ionic liquid gating Lei et al. (2016) or protonation Cui et al. (2019) without
introducing a structural phase transition.

To summarize, we carried out a detailed study on the optical properties of Sn-
BSTS, a TI with $E_{F}$ lying in the bulk band gap. The absence of Drude
response in $\sigma_{1}(\omega)$ attests the bulk-insulating nature of this
material. We observed a sharp peak at about 2 200 cm-1 in $\sigma_{1}(\omega)$
at 5 K. Further quantitative analyses of the line shape and $T$ dependence of
the peak, in conjunction with first-principles calculations, unambiguously
testify that the sharp peak in $\sigma_{1}(\omega)$ represents a VHS arising
from Mexican-hat-shaped inverted bands.

###### Acknowledgements.

We thank Bing Xu, Xiaoxiang Xi and Yuxin Zhao for illuminating discussions. We
gratefully acknowledge financial support from the MOST of China (grant nos.
2016YFA0300400 and 2015CB921202), the National Natural Science Foundation of
China (grant nos. 11874206, 11534005, 11374144, 11574133 and 11834006), and
the Fundamental Research Funds for the Central Universities with grant no.
020414380095. The calculations were carried out using supercomputers at the
High Performance Computing Center of Collaborative Innovation Center of
Advanced Microstructures, the high-performance supercomputing center of
Nanjing University and ‘Tianhe-2’ at NSCC-Guangzhou. Wenchao Jiang, Bowen Li
and Xiaomeng Wang contributed equally to this work.

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