# The Optimal Paper Moebius Band

Richard Evan Schwartz Supported by N.S.F. Grant DMS-2102802, a Simons
Sabbatical Fellowship, and a Mercator Fellowship.

###### Abstract

We prove that a smooth embedded paper Moebius band must have aspect ratio
greater than $\sqrt{3}$. We also prove that any sequence of smooth embedded
paper Moebius bands whose aspect ratio converges to $\sqrt{3}$ must converge,
up to isometry, to the triangular Moebius band. These results answer the
mimimum aspect ratio question discussed by W. Wunderlich in 1962 and prove the
more specific conjecture of B. Halpern and C. Weaver from 1977.

##  1 Introduction

###  1.1 The Triangular Moebius Band

To make a paper Moebius band you give a $1\times\lambda$ strip of paper an odd
number of twists and then join the ends together. For long strips this is easy
and for short strips it is difficult or impossible. Let me first discuss a
beautiful example, known as the triangular Moebius band. Figure 1a shows the
triangular Moebius band. It is based on a $1\times\sqrt{3}$ strip.

Figure 1a: The triangular Moebius band

The strip in Figure 1a is lightly shaded on one side and darkly shaded on the
other. First fold the flaps in to make a rhombus, then fold the rhombus in
half like a wallet. This folding brings the two ends together with a twist.
The dotted segment indicates where the ends are joined. The bold segment
indicates the “wallet fold”. The dotted and bold segments together make a
pattern like a T. The pinstriping exhibits the strip as a union of line
segments, disjoint except at the endpoints, which stay straight during the
folding.

Figure 1b: The triangular Moebius band: another view

Figure 1b shows another view. Here we start with a symmetric trapezoid rather
than a rectangle, but we get the same object when we fold and join the sides
together. The bold edge indicates where the sides are joined. The dotted and
bold segments again make a “T-pattern”.

The triangular Moebius band goes back at least to the 1930 paper [Sa] of M.
Sadowski. Technically, it does not quite fit the definition of a (smooth,
embedded) paper Moebius band that we give below, but it is the limit of such.
The triangular Moebius band is probably best considered as a folded ribbon
knot. See [DL] for this point of view.

###  1.2 The Minimum Aspect Ratio Question

The triangular Moebius band looks like an extremely efficient construction.
Can we do better in terms of making $\lambda$ smaller? To answer this question
in a meaninful way, we first need a formal definition.

Definition: A smooth paper Moebius band of aspect ratio $\lambda$ is an
infinitely differentiable isometric mapping
$I:M_{\lambda}\to\mbox{\boldmath{$R$}}^{3}$, where $M_{\lambda}$ is the flat
Mobius band obtained by identifying the length-$1$ sides of a $1\times\lambda$
rectangle. That is:

$M_{\lambda}=([0,\lambda]\times[0,1])\\!/\sim,\hskip
30.0pt(0,y)\sim(\lambda,1-y).$ (1)

An isometric mapping is a map which preserves arc-lengths. The map is an
embedding if it is injective, and an immersion in general. Let
$\Omega=I(M_{\lambda})$. We often write $I:M_{\lambda}\to\Omega$. We call
$\Omega$ embedded when $I$ is an embedding.

The early papers of M. Sadowsky [Sa] and W. Wunderlich [W] treat both the
existence and differential geometry of smooth paper Moebius bands. (See [HF]
and [T] respectively for modern English translations.) The paper [CF] gives a
modern differential geometric framework for developable surfaces like
$\Omega$. The papers [CK], [KU], [RR] and [Sab] are all studies of the
differential geometry of paper Moebius bands. I learned about paper Moebius
bands from the great expository article [FT, Chapter 14] by Dmitry Fuchs and
Sergei Tabachnikov.

Why bother with smooth maps? Well, if you just look at ways of folding paper
up to make a Moebius band you can get weird examples which render the main
question meaningless. For instance, you could fold any rectangle (e.g. a
square) like an accordion into a thin strip, twist, then tape. The smooth
formalism efficiently rules out the Moebius-accordion and other kinds of
origami monsters. They are not limits of smooth paper Moebius bands. In
contrast, the triangular Moebius band is the limit of smooth (embedded) paper
Moebius bands. See [Sa], [HW], and [FT]. Basically you just use a slightly
longer piece of paper and round out the sharp folds.

I imagine that the minimum aspect ratio question has been around for as long
as people have been making paper Moebius bands, but in any case W. Wunderlich
discusses this in the introduction of his 1962 paper [W]. In their 1977 paper
[HW], Ben Halpern and Charles Weaver study the minimum aspect ratio question
in detail. They prove two things.

  * •

For smooth immersed paper Moebius bands one has $\lambda>\pi/2$. Moreover, for
any $\epsilon>0$ one can find an immersed example with
$\lambda=\pi/2+\epsilon$.

  * •

There exists some $\epsilon_{0}>0$ such that $\lambda>\pi/2+\epsilon_{0}$ for
a smooth embedded paper Moebius band. This $\epsilon_{0}$ is not an explicit
constant.

On the last line of [HW], Halpern and Weaver conjecture that
$\lambda>\sqrt{3}$ for a smooth embedded paper Moebius band. In this paper I
will resolve the Halpern-Weaver conjecture.

######  Theorem 1.1 (Main)

A smooth embedded paper Moebius band has aspect ratio greater than $\sqrt{3}$.

######  Theorem 1.2 (Triangular Limit)

Let $\\{I_{n}:M_{\lambda_{n}}\to\Omega_{n}\\}$ be a sequence of smooth
embedded paper Moebius bands with $\lambda_{n}\to\sqrt{3}$. Then, up to
isometry, $I_{n}$ converges uniformly to the map giving the triangular Moebius
band.

###  1.3 Outline of the Proofs

Let $I:M_{\lambda}\to\Omega$ be a smooth embedded paper Moebius band. A bend
on $\Omega$ is a line segment $B^{\prime}$ which cuts across $\Omega$ and has
its endpoints in the boundary. It is a classic fact that $\Omega$ has a
continuous foliation by bends. See §2.1. The bend foliation may not be unique,
but we choose a bend foliation once and for all. The bends in the foliation
vary continuously and are pairwise disjoint.

Given a bend $B^{\prime}$, we call the inverse image $B=I^{-1}(B^{\prime})$ a
pre-bend. It is a classic fact that pre-bends are line segments. (Proof: If
not, then there are points $p,q\in B$ whose distance in $M_{\lambda}$ is less
than the distance in $\mbox{\boldmath{$R$}}^{3}$ between $I(p)$ and $I(q)$.
This contradicts the fact that $I:M_{\lambda}\to\mbox{\boldmath{$R$}}^{3}$ is
distance non-increasing.) The inverse image of our bend foliation is a
foliation of $M_{\lambda}$ into pre-bends. This is just like the pinstriping
in Figures 1a and 1b except that in the smooth case even the endpoints are
disjoint.

We say that a $T$-pattern on $\Omega$ is a pair of bends which lie in
perpendicular intersecting lines. Look again at the right sides of Figures 1a
and 1b. We call the $T$-pattern embedded if the two bends are disjoint. In
§2.2 we prove

######  Lemma 1.3 (T)

A smooth embedded paper Moebius band has an embedded $T$-pattern.

Here is the idea. The space of pairs of unequal bends in our bend foliation
has a $2$-point compactification which makes it into the $2$-sphere, $S^{2}$.
We define a pair of odd functions on $S^{2}$ which detect a $T$-pattern when
they have a common zero. We apply the Borsuk-Ulam Theorem to get a common
zero. In §2.3 we prove

######  Lemma 1.4 (G)

A smooth embedded paper Moebius band with an embedded $T$-pattern has aspect
ratio greater than $\sqrt{3}$.

Here is the idea. Let $(T^{\prime},B^{\prime})$ be the embedded $T$-pattern.
We cut open $M_{\lambda}$ along the pre-bend $T=I^{-1}(T^{\prime})$, and the
result is a bilaterally symmetric trapezoid $\tau$. See Figure 2 below. We
then solve an optimization problem which involves mapping $\tau$ into
$\mbox{\boldmath{$R$}}^{3}$ with constraints coming from the geometry of
trapezoids and $T$-patterns. The Main Theorem is an immediate consequence of
Lemma T and Lemma G.

In §2.4 we prove the Triangular Limit Theorem by examining what the proof of
Lemma G says about a minimizing sequence of examples.

###  1.4 Additional Material and Context

The proofs are done after §2, but I include some more material. In §3 I
comment on the proofs and also give additional context. In §4 I give an
elementary and self-contained proof of a special case of [HN, §3, Lemma 2],
the result that underlies the existence of the bend foliation.

The topic of paper Moebius bands is adjacent to a number of different
subjects. The paper [CKS] and [DDS] consider the related question of tying a
piece of rope into a knot using as little rope as possible. The papers [D] and
[DL] consider folded ribbon knots. [DL, Corollary 25] is in some sense a
special case of our two results, and [DL, Conjecture 26] is a variant of the
Halpern-Weaver Conjecture in the category of folded ribbon knots. Our Main
Theorem incidentally resolves this folded ribbon knot conjecture. Some authors
have considered “optimal Moebius bands” from other perspectives, either
algebraic [Sz] or physical [MK], [SH]. I will discuss some of these things in
§3.4.

In §3.5 I discuss multi-twist Moebius bands, and some new results about them,
namely [BS], [H], and [S3].

This paper is an outgrowth of my earlier paper [S1]. In [S1] I (correctly)
prove that $\lambda\geq\phi=(1+\sqrt{5})/2$, but I also make an idiotic
mistake: Somehow I thought that when you cut open $M_{\lambda}$ along a pre-
bend you get a parallelogram. This mistake invalidates my final bound, a weird
and forgettable algebraic number in $(\phi,\sqrt{3})$. This paper supersedes
[S1] and is independent from it.

###  1.5 Acknowledgements

I thank Brienne Brown, Matei Coiculescu, Robert Connelly, Dan Cristofaro-
Gardiner, Elizabeth Denne, Ben Halpern, Dmitry Fuchs, Javier Gomez-Serrano,
Aidan Hennessey, Anton Izosimov, Jeremy Kahn, Rick Kenyon, Stephen D. Miller,
Noah Montgomery, Sergei Tabachnikov, and Charles Weaver for helpful
discussions about this subject. I especially thank Matei for suggesting that I
try for a “mapping proof” of Lemma T as opposed to the kind of proof I had
previously. That suggestion led me to find a really nice proof of Lemma T that
greatly simplified this paper. Jeremy Kahn’s remarks about the Borsuk-Ulam
Theorem also helped streamline the proof of Lemma T. Finally, I thank the
anonymous referees for insightful and helpful comments which improved the
exposition of the paper.

##  2 Proofs of the Results

###  2.1 Existence of a Bend Foliation

Let $I:M_{\lambda}\to\Omega$ be a smooth embedded paper Moebius band. Recall
that a bend is a line segment that cuts across $\Omega$ and has its endpoints
in $\partial\Omega$.

###### Theorem 2.1

$\Omega$ has a foliation by bends.

We derive Theorem 2.1 from a classic result. As always for manifolds with
boundary, the smoothness of the map $I$ means that $I$ has a smooth extension
to a neighborhood of each boundary point. Thus, it makes sense to speak of the
mean curvature of $\Omega$ even at points of $\partial\Omega$. Let
$\Omega^{o}$ denote the interior of $\Omega$. Let $U\subset\Omega^{o}$ denote
the subset having nonzero mean curvature.

###### Lemma 2.2

Each $p\in U$ lies in a unique bend $\gamma$. Every point along $\gamma$ has
nonzero mean curvature, even the endpoints.

Lemma 2.2 follows from either of the two essentially identical results, [CL,
p. 314, Lemma 2] and [HN, §3, Lemma 2]. In §4 I give an elementary and self-
contained proof of Lemma 2.2.

Proof of Theorem 2.1: Let $U^{*}$ denote the union of all the bends through
points of $U$. We obtain $U^{*}$ from $U$ by adding in the endpoints of the
bends. No two bends of $U^{*}$ can intersect, even at the boundary. The
reason: If they did intersect, then the point of intersection would have two
distinct straight lines through it and hence zero mean curvature. This
contradicts Lemma 2.2. Hence $U^{*}$ actually has a foliation by bends. The
fact that no two bends in this foliation intersect implies that they vary
continuously.

Let $\tau^{\prime}$ be a component of $\Omega-U^{*}$. If $\tau^{\prime}$ has
empty interior then $\tau^{\prime}$ is a line segment, the limit of a sequence
of bends. In this case $\tau^{\prime}$ is also a bend. Suppose $\tau^{\prime}$
has non-empty interior. Since all points of $\tau^{\prime}$ have zero mean
curvature, $\tau^{\prime}$ lies in a single plane. (One way to see this is to
note that the Gauss map is constant in $\tau^{\prime}$.) Let
$\tau=I^{-1}(\tau^{\prime})$. The map $I|_{\tau}:\tau\to\tau^{\prime}$ is an
isometric embedding between planar regions, and also the domain $\tau$ is
convex – just a trapezoid. This implies that $I|_{\tau}$ is a global isometry
and in particular maps line segments to line segments. Now we foliate $\tau$
by line segments interpolating between the two pre-bends in $\partial\tau$.
The image under $I|_{\tau}$ of this foliation of $\tau$ is a bend foliation of
$\tau^{\prime}$. Doing this construction on all such components, we get our
bend foliation of $\Omega$. $\spadesuit$

###  2.2 Proof of Lemma T

Let $I:M_{\lambda}\to\Omega$ be a smooth embedded paper Moebius band. We first
give some definitions.

Recall from §1.3 that a pre-bend is the pre-image of a bend. The pre-bends are
embedded line segments in $M_{\lambda}$ which have their endpoints in
$\partial M_{\lambda}$.

The centerline of $M_{\lambda}$ is the circle
$([0,\lambda]\times\\{1/2\\})/\\!\sim$. The centerline of $\Omega$ is the
image of the centerline of $M_{\lambda}$ under the map $I$. These centerlines
are topological circles and $I$ is a diffeomorphism between them.

Each bend $u$ has exactly $2$ unit vectors $\pm\overrightarrow{u}$ parallel to
it. We call either one an orientation of $u$.

Intersection with the Centerline: We are in a Moebius band, so a pre-bend $u$,
being embedded, must intersect the centerline exactly once. Here is an
alternate proof: Let $\ell(\cdot)$ denote length. If
$\ell(u)<\sqrt{1+\lambda^{2}}$ we can move $u$ by an isometry so that it
misses the vertical sides of $[0,\lambda]\times[0,1]$. But then $u$ clearly
intersects the centerline exactly once. So, if $u$ intersects the centerline
more than once, we have $\ell(u)\geq\sqrt{1+\lambda^{2}}>\lambda$. But
$\partial\Omega=I(\partial M_{\lambda})$ is a loop that contains the endpoints
of the bend $I(u)$, and $\ell(I(u))=\ell(u)>\lambda$. Hence
$\ell(\partial\Omega)>2\lambda$. But $\ell(\partial\Omega)=\ell(\partial
M_{\lambda})=2\lambda$, a contradiction.

The Circle of Bends: The bend foliation of $\Omega$ guaranteed by Theorem 2.1
may not be unique. We simply choose one. Call it $\beta$. We mention again
that the bends in $\beta$ are segments which vary continuously and are
pairwise disjoint. We parametrize the bends of $\beta$ by
$\mbox{\boldmath{$R$}}/2\pi$, as follows: Since $I$ is an embedding, our
result about pre-bends implies that each bend of $\Omega$ intersects the
centerline of $\Omega$ exactly once. We associate to each bend the point where
it intersects the centerline, and then we identify the centerline with
$\mbox{\boldmath{$R$}}/2\pi$.

The Cylinder and the Sphere: Let $\Upsilon$ be the topological cylinder of
pairs $(x_{0},x_{1})\in(\mbox{\boldmath{$R$}}/2\pi)^{2}$ with
$x_{0}\not=x_{1}$. A point $(x_{0},x_{1})\in\Upsilon$ corresponds to a pair
$(u_{0},u_{1})$ of unequal bends. We let $\overline{\Upsilon}$ be the
compactification of $\Upsilon$ obtained by adding $2$ points: $\partial_{+}$
(respectively $\partial_{-}$) is the limit of pairs $(x_{0},x_{1})$ where
$x_{1}$ is just ahead (respectively just behind) $x_{0}$ in the cyclic order
on $\mbox{\boldmath{$R$}}/2\pi$. The space $\overline{\Upsilon}$ is
homeomorphic to $S^{2}$, the $2$-sphere. See §3.1 for an explicit
homeomorphism. The map $\Sigma(x_{0},x_{1})=(x_{1},x_{0})$ extends to a
continuous involution of $S^{2}$ that swaps the two points $\partial_{+}$ and
$\partial_{-}$. The explicit homeomorphism in §3.1 conjugates $\Sigma$ to the
antipodal map of $S^{2}$.

Propagating the Orientations: We orient $\mbox{\boldmath{$R$}}/2\pi$. Let
$(x_{0},x_{1})\in\Upsilon$. There is a unique embedded, locally linear,
positively oriented path $t\to x_{t}$ in $\mbox{\boldmath{$R$}}/2\pi$ which
joins $x_{0}$ and $x_{1}$. This path is short (respectively long) when
$(x_{0},x_{1})$ is near $\partial_{+}$ (respectively $\partial_{-}$). Let
$u_{t}$ be the bend associated to $x_{t}$. We write
$\overrightarrow{u}_{0}\leadsto\overrightarrow{u}_{1}$ when there is a
continuous orientation of the bends $\\{u_{t}\\}$ that restricts to
$\overrightarrow{u}_{0}$ and $\overrightarrow{u}_{1}$. Note that
$-\overrightarrow{u}_{0}\leadsto-\overrightarrow{u}_{1}$ and, since $\Omega$
is a Moebius band, $\overrightarrow{u}_{1}\leadsto-\overrightarrow{u}_{0}$.
Also $\overrightarrow{u}_{1}\to\pm\overrightarrow{u}_{0}$ when
$(x_{0},x_{1})\to\partial_{\pm}$.

The Map: Let $m_{j}$ be the midpoint of $u_{j}$. Using the dot product
$(\cdot)$ and the cross product $(\times)$ define
$F=(g,h):\Upsilon\to\mbox{\boldmath{$R$}}^{2}$, where

$g(x_{0},x_{1})=\overrightarrow{u}_{0}\cdot\overrightarrow{u}_{1},\hskip
30.0pth(x_{0},x_{1})=(m_{0}-m_{1})\cdot(\overrightarrow{u}_{0}\times\overrightarrow{u}_{1}).$
(2)

This definition is independent of the chosen orientation since
$-\overrightarrow{u}_{0}\leadsto-\overrightarrow{u}_{1}$. Also, $F$ extends
continuously $S^{2}$ with $F(\partial_{\pm})=(\pm 1,0)$. Since
$\overrightarrow{u}_{1}\leadsto-\overrightarrow{u}_{0}$ we have
$F\circ\Sigma=-F$. Why? Well, clearly $g(x_{1},x_{0})=-g(x_{0},x_{1})$, and

$h(x_{1},x_{0})=(m_{1}-m_{0})\cdot(\overrightarrow{u}_{1}\times(-\overrightarrow{u}_{0}))=(m_{1}-m_{0})\cdot(\overrightarrow{u}_{0}\times\overrightarrow{u}_{1})=-h(x_{0},x_{1}).$

The Borsuk-Ulam Theorem says that $(0,0)\in F(S^{2})$. Since
$F(\partial_{\pm})\not=(0,0)$ we have $(0,0)\in F(\Upsilon)$. See the remark
below for a self-contained proof.

Endgame: Let $(u_{0},u_{1})$ be the bends corresponding to $(x_{0},x_{1})\in
F^{-1}(0,0)$. First, $u_{0}$ and $u_{1}$ are disjoint because they belong to
the same foliation. Second, $\overrightarrow{u}_{0}$ and
$\overrightarrow{u}_{1}$ are orthogonal because $g(x_{0},x_{1})=0$. Third,
$\overrightarrow{u}_{0}$ and $\overrightarrow{u}_{1}$ and $m_{0}-m_{1}$ are
all orthogonal to $\overrightarrow{u}_{0}\times\overrightarrow{u}_{1}$ because
$h(x_{0},x_{1})=0$, and this easily implies that $u_{0},u_{1}$ are coplanar.
Hence $(u_{0},u_{1})$ is an embedded T-pattern. This proves Lemma T.

Remark: Here is a self-contained proof that $(0,0)\in F(\Upsilon)$. Suppose
not. If $\gamma$ is a continuous path in $S^{2}$ which goes from
$\partial_{+}$ to $\partial_{-}$, then $F(\gamma)$ goes from $(1,0)$ to
$(-1,0)$, misses $(0,0)$, and winds some half integer $w(\gamma)$ times around
the origin. All choices of $\gamma$ are homotopic to each other relative to
$\partial_{\pm}$, so $w(\gamma)$ is independent of $\gamma$. But consider
$\gamma^{\prime}=\Sigma(\gamma)$, re-oriented so that it goes from
$\partial_{+}$ to $\partial_{-}$. Since $F\circ\Sigma=-F$ the image
$F(\gamma^{\prime})$ is obtained by rotating $F(\gamma)$ by $180$ degrees
about $(0,0)$ then re-orienting it so that it goes from $(1,0)$ to $(-1,0)$.
But then $w(\gamma^{\prime})=-w(\gamma)$, a contradiction.

###  2.3 Proof of Lemma G

Let $\ell$ denote arc length. Let $\bigtriangledown$ be a triangle with
horizontal base. Let $\vee$ be the union of the two non-horizontal sides of
$\bigtriangledown$.

###### Lemma 2.3

If $\bigtriangledown$ has base $\sqrt{1+t^{2}}$ and height $h\geq 1$ then
$\ell(\vee)\geq\sqrt{5+t^{2}}$, with equality iff $\bigtriangledown$ is
isosceles and $h=1$.

Proof: This is an exercise in high school geometry. Let $p_{1},p_{2},q$ be the
vertices of $\bigtriangledown$, with $p_{1},p_{2}$ lying on the base. Let
$p_{2}^{\prime}$ be the reflection of $p_{2}$ in the horizontal line through
$q$. Note that $\bigtriangledown$ is isosceles iff $p_{1},q,p_{2}^{\prime}$
are collinear. By symmetry, the triangle inequality, and the Pythagorean
Theorem,

$\ell(\vee)=\|p_{1}-q\|+\|q-p_{2}^{\prime}\|\geq\|p_{1}-p_{2}^{\prime}\|=\sqrt{1+t^{2}+4h^{2}}\geq\sqrt{5+t^{2}}.$

We get equality if and only if $p_{1},q,p_{2}^{\prime}$ are collinear and
$h=1$. $\spadesuit$

Let $I:M_{\lambda}\to\Omega$ be a smooth embedded paper Moebius band with an
embedded $T$-pattern. Let $S^{\prime}=I(S)$ for any $S\subset M_{\lambda}$. We
have $\ell(\gamma)=\ell(\gamma^{\prime})$ for any curve $\gamma\subset
M_{\lambda}$.

We rotate $\Omega$ so that one of the bends of the $T$-pattern, $T^{\prime}$,
lies in $X$-axis and the other bend, $B^{\prime}$, lies in the negative ray of
the $Y$-axis. Next, we let $B=I^{-1}(B^{\prime})$ and $T=I^{-1}(T^{\prime})$
be the corresponding pre-bends. We emphasize that $B$ and $T$ are disjoint
embedded line segments.

We cut $M_{\lambda}$ open along $T$ to get a bilaterally symmetric trapezoid
$\tau$. We normalize $\tau$ so that the parallel sides are horizontal.
Reflecting $\tau$ in the coordinate axes if needed, we arrange that $u,v,w,x$
are mapped to $\Omega$ as in Figure 2. The quantities $t$ and $b$ (which are
both positive in the case depicted) respectively denote the horizontal
displacements of $T$ and $B$.

Figure 2: The trapezoid $\tau$ (left) and the T-pattern (right).

With Lemma 2.3 in mind, note that the shaded triangle $\bigtriangledown$ has

${\rm base\/}=\ell(T^{\prime})=\ell(T)=\sqrt{1+t^{2}},\hskip 20.0pt{\rm
height\/}>\ell(B^{\prime})=\ell(B)=\sqrt{1+b^{2}}\geq 1.$

Let $H=H_{1}\cup H_{2}$ and $D=D_{1}\cup D_{2}$. We have
$H^{\prime},D^{\prime}\subset\mbox{\boldmath{$R$}}^{3}$, so you should imagine
you are floating in space above Figure 2 and looking down. Now $H^{\prime}$
connects the endpoints of $T^{\prime}$. Also, $D^{\prime}$ connects the
endpoints of $T^{\prime}$ and contains $u$. From this structure, and from
Figure 2 (left), we have

$\displaystyle\ell(H)+\ell(D)=2\lambda.$ $\displaystyle\ell(D)-2t=\ell(H).$
$\displaystyle\sqrt{1+t^{2}}=\ell(T^{\prime})\leq\ell(H^{\prime})=\ell(H).$
$\displaystyle\sqrt{5+t^{2}}<^{*}\ell(\vee)\leq\ell(D^{\prime})=\ell(D).$ (3)

The starred inequality is Lemma 2.3. Equation 2.3 give us

$\displaystyle\alpha(t):=\sqrt{1+t^{2}}+\sqrt{5+t^{2}}<\ell(H)+\ell(D)=2\lambda.$
$\displaystyle\beta(t):=2\sqrt{5+t^{2}}-2t<2\ell(D)-2t=\ell(D)+\ell(H)=2\lambda.$
(4)

Hence

$2\lambda>\max(\alpha(t),\beta(t)).$ (5)

Let $t_{0}=1/\sqrt{3}$. We have

  * •

$\alpha(t_{0})=\beta(t_{0})=2\sqrt{3}$.

  * •

$\alpha$ is increasing on $(0,\infty)$. Hence $\alpha(t)>2\sqrt{3}$ if
$t>t_{0}$.

  * •

$\beta$ is decreasing on $R$. Hence $\beta(t)>2\sqrt{3}$ if $t<t_{0}$.

Hence $\lambda>\sqrt{3}$. This proves Lemma G.

Our proof of the Main Theorem is done.

###  2.4 Proof of the Triangular Limit Theorem

Figure 3 shows what Figure 2 looks like when we make the construction for the
triangular Moebius band with respect to the $T$-pattern depicted on the right
side of Figure 1b.

Figure 3: Figure 2 for the triangular Moebius band.

Let $\\{I_{n}:M_{\lambda_{n}}\to\Omega_{n}\\}$ be as in the Triangular Limit
Theorem. We run the construction of Lemma G for each $n$ and analyze what
happens as $n\to\infty$. We use the same notation as in the proof of Lemma G,
except that we add subscripts to indicate the dependence on $n$.

The Range: Since $\max(\alpha(t_{n}),\beta(t_{n}))\to 2\sqrt{3}$ we have
$t_{n}\to 1/\sqrt{3}$. Hence $\ell(T^{\prime}_{n})\to 2/\sqrt{3}$. Since
$\lambda_{n}\to\sqrt{3}$ and $\ell(\bigtriangledown_{n})\leq 2\lambda_{n}$, we
have

$\limsup\ell(\bigtriangledown_{n})\leq 2\sqrt{3}.$ (6)

The base of $\bigtriangledown_{n}$ converges to $2/\sqrt{3}$ and the height is
always greater than $1$. Lemma 2.3 combines with Equation 6 to show that
$\bigtriangledown_{n}$ converges (up to isometries) to an isosceles triangle
of base $2/\sqrt{3}$ and height $1$. This is the shaded equilateral triangle
$\bigtriangledown$ shown in Figure 3 (right). We normalize by isometries so
that we get actual convergence.

The Domain: Since $\ell(\partial\Omega_{n})-\ell(\bigtriangledown_{n})\to 0$
and also $v_{n}^{\prime}$ converges to the midpoint of $T^{\prime}_{n}$, all
the slack goes out of Equation 2.3, and

$\lim\ell(H_{n,1}^{\prime})=\lim\ell(H_{n,2}^{\prime})=1/\sqrt{3},\hskip
10.0pt\lim\ell(D_{n,1}^{\prime})=\lim\ell(D_{n,2}^{\prime})=2/\sqrt{3}.$ (7)

Since $\ell(H_{n,1})=\ell(H_{n,1}^{\prime})$, etc. $\tau_{n}$ converges (up to
isometries) to $\tau$, the trapezoid in Figure 3 (left). We normalize so that
we get actual convergence.

The Boundary Map: The arcs
$H_{n,1}^{\prime},H_{n,2}^{\prime},D_{n,1}^{\prime},D_{n,2}^{\prime}$ converge
as sets to the line segments connecting their endpoints because
$\ell(\partial\Omega_{n})-\ell(\bigtriangledown_{n})\to 0$. Since $I_{n}$ is
length preserving, $I_{n}$ converges uniformly to a linear isometry when
restricted to each of $H_{n,1},H_{n,2},D_{n,1},D_{n,2}$. Hence $I_{n}$
converges on $\partial M_{\lambda_{n}}$ to the piecewise linear isometry
associated to the triangular Moebius band.

The Whole Map: We divide $M_{\lambda}$ into $3$ triangles. See Figure 4.

Figure 4: Figure 2 revisited

Consider the restriction of $I_{n}$ to the left triangle $\bigtriangleup_{n}$.
First, $I_{n}$ is a linear isometry on $\overline{w_{n}x_{n}}=T_{n}$. Second,
$I_{n}$ converges to a linear isometry on $\overline{w_{n}u_{n}}=D_{n,1}$.
Third, $I_{n}$ converges to a linear isometry on $\overline{x_{n}u_{n}}$
because $\|x_{n}^{\prime}-u_{n}^{\prime}\|\to\|x_{n}-u_{n}\|$ and $I_{n}$ is
distance non-increasing. In summary, $I_{n}$ converges to a linear isometry on
$\partial\bigtriangleup_{n}$. Since $I_{n}$ is distance non-increasing, this
implies that $I_{n}$ converges to a linear isometry on $\bigtriangleup_{n}$.
The same argument works for the other $2$ triangles. Hence $I_{n}$ converges
to the piecewise linear isometry associated to the triangular Moebius band.

Our proof of the Triangular Limit Theorem is done.

Remark: The proof is done, but we have a bit more to say. Let $p_{n}$ be the
midpoint of $\overline{x_{n}u_{n}}$. The point $p_{n}^{\prime}$ converges to
the midpoint of $\overline{x_{n}^{\prime}u_{n}^{\prime}}$. The bend
$\beta_{n}^{\prime}$ through $p_{n}^{\prime}$ has its endpoints on
$H_{n,1}^{\prime}$ and $D_{n,1}^{\prime}$, and this forces one endpoint of
$\beta_{n}^{\prime}$ to converge to $x_{n}^{\prime}$ and the other to
$u_{n}^{\prime}$. This means the pre-bend
$\beta_{n}=I_{n}^{-1}(\beta_{n}^{\prime})$ converges to
$\overline{x_{n}u_{n}}$. In other words, the left dotted line in Figure 4 is,
for large $n$, quite close to a pre-bend. The same argument works for the
right dotted line. From this, we see that the pre-bend foliation of
$M_{\lambda_{n}}$ converges to the pinstriping shown in Figures 1a (left) and
1b (left). Likewise, the bend foliation of $\Omega_{n}$ converges to the
pinstriping shown in Figures 1a (right) and 1b (right).

##  3 Discussion

###  3.1 An Explicit Homeomorphism

Referring to the proof of Lemma T, here we give an explicit homeomorphism from
$S^{2}$ to $\overline{\Upsilon}$. We identify $\partial_{+}$ and
$\partial_{-}$ respectively with the north and south pole of $S_{2}$. We
parametrize $S^{2}-\partial_{\pm}$ by $(\theta,\phi)$, where
$\theta\in\mbox{\boldmath{$R$}}/2\pi$ is the longitude and $\phi\in(0,\pi)$,
the angle with $\partial_{+}$, is the latitude. The antipodal map is
$(\theta,\phi)\to(\pi+\theta,\pi-\phi)$. We get a homeomorphism between
$S^{2}-\partial_{\pm}$ and $\Upsilon$ with the correspondence
$(\theta,\phi)\leftrightarrow(X_{\theta-\phi},X_{\theta+\phi})$. This
conjugates $\Sigma$ to the antipodal map, and extends to a homeomorphism
between $S^{2}$ and $\overline{\Upsilon}$.

###  3.2 Paths of Oriented Lines

Anton Izosimov and Sergei Tabachnikov independently suggested to me the
following generalization of Lemma T.

###### Lemma 3.1

Suppose $\\{L_{t}|\ t\in[0,1]\\}$ is a continuous family of oriented lines in
$\mbox{\boldmath{$R$}}^{3}$ such that $L_{1}=L_{0}^{\rm opp\/}$, the same line
as $L_{0}$ but with the opposite orientation. Then there exist parameters
$r,s\in[0,1]$ such that $L_{r}$ and $L_{s}$ are perpendicular intersecting
lines.

Proof: This has the same proof as Lemma T, once we observe that our function
$h$, defined in Equation 2, is more natural than we have let on. The points
$m_{0},m_{1}$ in the definition of $h$ could be any points on $u_{0},u_{1}$
and we would get the same result. $g$ and $h$ are invariants of pairs of
oriented lines. $\spadesuit$

Sergei also suggested to me a beautiful alternate formalism for Lemma T: the
dual numbers. These have the form $x+\epsilon y$ where
$x,y\in\mbox{\boldmath{$R$}}$ and $\epsilon^{2}=0$. Relatedly, the dual
vectors have the form $\overrightarrow{a}+\epsilon\overrightarrow{b}$, where
$\overrightarrow{a},\overrightarrow{b}\in\mbox{\boldmath{$R$}}^{3}$ and again
$\epsilon^{2}=0$. In this context, the dot product of two dual vectors makes
sense as a dual number. See [HH] for an exposition.

Each oriented line $\ell\subset\mbox{\boldmath{$R$}}^{3}$ gives rise to a dual
vector $\xi_{\ell}=\overrightarrow{a}+\epsilon\overrightarrow{b}$ where
$\overrightarrow{a}$ is the unit vector pointing in the direction of $\ell$
and $\overrightarrow{b}=\ell^{\prime}\times\overrightarrow{a}$. Here
$\ell^{\prime}\in\ell$ is any point. All choices of $\ell^{\prime}$ give rise
to the same $\overrightarrow{b}$; this vector is called the moment vector of
$\ell$. This formalism identifies the space of oriented lines in
$\mbox{\boldmath{$R$}}^{3}$ with the so-called study sphere consisting of dual
vectors $\xi$ such that $\xi\cdot\xi=1$. The dual dot product
$\xi_{\ell}\cdot\xi_{m}$ vanishes if and only if $\ell$ and $m$ are
perpendicular and intersect.

###  3.3 Interpretations of Smoothness at the Boundary

Our map $I:M_{\lambda}\to\Omega$ is a smooth isometric embedding. The domain
of $I$ is a smooth manifold with boundary. Whenever we have a situation like
this, the notion smoothness at the boundary really means that we can extend
the definition of $I$ at each boundary point to a larger open manifold so that
the extension is smooth. In our situation we can use compactness to say that
there is some $\epsilon>0$ such that $I$ extends smoothly to

$M_{\lambda}^{\epsilon}=[0,\lambda]\times(-\epsilon,1+\epsilon)/\\!\sim,\hskip
30.0pt(0,y)\sim(\lambda,1-y).$ (8)

A stronger interpretation would be that for some $\epsilon$ the extension is
also an isometric embedding. With this interpretation the proof of Lemma 2.2
(given below in §4) would be easier. We wouldn’t need to treat boundary points
in a special way. A weaker interpretation would be that $I$ is merely
continuous on $M_{\lambda}$ and a smooth isometric embedding on the interior
of $M_{\lambda}$. Our standard interpretation steers a middle ground between
these extremes.

Let us say that the Main Theorem♭ is the Main Theorem for the strong version
kind of smooth embedded paper Moebius bands. Let us say that the Main Theorem♯
is the Main Theorem for the weak version kind of smooth embedded paper Moebius
bands. We use similar notation for other concepts. Obviously, our theorems
imply our theorems♭, and in turn our theorems♯ would imply our theorems.
Actually the reverse is also true.

Given a smooth♯ embedded paper Moebius band we have a trimming operation: We
can restrict $I$ to the thinner Moebius band

$[0,\lambda]\times(\epsilon,1-\epsilon)/\\!\sim,\hskip
30.0pt(0,y)\sim(\lambda,1-y).$ (9)

Then we can rescale and domain and range to get a smooth${}^{\flat\/}$
embedded paper Moebius band of aspect ratio $\sqrt{3}/(1-2\epsilon)$. Applying
the Main Theorem♭ we see that $\lambda>\sqrt{3}(1-2\epsilon)$. Letting
$\epsilon\to 0$ we get $\lambda\geq\sqrt{3}$. Suppose $\lambda=\sqrt{3}$. Then
we consider a sequence of trimmings with $\epsilon\to 0$. The Triangular Limit
Theorem♭ says that these converge to the triangular Moebius band. But this
implies that our map $I:M_{\lambda}\to\Omega$ is the triangular Moebius band.
This is a contradiction. Hence $\lambda>\sqrt{3}$. Thus, the Main Theorem♭ and
the Triangular Limit Theorem♭ imply the Main Theorem♯. A similar argument
shows that the Triangular Limit Theorem♭ implies the Triangular Limit
Theorem♯.

In short, we really don’t need to fuss about the exactly what happens at the
boundary of a smooth embedded paper Moebius band.

###  3.4 Related Topics

Square Peg: Around the time I got interested in the Halpern-Weaver Conjecture
I had been thinking about the Toeplitz Square Peg Conjecture, which asks if
every continuous loop in the plane contains $4$ points which make the vertices
of a square. See [Mat] for a fairly recent survey. One can view a $T$-pattern
as a collection of $4$ points in the boundary of the Moebius band which
satisfy certain additional constraints – e.g. they are coplanar. Put this way,
a $T$-pattern is sort of like a square inscribed in a Jordan loop.

Quadrisecants: The idea for Lemma T is also similar in spirit for the idea
developed in [DDS] concerning $4$ collinear points on a knotted loop. These
so-called quadrisecants play a role similar to Lemma T in getting a lower
bound for the length of a knotted rope.

Folded Ribbon Knots: Elizabeth Denne pointed out to me the connection between
paper Moebius bands and folded ribbon knots. Her paper with Troy Larsen [DL]
gives a formal definition of a folded ribbon knot and has a wealth of
interesting constructions, results, and conjectures. See also [D], a survey
article.

Informally, folded ribbon knots are the objects you get when you take a flat
cylinder or Moebius band, fold it into a knot, and then press it into the
plane. Associated to a folded ribbon knot is a polygon, which comes from the
centerline of the object. Even though the ribbon knot lies entirely in the
plane, one assigns additional combinatorial data which keeps track of
“infinitesimal” under and over crossings as in a knot diagram. So the
associated centerline is really a knot (or possibly the unknot).

[DL, Corollary 25] proves our Main Theorem in the category folded ribbon
Moebius bands whose associated centerline is a triangle. This is a finite
dimensional problem. [DL, Conjecture 26] says that [DL, Corollary 25] is true
without the very strong triangle restriction. This is an infinite dimensional
problem like the Halpern-Weaver Conjecture. The combination of our Main
Theorem and the Triangular Limit Theorem implies [DL, Conjecture 26]. One
takes arbitrarily nearby smooth approximations, as in [HW], and then applies
our results to them.

###  3.5 More Twists

The Halpern-Weaver Conjecture is one of infinitely many similar kinds of
questions one can ask about paper Moebius bands. For instance, one translate
the many conjectures made in [DL] about folded ribbon knots into the broader
language of paper geometry.

Twisted Cylinders: One can make a twisted cylinder by taking a
$1\times\lambda$ strip of paper, giving it an even and nonzero number of
twists, and then joining the ends together. The essential feature of twisted
cylinders is that their two boundary components make a nontrivial link.

There are two optimal limiting shapes which have interpretations as folded
ribbon knots. Both are folding patterns which wrap a $1\times 2$ strip $4$
times around a right-angled isosceles triangle. In [S3] I prove that a twisted
cylinder has aspect ratio greater than $2$ and that any minimizing sequence
converges on a subsequence to one of the two optimal models. This result also
confirms the $n=1$ case of [DL, Conjecture 39]. The proof is somewhat similar
to what I do in this paper, though the fine-scale details are different. Noah
Montgomery (private communication) independently came up with a proof of the
cylinder result. His elegant proof is different than mine.

Three Twist Moebius Bands: What happens when you insist that a paper Moebius
band have least $3$ twists? An essential feature of these objects is that
their boundaries are knotted. I think it follows from the Triangular Limit
Theorem and from compactness that there is some $\epsilon_{0}$ such that the
aspect ratio of a multi-twisted paper Moebius band is at least
$\sqrt{3}+\epsilon_{0}$.

Brienne Brown did some experiments with these objects and found two candidate
optimal models. We call these the crisscross and the cup. Both are made from a
$1\times 3$ strip of paper. The crisscross is planar, and has an
interpretation as a folded ribbon knot. The cup is not-planar: It is a double
wrap of $3$ mutually orthogonal right-angled isoceles triangles arranged like
$3$ faces of a tetrahedron. We wrote about this in [BS], and conjecture there
that $\lambda>3$ for an embedded multi-twisted paper Moebius band.

Multi-Twist Examples: Recently, Aidan Hennessey [H] proved the very surprising
result that one can make a cylinder or a Moebius band with any number of
twists using a $1\times 8$ strip. For Moebius bands only, his bound is better:
you can do it with a $1\times 6.25$ strip of paper.

##  4 Appendix: Proof of Lemma 2.2

We give a self-contained proof of Lemma 2.2. Let $S^{2}$ be the unit
$2$-sphere. The Gauss map, which is well defined and smooth on any simply-
connected subset of $\Omega$, associates to each point $p\in\Omega$ a unit
normal vector $n_{p}\in S^{2}$. Let $dn_{p}$ be the differential of the Gauss
map at $p$. Since the curvature $\Omega$ is $0$ everywhere, $dn_{p}$ has a
nontrivial kernel. The point $p$ has nonzero mean curvature if and only if
$dn_{p}$ has nontrivial image.

Let $U$ be as in Lemma 2.2, the subset of $\Omega^{o}$ with nonzero mean
curvature. Let $p\to n_{p}$ be a local choice of the Gauss map. We can rotate
and translate so that near the origin $U$ is the graph of a function

$F(x,y)=Cy^{2}+{\rm higher\ order\ terms\/}.$ (10)

Here $C>0$ is some constant. The normal vector at the origin is
$n_{0}=(0,0,1)$. The vector $v_{0}=(1,0,0)$ lies in the kernel of $dn_{0}$.
Let $w_{0}=(0,1,0)$. Let $\Pi_{0}$ be the plane spanned by $w_{0}$ and
$n_{0}$. The image of $\Pi_{0}\cap U$ under the Gauss map is (near $n_{0}$) a
smooth regular curve tangent to $w_{0}$ at $n_{0}$. The sign depends on the
choice of local Gauss map.

Working locally, we have three smooth vectorfields:

$p\to n_{p},\hskip 30.0ptp\to v_{p},\hskip 30.0ptp\to w_{p}=v_{p}\times
n_{p}.$ (11)

Here $v_{p}$ is the kernel of $dn_{p}$ and $\times$ denote the cross product.
Let $\Pi_{p}$ be the plane through $p$ and spanned by $w_{p}$ and $n_{p}$.
From our analysis of the special case, and from symmetry, the image of
$\Pi_{p}\cap U$ under the Gauss map is (near $n_{p}$) a smooth regular curve
tangent to $w_{p}$ at $n_{p}$. The asymptotic curves are the smooth curves
everywhere tangent to the $v$ vector field.

###### Lemma 4.1

The asymptotic curves are line segments.

Proof: Let $\gamma$ be an asymptotic curve. The Gauss map is constant along
$\gamma$. Each point in $\gamma$ has a small neighborhood $V$ which is
foliated by asymptotic curves that transversely intersect each plane $\Pi_{p}$
when $p\in\gamma\cap V$. Hence the image of $V$ under the Gauss map equals the
image of $\Pi_{p}\cap V$ under the Gauss map. This latter image is a smooth
regular curve tangent to $w_{p}$ at $n_{p}$. Since this is true for all
$p\in\gamma\cap V$ and since $n_{p}$ is constant along $\gamma$, $w_{p}$ is
constant along $\gamma$. Hence $v_{p}$ is constant along $\gamma$. Hence
$\gamma$ is a line segment. $\spadesuit$

The nonzero mean curvature implies that $\gamma$ is the unique line segment
through any of its interior points. We first rule out the possibility that
$\gamma$ reaches $\partial U$ before it reaches $\partial\Omega$. Assume for
the sake of contradiction that this happens. We normalize as in Equation 10.

We now allow ourselves the liberty of dilating our surface. This dilation
preserves all the properties we have discussed above. By focusing on a point
of $\gamma$ sufficiently close to $\partial U$ and dilating, we arrange the
following:

  * •

A neighborhood $\cal V$ of $\Omega^{o}$ is the graph of a function over the
disk of radius $3$ centered at the origin.

  * •

Given $p\in\cal V$ let $p^{\prime}$ be the projection of $p$ to the
$XY$-plane. We have $|p_{1}^{\prime}-p_{2}^{\prime}|>(2/3)|p_{1}-p_{2}|$ for
all $p_{1},p_{2}\in\cal V$.

  * •

$\gamma\subset U$ contains the arc connecting $(0,0,0)$ to $(3,0,0)$, but
$(0,0,0)\not\in U$.

Let $a\in(0,3)$. At $(a,0,0)$ we have $v_{a}=(1,0,0)$ and $w_{a}=(0,1,0)$ and
$n_{a}=(0,0,1)$. Let $\Pi_{a}$ be the plane $\\{X=a\\}$. Near $(a,0,0)$, the
intersection $U_{a}=U\cap\Pi_{a}$ is a smooth curve tangent to $w_{a}$ at
$(a,0,0)$.

Let $\zeta=(1,0,0)$. Fix $\delta>0$. By continuity and compactness, the
asymptotic curves through points of $U_{1}$ sufficiently near $\zeta$ contain
line segments connecting points on $U_{2}$ to points on $U_{\delta}$. Call
these connectors. There exists a canonical map $\Phi_{\delta}:U_{1}\to
U_{\delta}$ defined in a neighborhood of $\zeta$: The points $q\in U_{1}$ and
$\Phi_{\delta}(q)\in U_{\delta}$ lie in the same connector.

###### Lemma 4.2

$\Phi_{\delta}$ expands distances by less than a factor of $3$.

Proof: Let $\ell_{1}$ and $\ell_{2}$ be two connectors. Let
$a_{j}=\ell_{j}\cap U_{1}$. Let $b_{j}=\ell_{j}\cap U_{\delta}$. For any set
$S$ let $S^{\prime}$ be the projection of $S$ to $\mbox{\boldmath{$R$}}^{2}$.
We have the bounds

$\frac{|a_{1}^{\prime}-a_{2}^{\prime}|}{|a_{1}-a_{2}|},\frac{|b_{1}^{\prime}-b_{2}^{\prime}|}{|b_{1}-b_{2}|}\in\bigg{[}\frac{2}{3},1\bigg{]},\hskip
30.0pt\frac{|a_{j}^{\prime}-b_{j}^{\prime}|}{{\rm
length\/}(\ell_{j}^{\prime})}<2.$

Geometrically, $a_{j}^{\prime}$ is very nearly the midpoint of
$\ell_{j}^{\prime}$ and $b_{j}^{\prime}$ is the closer of the two endpoints.
Since $\ell_{1}^{\prime}$ and $\ell_{2}^{\prime}$ are planar and disjoint, our
last inequality (and essentially a similar-triangles argument) gives
$|b_{1}^{\prime}-b_{2}^{\prime}|/|a_{1}^{\prime}-a_{2}^{\prime}|<2.$ Putting
everything together, we have $|b_{1}-b_{2}|/|a_{1}-a_{2}|<3$. $\spadesuit$

Fix $\epsilon>0$. The mean curvature along $U_{\delta}$ tends to $0$ as
$\delta\to 0$. If we choose $\delta$ sufficiently small then the Gauss map
expands distances along $U_{\delta}$ in a neighborhood of $(\delta,0,0)$ by a
factor of less than $\epsilon$. Combining Lemma 4.2 and the fact that
$n_{q}=n_{\Phi_{\delta}(q)}$, we see that the Gauss map expands distances by
at most a factor of $3\epsilon$ along $U_{1}$ in a small neighborhood of
$\zeta$. Since $\epsilon$ is arbitrary, $w_{1}\in{\rm ker\/}(dn_{\zeta})$. But
$v_{1}\in{\rm ker\/}(dn_{\zeta})$ by definition. Hence $dn_{\zeta}$ is the
trivial map. The contradicts the fact that $\zeta\in U$.

Now we know that the interior of $\gamma$ lies in $U$. Next, we show that the
mean curvature cannot vanish at an endpoint of $\gamma$. As discussed in §3.3,
there is some $\epsilon>0$ so that $I$ has a smooth extension to the fatter
open Moebius band $M^{\epsilon}_{\lambda}$ described in Equation 8. If we knew
that this smooth extension was also an isometric embedding, we could just use
our previous argument exactly as is. (Compare the discussion in §3.3.) But,
even if the extension is not an isometric map, we can still use the
essentially the same argument. This time our open disk of radius $3$ is
divided into two sides, a lightly shaded familiar side consisting of
projections of points of $\Omega$ and a darkly shaded mystery side consisting
of projections of points of $I(M_{\lambda}^{\epsilon})-I(M_{\lambda})$. The
common boundary of these regions is curve, transverse to the $x$-axis, which
is a projection of $I(\partial M_{\lambda})$. Figure 5 shows what we are
taking about.

Figure 5: The connectors lie on the familiar side.

In the argument above, we can take the connectors to all lie in the familiar
side, as shown in Figure 5. Then we can run exactly the same argument without
ever entering into the mystery side. We then get the same contradiction as
above. Hence the mean curvature does not vanish at an endpoint.

This completes the proof of Lemma 2.2.

##  5 References

[BS] B. E. Brown and R. E. Schwartz, The crisscross and the cup: Two short
$3$-twist paper Moebius bands, preprint 2023, arXiv:2310.10000 [CF] Y. Chen
and E. Fried, Mobius bands, unstretchable material sheets and developable
surfaces, Proceedings of the Royal Society A, (2016) [CK] Carmen Chicone, N.
J. Kalton, Flat Embeddings of the Moebius Strip in
$\mbox{\boldmath{$R$}}^{3}$, Communications in Nonlinear Analysis 9 (2002),
no. 2, pp 31-50

[CKS] J. Cantarella, R. Kusner, J. Sullivan, On the minimum ropelength of
knots and links, Invent. Math. 150 (2) pp 257-286 (2003) [CL], S.-S. Chern and
R. K. Lashof, On the total curvature of immersed manifolds, Amer. J. Math. 79
(1957) pp 306–318 [D] E. Denne, Folded Ribbon Knots in the Plane, The
Encyclopedia of Knot Theory (ed. Colin Adams, Erica Flapan, Allison Henrich,
Louis H. Kauffman, Lewis D. Ludwig, Sam Nelson) Chapter 88, CRC Press (2021)
[DL] E. Denne, T. Larsen, Linking number and folded ribbon unknots, Journal of
Knot Theory and Its Ramifications, Vol. 32 No. 1 (2023) [DDS] E. Denne, Y.
Diao, J. M. Sullivan, Quaadrisecants give new lower bounds for the ropelength
of a knot, Geometry$\&$Topology 19 (2006) pp 1–26 [FT], D. Fuchs, S.
Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, AMS
2007 [H] A. Hennessey, Constructing many-twist Möbius bands with small aspect
ratios. arXiv:2401:14639 (2024) [HF], D.F. Hinz, E. Fried, Translation of
Michael Sadowsky’s paper ‘An elementary proof for the existence of a
developable MÖBIUS band and the attribution of the geometric problem to a
variational problem’. J. Elast. 119, 3–6 (2015) [HH], A. A. Harkin and J. B.
Harkin, Geometry of Generalized Complex Numbers, Mathematics Magazine 77 (2),
pp 118-129 (2004) [HL], P. Hartman and L. Nirenberg, On spherical maps whose
Jacobians do not change sign, Amer. J. Math. 81 (1959) pp 901–920 [HW], B.
Halpern and C. Weaver, Inverting a cylinder through isometric immersions and
embeddings, Trans. Am. Math. Soc 230, pp 41–70 (1977) [KU], Kurono, Yasuhiro
and Umehara, Masaaki, Flat Möbius strips of given isotopy type in
$\mbox{\boldmath{$R$}}^{3}$ whose centerlines are geodesics or lines of
curvature, Geom. Dedicata 134 (2008), pp 109-130 [MK] L. Mahadevan and J. B.
Keller, The shape of a Möbius band, Proceedings of the Royal Society A (1993)
[Ma1] B. Matschke, A survey on the Square Peg Problem, Notices of the A.M.S.
Vol 61.4, April 2014, pp 346-351. [RR] T. Randrup, P. Rogan, Sides of the
Möbius Strip, Arch. Math. 66 (1996) pp 511-521

[Sa], M. Sadowski, Ein elementarer Beweis für die Existenz eines abwickelbaren
MÖBIUSschen Bandes und die Zurückführung des geometrischen Problems auf
einVariationsproblem. Sitzungsberichte der Preussischen Akad. der
Wissenschaften, physikalisch-mathematische Klasse 22, 412–415.2 (1930) [Sab]
I. Kh. Sabitov, Isometric immersions and embeddings of a flat Möbius strip
into Euclidean spaces, Izv. Math. 71 (2007), pp 1049-1078 [S1] R. E. Schwartz,
An improved bound on the optimal paper Moebius band, Geometriae Dedicata, 2021
[S2] R. E. Schwartz, The Optimal Paper Moebius Band: A Friendly Account,
preprint, 2023 [S3] R. E. Schwartz, The Optimal Twisted Paper Cylinder,
preprint 2023, arXiv:2309.14033 [SH] E. L. Starostin, G. H. M. van der
Heijden, The shape of a Möbius Strip, Nature Materials 6 (2007) pp 563 – 567
[Sz] G. Schwarz, A pretender to the title “canonical Moebius strip”, Pacific
J. of Math., 143 (1) pp. 195-200, (1990) [T] Todres, R. E., Translation of W.
Wunderlich’s On a Developable Möbius band, Journal of Elasticity 119 pp 23–34
(2015) [W] W. Wunderlich, Über ein abwickelbares Möbiusband, Monatshefte für
Mathematik 66 pp 276–289 (1962)


Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

# Directly Probing the CP-structure of the Higgs-Top Yukawa at HL-LHC and
Future Colliders

Rahool Kumar Barman  Department of Physics, Oklahoma State University,
Stillwater, OK, 74078, USA  Morgan E. Cassidy  Department of Physics and
Astronomy, University of Kansas, Lawrence, Kansas 66045 U.S.A.  Zhongtian Dong
Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas
66045 U.S.A.  Dorival Gonçalves  Department of Physics, Oklahoma State
University, Stillwater, OK, 74078, USA  Jeong Han Kim  Department of Physics,
Chungbuk National University, Cheongju, 28644, Korea  Felix Kling  Deutsches
Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Kyoungchul
Kong  Department of Physics and Astronomy, University of Kansas, Lawrence,
Kansas 66045 U.S.A.  Ian M. Lewis  Department of Physics and Astronomy,
University of Kansas, Lawrence, Kansas 66045 U.S.A.  Yongcheng Wu  Department
of Physics, Oklahoma State University, Stillwater, OK, 74078, USA  Yanzhe
Zhang  Department of Physics and Astronomy, University of Kansas, Lawrence,
Kansas 66045 U.S.A.  Ya-Juan Zheng  Department of Physics and Astronomy,
University of Kansas, Lawrence, Kansas 66045 U.S.A.

###### Abstract

Executive Summary

Constraining the Higgs boson properties is a cornerstone of the LHC program
and future colliders. In this Snowmass contribution, we study the potential to
directly probe the Higgs-top CP-structure via the $t\bar{t}h$ production at
the HL-LHC, 100 TeV FCC and muon colliders. We find the limits on the CP phase
($\alpha$) at 95% CL are $|\alpha|\lesssim 36^{\circ}$ with dileptonic
$t\bar{t}(h\to b\bar{b})$ and $|\alpha|\lesssim 25^{\circ}$ with combined
$t\bar{t}(h\to\gamma\gamma)$ at the HL-LHC. The 100 TeV FCC brings a
significant improvement in sensitivity with $|\alpha|\lesssim 3^{\circ}$ for
the dileptonic $t\bar{t}(h\to b\bar{b})$, due to the remarkable gain in the
signal cross-section and the increased luminosity. At future muon colliders,
we find that the bounds with semileptonic $t\bar{t}(h\to
b\bar{b})\nu\bar{\nu}$ are $|\alpha|\lesssim 9^{\circ}$ for 10 TeV and
$|\alpha|\lesssim 3^{\circ}$ for 30 TeV, respectively.

††preprint: Contribution to Snowmass 2021: EF01, TF07

Bounds on $\alpha$ at 95% CL ($\kappa_{t}=1$) | Channel | Collider |  Luminosity
---|---|---|---
$|\alpha|\lesssim 36^{\circ}$ Goncalves:2021dcu  | dileptonic $t\bar{t}(h\to b\bar{b})$ | HL-LHC | 3 ab-1
$|\alpha|\lesssim 25^{\circ}$ Barman:2021yfh  |  $t\bar{t}(h\to\gamma\gamma)$ combination | HL-LHC | 3 ab-1
$|\alpha|\lesssim 3^{\circ}$ Goncalves:2021dcu  | dileptonic $t\bar{t}(h\to b\bar{b})$ | 100 TeV FCC | 30 ab-1
$|\alpha|\lesssim 9^{\circ}$ ToAppear  | semileptonic $t\bar{t}(h\to b\bar{b})$ | 10 TeV $\mu^{+}\mu^{-}$ | 10 ab-1
$|\alpha|\lesssim 3^{\circ}$ ToAppear  | semileptonic $t\bar{t}(h\to b\bar{b})$ |  30 TeV $\mu^{+}\mu^{-}$ | 10 ab-1

##  I Introduction

Beyond the Standard Model CP effects to the top-quark Yukawa can be
parameterized as

$\displaystyle\mathcal{L}\supset-\frac{m_{t}}{v}\kappa_{t}\bar{t}\left(\cos\alpha+i\gamma_{5}\sin\alpha\right)t\,h\,,$
(1)

where $\alpha$ is the CP-phase, $\kappa_{t}$ is a real number that controls
the interaction strength, and ${v=246}$ GeV. Following this parametrization,
the SM is described with $\kappa_{t}=1$ and $\alpha=0$. In contrast, a purely
CP-odd interaction would display $\alpha=\pi/2$.

Numerous kinematic observables have been proposed in the literature to probe
the CP-structure of the Higgs-top interaction Ellis:2013yxa ; Boudjema:2015nda
; Buckley:2015vsa ; Buckley:2015ctj ; Demartin:2014fia ; Gritsan:2016hjl ;
Goncalves:2016qhh ; Mileo:2016mxg ; AmorDosSantos:2017ayi ; Azevedo:2017qiz ;
Li:2017dyz ; Goncalves:2018agy ; ATLAS:2018mme ; CMS:2018uxb ; Ren:2019xhp ;
Bortolato:2020zcg ; Cao:2020hhb ; Martini:2021uey ; Goncalves:2021dcu ;
Barman:2021yfh ; Bahl:2022yrs . Some illustrative examples are the transverse
momentum of the Higgs boson $p_{Th}$, the invariant mass of the top-quark pair
$m_{tt}$, and the angle between the beam direction and the top-quark in the
$t\bar{t}$ center of mass frame $\theta^{*}$ also known as Collins-Soper
angle. These variables are sensitive to the squared terms $\cos^{2}\alpha$ and
$\sin^{2}\alpha$, being CP-even. In particular, they are insensitive to the
sign of the CP-phase. CP-odd observables can be constructed from antisymmetric
tensor products that demand four four-momenta. Owning to the top-quark short
lifetime, the top-quark polarization is passed to its decay products. Thus, it
is possible to construct such tensor product with the top pair and their decay
products, such as
$\epsilon(p_{t},p_{\bar{t}},p_{i},p_{k})\equiv\epsilon_{\mu\nu\rho\sigma}p_{t}^{\mu}p_{\bar{t}}^{\nu}p_{i}^{\rho}p_{k}^{\sigma}$
Boudjema:2015nda ; Mileo:2016mxg ; Goncalves:2018agy . This tensor product can
be simplified to $\vec{p}_{t}\cdot(\vec{p}_{i}\times\vec{p}_{k})$ in the
$t\bar{t}$ CM frame, granting the definition of azimuthal angle differences
that are odd under CP transformation

$\displaystyle\\!\\!\\!\Delta\phi_{ik}^{t\bar{t}}\\!=\\!\text{sgn}\left[\vec{p}_{t}\\!\cdot\\!(\vec{p}_{i}\\!\times\\!\vec{p}_{k})\right]\arccos\\!\left(\frac{\vec{p}_{t}\\!\times\\!\vec{p}_{i}}{|\vec{p}_{t}\\!\times\\!\vec{p}_{i}|}\\!\cdot\\!\frac{\vec{p}_{t}\\!\times\\!\vec{p}_{k}}{|\vec{p}_{t}\\!\times\\!\vec{p}_{k}|}\right)\\!.$
(2)

Given the complex multiparticle $t\bar{t}h$ phase space, it is enlightening to
quantify and compare how much information on the CP-phase $\alpha$ is
available using the distinct CP-even or CP-odd observables. Ref.
Barman:2021yfh  addressed this task, using the Fisher information as a
convenient metric. The results are reproduced in Fig. 1. The
$\Delta\eta_{t\bar{t}}$ and $\theta^{*}$ are the most sensitive CP-even
observables, carrying approximately 60% of the full information. Successively
adding further observables augment the information. This highlights that it is
crucial to perform a multivariate analysis to maximize the CP sensitivity.
Efficient kinematic reconstruction methods will play a crucial role in this
task. In particular, it will grant the reconstruction of the top-quark pairs
required for the CP-odd observables in Eq. 2. This is notably challenging for
the dileptonic top-pair final state, that presents two missing neutrinos.

Figure 1: Left panel: Fisher information $I$ from the CP-odd observables for
the di-leptonic (red), semi-leptonic (gray), and hadronic (blue) top pair
final state. Right panel: Modified Fisher information $I^{\prime}$ sensitive
to the CP-even observables. These results are from Ref. Barman:2021yfh .

##  II HL-LHC

###  II.1 $pp\to t\bar{t}h(\gamma\gamma)$

ATLAS and CMS high-luminosity LHC (HL-LHC) projections indicate that the
$pp\to t\bar{t}h$ channel, in the diphoton $h\to\gamma\gamma$ final state,
will display dominant sensitivities to the top-quark Yukawa strength
$\kappa_{t}$ Cepeda:2019klc . Despite the limited statistics, the diphoton
final state analysis strongly benefits from controlled backgrounds from the
side-bands. Exploring this fact, Ref. Barman:2021yfh  shows that a combination
of machine learning techniques and efficient kinematic reconstruction methods
can boost new physics sensitivity on the CP-phase $\alpha$, effectively
probing the complex $t\bar{t}h$ multi-particle phase space. Special attention
is committed to top quark polarization observables, uplifting the study from a
raw rate to a polarization analysis.

In Fig. 2, we summarize the projected sensitivity in the $(\alpha,\kappa_{t})$
plane. In the left panel we show the projected $68\%$ CL contours from direct
Higgs-top searches in the semi-leptonic (blue), di-leptonic (green), hadronic
(red) $t\bar{t}h$ channels, and their combination (black), considering all
input observables. The right panel shows the projected $68\%$ CL (dashed) and
$95\%$ CL (solid) contours from the combination of the three channels,
considering all input observables. The color palette illustrates the expected
p-value of the estimated log-likelihood ratio. The projections are derived for
14 TeV LHC assuming $\mathcal{L}=3~{}\text{ab}^{-1}$.

Figure 2: Projected sensitivity in the $(\alpha,\kappa_{t})$ plane. In the
left panel we show the projected $68\%$ CL contours from direct Higgs-top
searches in the semi-leptonic (blue), di-leptonic (green), hadronic (red)
$t\bar{t}h$ channels, and their combination (black), considering all input
observables. The right panel shows the projected $68\%$ CL (dashed) and $95\%$
CL (solid) contours from the combination of the three channels. The color
palette illustrates the expected p-value of the estimated log-likelihood
ratio. The projections are derived for 14 TeV LHC assuming
$\mathcal{L}=3~{}\text{ab}^{-1}$. The results are from Ref. Barman:2021yfh .

###  II.2 $pp\to t\bar{t}h(b\bar{b})$

We perform a similar analysis with the Higgs decay to $b\bar{b}$ and two top
quarks to dilepton (see Ref. Goncalves:2021dcu  for details), which provides
the extra background suppression as well as a better probe to the top
polarization, using the charged leptons. The larger spin analyzing power
associated with the charged leptons results in the stronger CP-violation
observables, such as $\Delta\phi_{\ell\ell}^{t\bar{t}}$, strengthening the CP-
sensitivity. We adopt a binned log-likelihood analysis exploring the Higgs
candidate invariant mass profile in the boosted regime, for the signal range
${m_{J}^{\text{BDRS}}\in[110,135]\,\rm GeV}$, together with the CP-sensitive
observable $\theta^{*}$. Since the considered $t\bar{t}h$ channel with $h\to
b\bar{b}$ typically encounters a large $t\bar{t}b\bar{b}$ background, which
has a significant uncertainty ATLAS:2017fak ; CMS:2018hnq , the final result
displays relevant correlation with the considered background uncertainties. To
estimate this effect, we derive the new physics sensitivity on the
$(\alpha,\kappa_{t})$ plane for two scenarios. In the first case, we assume
that $t\bar{t}b\bar{b}$ background rate has 20% of uncertainty, which is
included as a nuisance parameter. The magnitude of the considered error is
comparable to that used in the current experimental analyses ATLAS:2017fak ;
CMS:2018hnq . For the second case, we assume an optimistic scenario with 5%
error. The uncertainties on the $t\bar{t}h$ and $t\bar{t}Z$ samples are
assumed to be 10% for both scenarios. The result of this analysis is presented
in the left panel of Fig. 3. We obtain that the CP-mixing angle can be
constrained to $|\alpha|\lesssim 32^{\circ}$ at 68% CL at the HL-LHC for both
scenarios. At the same time, we find that the sensitivity from $\kappa_{t}$ to
the systematic error is more pronounced. While in the first scenario we can
constrain the top Yukawa strength to $\delta\kappa_{t}\lesssim 0.3$, the more
optimistic case leads to $\delta\kappa_{t}\lesssim 0.15$.

Figure 3: The exclusion at 68% (red) and 95% (green) CL in the
$\alpha$-$\kappa_{t}$ plane at the 14 TeV LHC with $3\,{\rm ab}^{-1}$ for a
narrow (left) and wide (right) mass window. 20% systematics (5%) for
$t\bar{t}b\bar{b}$ is assumed in solid (dotted) curves, while 10% systematics
is used for both $t\bar{t}Z$ and $t\bar{t}h$. The results are from Ref.
Goncalves:2021dcu .

To illustrate how to reduce the systematics for $t\bar{t}b\bar{b}$ in a
realistic measurement, we enlarge the mass range of the Higgs candidate to
${m_{J}^{\rm BDRS}\in[50,150]\,\rm GeV}$. In this case, the events outside the
Higgs peak, which mainly arise due to $t\bar{t}b\bar{b}$ production, can be
used together with the shape of $m_{J}^{\rm BDRS}$ distribution of
$t\bar{t}b\bar{b}$ from MC simulation within the binned log-likelihood method.
Fitting to a broader range of $m_{J}^{\rm BDRS}$, we obtain a better control
of the uncertainties of $t\bar{t}b\bar{b}$ and show the results in the right
panel of Fig. 3. We find that this analysis depletes the influence of the
systematic uncertainties, leading to similar results for the two considered
systematic uncertainty scenarios. The obtained limits are $|\alpha|\lesssim
26^{\circ}~{}(36^{\circ})$ and $\delta\kappa_{t}\lesssim 0.12~{}(0.2)$ at 68%
(95%) CL. Using the wider mass window, the log-likelihood analysis takes full
advantage of the shape information of $t\bar{t}h$ and $t\bar{t}b\bar{b}$
events.

##  III 100 TeV FCC

Figure 4: Production cross-section for $pp\to t\bar{t}h$ and $pp\to t\bar{t}Z$
at the parton level as a function of the hadron collider energy. We consider
the Higgs and Z bosons in the boosted regime, $p_{Th,Z}>200$ GeV. Their
branching ratios to a bottom-quark pair $\mathcal{BR}(h,Z\to b\bar{b})$ are
accounted for. Top quarks are set stable. The results are from Ref.
Goncalves:2021dcu .

The Higgs-top CP-phase measurement would obtain remarkable gains at a future
100 TeV collider due to the immensely increased statistics. As shown in Fig.
4, while the $t\bar{t}(h\to b\bar{b})$ and $t\bar{t}(Z\to b\bar{b})$ processes
are phase space suppressed at the 14 TeV LHC, leading to 0.04 pb and 0.02 pb
for their cross-sections, the 100 TeV collider would result in one hundred-
fold enhancement, with a cross-section of 3.8 pb and 2.1 pb, respectively.
Considering the leptonic top pair decay, this corresponds to a significant
increase in the number of events (after top quark decays) for the $t\bar{t}h$
signal from $5.8\times 10^{3}$ at the HL-LHC with 3 ab-1 to $5.5\times 10^{6}$
at 100 TeV with 30 ab-1.

Performing a similar analysis with the uplifted cross-section and enlarged
luminosity, the 100 TeV FCC can boost the sensitivities on ($\alpha$,
$\kappa_{t}$), using the binned log-likelihood method, as summarized in Fig.
5. We choose a wide mass window, $m_{\rm BDRS}\in[50,150]$ GeV for better
control of the continuum $t\bar{t}b\bar{b}$ background, along with
$\theta^{\ast}$ in the left panel. In both panels, the solid curves correspond
to the case with 20% systematics for $t\bar{t}b\bar{b}$ and 10% systematics
for $t\bar{t}h$ and $t\bar{t}Z$, while we assume $t\bar{t}h$ and $t\bar{t}Z$
uncertainties are correlated for the dashed curves. It is clear that, at high
luminosities, the solid curves are limited by the systematic uncertainties.
However, by assuming that the systematics of $t\bar{t}h$ is correlated with
$t\bar{t}Z$, the precision can be improved, as shown by the dashed curves,
which can achieve $\delta\kappa_{t}\lesssim 1\%$ and $|\alpha|\lesssim
3^{\circ}$ at 95% CL. Finally, extending the analysis to the $(m_{\rm
BDRS},\theta^{*},\Delta\phi_{\ell\ell}^{t\bar{t}})$ plane, we find that the
CP-odd observable $\Delta\phi_{\ell\ell}^{t\bar{t}}$ ($\Delta\phi$ between two
leptons in the $t\bar{t}$ rest frame) brings additional improvement on the
measurement of $\alpha$ by a factor of 2, $|\alpha|\lesssim 1.5^{\circ}$, as
shown in the right panel of Fig. 5, which highlights the importance of the CP-
odd observable in the $t\bar{t}$ rest frame.

Figure 5: The exclusion at 68% (red) and 95% (green) CL limits on the
$\alpha$-$\kappa_{t}$ plane at the 100 TeV FCC with $30\ {\rm ab}^{-1}$
without (left) and with (right) $\Delta\phi_{\ell\ell}^{t\bar{t}}$, which is
the azimuthal angle between two leptons in the $t\bar{t}$ rest frame. For the
solid curves, 10% systematics is used for both $t\bar{t}h$ and $t\bar{t}Z$
individually, while for the dashed curves, the uncertainties for $t\bar{t}h$
and $t\bar{t}Z$ are assumed to be correlated. 20% systematics is used for
$t\bar{t}b\bar{b}$ for both scenarios. The results are from Ref.
Goncalves:2021dcu .

A recent study on ML-inspired reconstruction algorithm can improve our
results. As discussed in Refs. Goncalves:2018agy ; Goncalves:2021dcu , top
quark reconstruction in the dilepton channel plays an important role. These
studies show that the endpoints method gives $\sim$74% efficiency and
$\sim$87% purity at detector-level. On the other hand, Ref. Alhazmi:2022qbf
shows that the same purity ($\sim$87%) can be obtained at a higher efficiency
using deep neural networks (with 92% efficiency) and Lorentz Boost Networks
(with nearly 100% efficiency), which would lead to a gain of 24% and 35% more
events, respectively. The ML methods should improve further on the precision
measurement of $\kappa_{t}$ and $\alpha$. More challenging use of ML methods
would be direct reconstruction of the kinematic variables that are sensitive
to the CP angle Kim:2021pcz .

##  IV Muon Collider

In this section we summarize the results of Ref. ToAppear . Throughout this
section we will assume $\kappa_{t}=1$. We show in Fig. 6 the SM ($\alpha=0$)
cross-section versus $\sqrt{s}$ for

$\displaystyle\mu^{+}\mu^{-}$ $\displaystyle\to$ $\displaystyle
t\bar{t}h,\,t\bar{t}h\nu{\bar{\nu}},\,tbh\mu\nu,~{}{\rm where}$ $\displaystyle
t\bar{t}h\nu{\bar{\nu}}$ $\displaystyle\equiv$ $\displaystyle
t\bar{t}h\nu_{\ell}{\bar{\nu}}_{\ell}\quad(\ell=e~{}{\rm and}~{}\mu)$
$\displaystyle tbh\mu\nu$ $\displaystyle\equiv$ $\displaystyle
t\bar{b}h\mu^{-}\nu_{\mu}+\bar{t}{b}h\mu^{+}\bar{\nu}_{\mu}$ (3)

with $\sqrt{s}$ from 500 GeV to 30 TeV. To consistently keep gauge invariance,
the vector boson fusion (VBF) subprocess contributions are shown separately as
dashed lines for $tbh\mu\nu$ and $t\bar{t}h\nu\bar{\nu}$ by replacing the
$\mu^{+}$ with $e^{+}$ as adopted in Ref. Costantini:2020stv . All events are
generated by implementing the CP-violating model in MadGraph5_aMC@NLO
Alwall:2014hca  via FeynRules Christensen:2008py ; Alloul:2013bka .

At low energies, the $t\bar{t}h$ production is dominant, while around
$\sqrt{s}\sim 7-8$ TeV the $t\bar{t}h\nu\bar{\nu}$ and $tbh\mu\nu$ take over.
This can be understood by noting that the $t\bar{t}h$ production only occurs
through $s$-channel off-shell $\gamma/Z$ contributions. Hence, the $t\bar{t}h$
cross-section decreases with $\sqrt{s}$ above threshold, from about 2.0 fb at
1 TeV to $7.8\times 10^{-3}$ fb at 30 TeV. However, for the
$t\bar{t}h\nu\bar{\nu}$ and $tbh\mu\nu$ production, the VBF contribution is
dominant for $\sqrt{s}\gtrsim 3$ TeV causing the cross-sections to grow with
energy. For $tbh\mu\nu$ production, there is a VBF diagram where one muon
radiates a photon and other a $W$. For example, a representative process is

$\displaystyle\mu^{-}\mu^{+}\rightarrow\mu^{-}\bar{\nu}(\gamma^{*}W^{+*}\rightarrow
t\bar{b}h),$ (4)

where the parenthesis indicates the radiated photon and $W$ fuse into $tbh$.
The massless photon mediator in this case causes a singularity when the final
and initial state muons are collinear. We impose a cut on the transverse
momentum of the outgoing muon $p_{T}^{\mu}>10$ GeV to avoid the singularity
and generate numerically stable results111A proper treatment involves the
effective vector boson approximation vonWeizsacker:1934nji ; Williams:1935dka
; Dawson:1984gx ; Ruiz:2021tdt . For $\sqrt{s}\gtrsim 30$ TeV, all VBF
processes should incorporate the effective vector boson approximation to resum
the large logs Costantini:2020stv . .

Figure 6: The SM ($\alpha=0$) cross-sections for (blue) $\mu^{+}\mu^{-}\to
t\bar{t}h$, (red) $tbh\mu\nu$ and (green) $t\bar{t}h\nu\bar{\nu}$ processes as
a function of $\sqrt{s}$. VBF contributions are shown as dashed lines and
generated by replacing the $\mu^{+}$ with a positron. For $tbh\mu\nu$ a cut of
$p_{T}^{\mu}>10$ TeV is applied.

In Fig. 7, we show the dependence of the signal cross-sections on the CP-
violating angle $\alpha$ for $\sqrt{s}=1$ TeV (left), $\sqrt{s}=10$ TeV
(middle), and $\sqrt{s}=30$ TeV (right). Even though the muon collider is
designed for better sensitivity reach for $3$ TeV and above, we start from $1$
TeV to gauge our understanding at lower energy. It is interesting to note that
the dependence on $\alpha$ is significantly different between the signal
processes and different collider energies. Hence, it could be hoped that
making a measurement of $\alpha$ at different energies and in different
processes could give complementary information. At $\sqrt{s}=1$ TeV where VBF
is subdominant, each signal cross-section shows a similar dependence on
$\alpha$. For $\sqrt{s}=10$ and $30$ TeV, the dependencies of $\alpha$ vary
significantly between each process. While $t\bar{t}h$ has a maximum cross-
section at the SM point $\alpha=0$, both $t\bar{t}h\nu\bar{\nu}$ and
$tbh\mu\nu$ have minimum cross-sections at $\alpha=0$. Also, while
$t\bar{t}h\nu\bar{\nu}$ and $tbh\mu\nu$ has similar SM cross-sections, as the
CP violating phase moves away from zero $t\bar{t}h\nu\bar{\nu}$ becomes the
dominant signal.

Figure 7: Cross-sections for $\mu^{+}\mu^{-}\to t\bar{t}h$ (blue),
$t\bar{t}h\nu{\bar{\nu}}$ (green), and $tbh\mu\nu$ (red) at 1 TeV (left), 10
TeV (middle) and 30 TeV muon colliders (right) with CP violating phase
$\alpha$ from $-\pi$ to $\pi$. For $tbh\mu\nu$ a cut of $p_{T}^{\mu}>10$ GeV
has been applied. Dashed lines are for the VBF-like contributions.

Now we move onto a collider analysis with a full signal vs. background
simulation to extrapolate how well a future muon collider can constrain the CP
violating phase. We consider the semileptonic decay of the top quarks with
$h\to b\bar{b}$. The backgrounds we include are $b\bar{b}(g\to t\bar{t})$,
$t\bar{t}(g\to b\bar{b})$, $t\bar{t}(Z/\gamma^{*}\to b\bar{b})$,
$W^{*}W^{*}\to t\bar{b}\bar{t}b$, and $t\bar{t}b\bar{b}\nu\bar{\nu}$, where
paranthesis are used to indicate gluon, photon, or $Z$ splitting. We also
include detector effects with a Gaussian smearing of jet energies. Before cuts
$t\bar{t}h\bar{\nu}\nu$ and $tbh\mu\nu$ have similar cross-sections. However,
$tbh\mu\nu$ has a very low cut efficiency, so $t\bar{t}h\bar{\nu}\nu$ gives by
far the strongest constraints at $\sqrt{s}=10$ and $\sqrt{s}=30$ TeV.

In Fig. 8, we show the resulting 95% CL confidence level bounds that result
from the collider analysis, the results of all signal channels are combined.
These bounds are statistical only. Systematic uncertainties can be added in
quadrature. We overlay results for (red) a 1 TeV muon collider with 100 fb-1
of data, (blue) 10 TeV with 10 ab-1, and (black) 30 TeV with 10 ab-1. The
horizontal lines represent the bounds on the cross-section normalized to SM
production cross-section for each energy. The bounds on the CP-violating angle
are $|\alpha|\lesssim 55^{\circ}$, $|\alpha|\lesssim 9^{\circ}$ and
$\alpha\lesssim 3^{\circ}$ for 1, 10 and 30 TeV, respectively.

Figure 8: 2$\sigma$ exclusion on $\alpha$ at (red) 1 TeV, (blue) 10 TeV and
(black) 30 TeV muon colliders with luminosities of 100 fb-1, 10 ab-1, and 10
ab-1, respectively. The solid lines show the combined signal cross-section
before cuts normalized to the SM prediction. The horizontal lines represent
the projected bounds on the cross-section normalized to the SM production
cross-section for each energy. These bounds are statistical only.

## Acknowledgements

R.K.B and D.G. are supported by the United States Department of Energy grant
number DE-SC0016013. F.K. is supported by the DFG under Germany’s Excellence
Strategy – EXC 2121 Quantum Universe – 390833306. K.K., I.M.L., and Y-J.Z. are
supported in part by the United States Department of Energy grant number DE-
SC0019474. M.C. and Y.Z. are by the State of Kansas EPSCoR grant program. C.D.
is supported in part by the US DOE under grant No. DE-SC0021447.

## References

  * (1) D. Gonçalves, J. H. Kim, K. Kong, and Y. Wu, “Direct Higgs-top CP-phase measurement with $t\overline{t}h$ at the 14 TeV LHC and 100 TeV FCC,” JHEP 01 (2022) 158, arXiv:2108.01083 [hep-ph].
  * (2) R. K. Barman, D. Gonçalves, and F. Kling, “Machine learning the Higgs boson-top quark CP phase,” Phys. Rev. D 105 no. 3, (2022) 035023, arXiv:2110.07635 [hep-ph].
  * (3) M. Cassidy, Z. Dong, K. Kong, I. M. Lewis, Y. Zhang, and Y.-J. Zheng, “CP Violating Top Yukawa Coupling at the Multi-TeV Muon Collider,” (2022) to appear.
  * (4) J. Ellis, D. S. Hwang, K. Sakurai, and M. Takeuchi, “Disentangling Higgs-Top Couplings in Associated Production,” JHEP 04 (2014) 004, arXiv:1312.5736 [hep-ph].
  * (5) F. Boudjema, R. M. Godbole, D. Guadagnoli, and K. A. Mohan, “Lab-frame observables for probing the top-Higgs interaction,” Phys. Rev. D 92 no. 1, (2015) 015019, arXiv:1501.03157 [hep-ph].
  * (6) M. R. Buckley and D. Goncalves, “Boosting the Direct CP Measurement of the Higgs-Top Coupling,” Phys. Rev. Lett. 116 no. 9, (2016) 091801, arXiv:1507.07926 [hep-ph].
  * (7) M. R. Buckley and D. Goncalves, “Constraining the Strength and CP Structure of Dark Production at the LHC: the Associated Top-Pair Channel,” Phys. Rev. D 93 no. 3, (2016) 034003, arXiv:1511.06451 [hep-ph].
  * (8) F. Demartin, F. Maltoni, K. Mawatari, B. Page, and M. Zaro, “Higgs characterisation at NLO in QCD: CP properties of the top-quark Yukawa interaction,” Eur. Phys. J. C 74 no. 9, (2014) 3065, arXiv:1407.5089 [hep-ph].
  * (9) A. V. Gritsan, R. Röntsch, M. Schulze, and M. Xiao, “Constraining anomalous Higgs boson couplings to the heavy flavor fermions using matrix element techniques,” Phys. Rev. D94 no. 5, (2016) 055023, arXiv:1606.03107 [hep-ph].
  * (10) D. Goncalves and D. Lopez-Val, “Pseudoscalar searches with dileptonic tops and jet substructure,” Phys. Rev. D94 no. 9, (2016) 095005, arXiv:1607.08614 [hep-ph].
  * (11) N. Mileo, K. Kiers, A. Szynkman, D. Crane, and E. Gegner, “Pseudoscalar top-Higgs coupling: exploration of CP-odd observables to resolve the sign ambiguity,” JHEP 07 (2016) 056, arXiv:1603.03632 [hep-ph].
  * (12) S. Amor Dos Santos et al., “Probing the CP nature of the Higgs coupling in $t{\bar{t}}h$ events at the LHC,” Phys. Rev. D96 no. 1, (2017) 013004, arXiv:1704.03565 [hep-ph].
  * (13) D. Azevedo, A. Onofre, F. Filthaut, and R. Gonçalo, “CP tests of Higgs couplings in $t\bar{t}h$ semileptonic events at the LHC,” Phys. Rev. D 98 no. 3, (2018) 033004, arXiv:1711.05292 [hep-ph].
  * (14) J. Li, Z.-g. Si, L. Wu, and J. Yue, “Central-edge asymmetry as a probe of Higgs-top coupling in $t\bar{t}h$ production at the LHC,” Phys. Lett. B 779 (2018) 72–76, arXiv:1701.00224 [hep-ph].
  * (15) D. Gonçalves, K. Kong, and J. H. Kim, “Probing the top-Higgs Yukawa CP structure in dileptonic $t\overline{t}h$ with M2-assisted reconstruction,” JHEP 06 (2018) 079, arXiv:1804.05874 [hep-ph].
  * (16) ATLAS Collaboration, M. Aaboud et al., “Observation of Higgs boson production in association with a top quark pair at the LHC with the ATLAS detector,” Phys. Lett. B 784 (2018) 173–191, arXiv:1806.00425 [hep-ex].
  * (17) CMS Collaboration, A. M. Sirunyan et al., “Observation of $\mathrm{t\overline{t}}$H production,” Phys. Rev. Lett. 120 no. 23, (2018) 231801, arXiv:1804.02610 [hep-ex].
  * (18) J. Ren, L. Wu, and J. M. Yang, “Unveiling CP property of top-Higgs coupling with graph neural networks at the LHC,” Phys. Lett. B 802 (2020) 135198, arXiv:1901.05627 [hep-ph].
  * (19) B. Bortolato, J. F. Kamenik, N. Košnik, and A. Smolkovič, “Optimized probes of $CP$ -odd effects in the $t\bar{t}h$ process at hadron colliders,” Nucl. Phys. B 964 (2021) 115328, arXiv:2006.13110 [hep-ph].
  * (20) Q.-H. Cao, K.-P. Xie, H. Zhang, and R. Zhang, “A New Observable for Measuring CP Property of Top-Higgs Interaction,” Chin. Phys. C 45 no. 2, (2021) 023117, arXiv:2008.13442 [hep-ph].
  * (21) T. Martini, R.-Q. Pan, M. Schulze, and M. Xiao, “Probing the CP structure of the top quark Yukawa coupling: Loop sensitivity versus on-shell sensitivity,” Phys. Rev. D 104 no. 5, (2021) 055045, arXiv:2104.04277 [hep-ph].
  * (22) H. Bahl, E. Fuchs, S. Heinemeyer, J. Katzy, M. Menen, K. Peters, M. Saimpert, and G. Weiglein, “Constraining the CP structure of Higgs-fermion couplings with a global LHC fit, the electron EDM and baryogenesis,” arXiv:2202.11753 [hep-ph].
  * (23) M. Cepeda et al., “Report from Working Group 2: Higgs Physics at the HL-LHC and HE-LHC,” CERN Yellow Rep. Monogr. 7 (2019) 221–584, arXiv:1902.00134 [hep-ph].
  * (24) ATLAS Collaboration, M. Aaboud et al., “Search for the standard model Higgs boson produced in association with top quarks and decaying into a $b\bar{b}$ pair in $pp$ collisions at $\sqrt{s}$ = 13 TeV with the ATLAS detector,” Phys. Rev. D 97 no. 7, (2018) 072016, arXiv:1712.08895 [hep-ex].
  * (25) CMS Collaboration, A. M. Sirunyan et al., “Search for $\mathrm{t}\overline{\mathrm{t}}\mathrm{H}$ production in the $\mathrm{H}\to\mathrm{b}\overline{\mathrm{b}}$ decay channel with leptonic $\mathrm{t}\overline{\mathrm{t}}$ decays in proton-proton collisions at $\sqrt{s}=13$ TeV,” JHEP 03 (2019) 026, arXiv:1804.03682 [hep-ex].
  * (26) H. Alhazmi, Z. Dong, L. Huang, J. H. Kim, K. Kong, and D. Shih, “Resolving Combinatorial Ambiguities in Dilepton $t\bar{t}$ Event Topologies with Neural Networks,” arXiv:2202.05849 [hep-ph].
  * (27) D. Kim, K. Kong, K. T. Matchev, M. Park, and P. Shyamsundar, “Deep-Learned Event Variables for Collider Phenomenology,” arXiv:2105.10126 [hep-ph].
  * (28) A. Costantini, F. De Lillo, F. Maltoni, L. Mantani, O. Mattelaer, R. Ruiz, and X. Zhao, “Vector boson fusion at multi-TeV muon colliders,” JHEP 09 (2020) 080, arXiv:2005.10289 [hep-ph].
  * (29) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” JHEP 07 (2014) 079, arXiv:1405.0301 [hep-ph].
  * (30) N. D. Christensen and C. Duhr, “FeynRules - Feynman rules made easy,” Comput. Phys. Commun. 180 (2009) 1614–1641, arXiv:0806.4194 [hep-ph].
  * (31) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, “FeynRules 2.0 - A complete toolbox for tree-level phenomenology,” Comput. Phys. Commun. 185 (2014) 2250–2300, arXiv:1310.1921 [hep-ph].
  * (32) C. F. von Weizsacker, “Radiation emitted in collisions of very fast electrons,” Z. Phys. 88 (1934) 612–625.
  * (33) E. J. Williams, “Correlation of certain collision problems with radiation theory,” Kong. Dan. Vid. Sel. Mat. Fys. Med. 13N4 no. 4, (1935) 1–50.
  * (34) S. Dawson, “The Effective W Approximation,” Nucl. Phys. B 249 (1985) 42–60.
  * (35) R. Ruiz, A. Costantini, F. Maltoni, and O. Mattelaer, “The Effective Vector Boson Approximation in High-Energy Muon Collisions,” arXiv:2111.02442 [hep-ph].