# Higher order derivations on $C^{*}$-algebras and applications to smooth
functional calculus and Schwartz functions on the tangent groupoid

Omar Mohsen

###### Abstract

We introduce higher order derivations on $C^{*}$-algebras which are a
noncommutative analogue of differential operators for commutative
$C^{*}$-algebras. We show that the domain of higher derivations is closed
under smooth functional calculus. As a corollary, we show that Schwartz
functions on Connes tangent groupoid are closed under smooth functional
calculus.

###### Contents

  1. 1 Higher derivations on $C^{*}$-algebras
  2. 2 Schwartz functions on Connes’s tangent groupoid

## Introduction

Let $M$ be a compact smooth manifold. In [Con94a, Chapter 2.5], Connes
introduced the tangent groupoid

$\mathbb{T}M:=M\times M\times\mathbb{R}_{+}^{\times}\sqcup TM\times\\{0\\}.$

The space $\mathbb{T}M$ is equipped with a smooth manifold structure which is
a special case of the deformation to the normal cone construction, see
[Ful84a, Chapter 5]. The space $C^{\infty}_{c}(\mathbb{T}M)$ is equipped with
a convolution law making it a $*$-algebra. The convolution law is a
deformation of the usual convolution law on $M\times M$ which is used in
Schwartz kernels of operators acting on $L^{2}M$ and the convolution law on
$T_{x}M$ for $x\in M$ which comes from the commutative group structure. As
Connes shows, the space $\mathbb{T}M$ very naturally gives a canonical
deformation from a differential operator to its principal symbol. Using this
deformation Connes gives a conceptual proof of the Atiyah-Singer index
theorem.

The tangent groupoid has been studied and generalised by many authors. In this
article, we will be interested in the space of Schwartz functions
$\mathcal{S}(\mathbb{T}M)$ on $\mathbb{T}M$ introduced by Carrillo Rouse
[Car08a]. The $*$-algebra $C^{\infty}_{c}(\mathbb{T}M)$ can be naturally
completed into a $C^{*}$-algebra $C^{*}\mathbb{T}M$. In [Car08a], Carrillo
Rouse proves that $\mathcal{S}(\mathbb{T}M)$ is a $*$-subalgebra of
$C^{*}\mathbb{T}M$. We prove that $\mathcal{S}(\mathbb{T}M)$ is closed under
smooth functional calculus.

######  Theorem A.

  1. 1.

If $a\in\mathcal{S}(\mathbb{T}M)$ is a Schwartz function and $f$ is a
holomorphic function on some open neighbourhood of $\mathrm{spec}(a)$ with
$f(0)=0$, then $f(a)\in\mathcal{S}(\mathbb{T}M)$.

  2. 2.

If $a\in\mathcal{S}(\mathbb{T}M)$ is a Schwartz function and $a$ is normal
($a^{*}a=aa^{*}$) and $f$ is a smooth function on some open neighbourhood of
$\mathrm{spec}(a)$ with $f(0)=0$, then $f(a)\in\mathcal{S}(\mathbb{T}M)$.

Of course $f(a)$ in Theorem A means the functional calculus in
$C^{*}\mathbb{T}M$ applied to $a$. The main difficulty in the above theorem is
the derivability of $f(a)$ in the $\mathbb{R}_{+}$ direction of $\mathbb{T}M$
because as $t\to 0^{+}$, the structure of $C^{*}\mathbb{T}M$ changes from
compact operators on $L^{2}M$ to $C_{0}(T^{*}M)$.

To prove Theorem A we introduce higher derivations on an arbitrary
$C^{*}$-algebra $A$. Recall that a derivation on $A$ is a map
$\delta:\operatorname{Dom}(\delta)\subseteq A\to A$ defined on a
$*$-subalgebra $\operatorname{Dom}(\delta)\subseteq A$ such that

$\delta(ab)=\delta(a)b+a\delta(b).$

Derivations usually come from the derivative of an $\mathbb{R}$ action on $A$.
For a commutative $C^{*}$-algebra $A=C(N)$ for a smooth manifold $N$, a
derivation (with domain $C^{1}(N)$) is a vector field on $N$. We introduce the
notion of higher derivations which correspond to higher order differential
operators on $N$ if $A=C(N)$. We show that the derivative in the direction of
$\mathbb{R}_{+}$ is a derivation on $C^{*}\mathbb{T}M$ of order $2$. The main
theorem of this article is the following, see Definition 1.8 for the precise
definition of $\mathcal{D}(\mathcal{A})$.

######  Theorem B.

Let $\mathcal{A}\subseteq A$ be a dense $*$-algebra and
$\mathcal{D}(\mathcal{A})$ be the intersection of the domain of the closure of
all higher order derivations with domain $\mathcal{A}$. Then

  1. 1.

if $a\in\mathcal{D}(\mathcal{A})$ and $f$ is a holomorphic function on some
open neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$, then
$f(a)\in\mathcal{D}(\mathcal{A})$.

  2. 2.

if $a\in\mathcal{D}(\mathcal{A})$ and $a$ is normal ($aa^{*}=a^{*}a$) and $f$
is a smooth function on some open neighbourhood of $\mathrm{spec}(a)$ with
$f(0)=0$, then $f(a)\in\mathcal{D}(\mathcal{A})$.

The methods we use to prove B are based on arguments by Bratteli and Robinson
[BR75a, BR76a] and Pedersen [Ped00a]. We obtain Theorem A as a corollary of
Theorem B because $\mathcal{D}(\mathcal{S}(TM))=\mathcal{S}(TM)$.

We remark that our approach to Schwartz functions is different from that of
Carrillo Rouse. His approach is based on using local coordinates for
$\mathbb{T}M$ and demanding growth conditions in such coordinates. He then
shows that this is independent of local coordinates. Our approach is more
global in nature. Recall that Schwartz functions on a vector space are smooth
functions $f$ such that $Df$ is bounded whenever $D$ is a differential
operator with polynomial coefficients. Analogously, on the space
$\mathbb{T}M$, we define differential operators which play the role of
differential operators with polynomial coefficients. We then define Schwartz
functions to be functions whose derivative by such operators is bounded.

### Organization of the article.

The article is organized as follows.

  * •

In Section 1, we define higher derivations on $C^{*}$-algebras and prove
Theorem B.

  * •

In Section 2, we give a quick introduction to Connes’s tangent groupoid. We
then prove Theorem A.

##  1 Higher derivations on $C^{*}$-algebras

Throughout this article $\mathbb{N}$ stands for $1,2,\cdots$ and
$\mathbb{Z}^{+}$ stands for $0,1,\cdots$ and $\mathbb{R}_{+}^{\times}$ for
$x\in\mathbb{R}$ with $x>0$ and $\mathbb{R}_{+}$ for
$\mathbb{R}_{+}^{\times}\sqcup\\{0\\}$.

######  Example 1.1.

Let

$\mathbb{H}:=\mathbb{R}^{3},\quad(x,y,z)\cdot(x^{\prime},y^{\prime},z^{\prime})=(x+x^{\prime},y+y^{\prime},z+z^{\prime}+xy^{\prime}-yx^{\prime})$

be the Heisenberg group and

$\delta_{x},\delta_{y},\delta_{z}:C^{\infty}_{c}(\mathbb{H})\subseteq
C^{*}\mathbb{H}^{3}\to C^{*}\mathbb{H}^{3}$

given by $f\mapsto xf,yf,zf$ respectively. The maps $\delta_{x}$ and
$\delta_{y}$ satisfy the Leibniz rule

$\displaystyle\delta_{x}(f\star g)=\delta_{x}(f)\star
g+f\star\delta_{x}(g),\quad\delta_{y}(f\star g)=\delta_{y}(f)\star
g+f\star\delta_{y}(g)$

but $\delta_{z}$ satisfies

$\displaystyle\delta_{z}(f\star g)=\delta_{z}(f)\star
g+f\star\delta_{z}(g)+\delta_{x}(f)\star\delta_{y}(g)-\delta_{y}(f)\star\delta_{x}(g)$
(1)

We define higher derivations to be maps which satisfy a Leibniz rule as in
(1).

######  Definition 1.2.

Let $A$ be a $C^{*}$-algebra, $\mathcal{A}$ a dense $*$-subalgebra,
$\delta:\mathcal{A}\to A$ a $\mathbb{C}$-linear map.

  * •

We say that $\delta$ is a derivation of order $1$ if

$\displaystyle\delta(ab)=\delta(a)b+a\delta(b)\quad\forall a,b\in\mathcal{A}$
(2)

We say that $\delta$ leaves $\mathcal{A}$ invariant if
$\delta(\mathcal{A})\subseteq\mathcal{A}$.

  * •

For $n\geq 2$, we define derivations of order $n$ inductively as follows. We
say that $\delta$ is a derivation of order $n$ if there exists a finite family
of derivations
$\delta_{11},\cdots,\delta_{1l},\delta_{21},\cdots,\delta_{2l}:\mathcal{A}\to
A$ of order $<n$ such that

$\delta(ab)=\delta(a)b+a\delta(b)+\sum_{i=1}^{l}\delta_{1i}(a)\delta_{2i}(b),\quad\forall
a,b\in\mathcal{A}.$ (3)

We say that $\delta$ leaves $\mathcal{A}$ invariant if
$\delta(\mathcal{A})\subseteq\mathcal{A}$ and
$\delta_{11},\cdots,\delta_{1l},\delta_{21},\cdots,\delta_{2l}$ leave
$\mathcal{A}$ invariant.

Let $\operatorname{\mathrm{Der}}_{n}(\mathcal{A})$ be the space of derivations
of order $n$ which leave $\mathcal{A}$ invariant. Clearly
$\operatorname{\mathrm{Der}}_{n}(\mathcal{A})\subseteq\operatorname{\mathrm{Der}}_{m}(\mathcal{A})$
any $n\leq m$. Let
$\operatorname{\mathrm{Der}}(\mathcal{A}):=\bigcup_{n\in\mathbb{N}}\operatorname{\mathrm{Der}}_{n}(\mathcal{A})$.

######  Proposition 1.3.

The space $\operatorname{\mathrm{Der}}(\mathcal{A})$ is a filtered algebra,
i.e., if $\delta\in\operatorname{\mathrm{Der}}_{n}(\mathcal{A})$ and
$\delta^{\prime}\in\operatorname{\mathrm{Der}}_{m}(\mathcal{A})$, then

$\displaystyle\delta+\delta^{\prime}\in\operatorname{\mathrm{Der}}_{\max(n,m)}(\mathcal{A}),\quad\delta\circ\delta^{\prime}\in\operatorname{\mathrm{Der}}_{n+m}(\mathcal{A})$

###### Proof.

The statement
$\delta+\delta^{\prime}\in\operatorname{\mathrm{Der}}_{\max(n,m)}(\mathcal{A})$
is straightforward. We prove that $\delta\circ\delta^{\prime}$ is a derivation
by induction on the order. We have

$\displaystyle\delta\circ\delta^{\prime}(ab)$
$\displaystyle=\delta\left(\delta^{\prime}(a)b+a\delta^{\prime}(b)+\sum_{j=1}^{k^{\prime}}\delta^{\prime}_{1j}(a)\delta^{\prime}_{2j}(b)\right)$
$\displaystyle=\delta\circ\delta^{\prime}(a)b+\delta^{\prime}(a)\delta(b)+\sum_{i=1}^{k}\delta_{1i}\circ\delta^{\prime}(a)\delta_{2i}(b)$
$\displaystyle+\delta(a)\delta^{\prime}(b)+a\delta\circ\delta^{\prime}(b)+\sum_{i=1}^{k}\delta_{1i}(a)\delta_{2i}\circ\delta^{\prime}(b)$
$\displaystyle+\sum_{j=1}^{k^{\prime}}\delta\circ\delta^{\prime}_{1j}(a)\delta^{\prime}_{2j}(b)+\delta_{1j}^{\prime}(a)\delta\circ\delta^{\prime}_{2j}(b)+\sum_{i=1}^{k}\delta_{1i}\circ\delta_{1j}^{\prime}(a)\delta_{2i}\circ\delta^{\prime}_{2j}(b)$

It follows that $\delta\circ\delta^{\prime}$ is a derivation of order $n+m$.∎

We can now give examples of higher order derivations

######  Proposition 1.4.

Let $\mathfrak{g}$ be a nilpotent Lie group equipped with a Euclidean metric
and $G$ is the space $\mathfrak{g}$ seen as a Lie group,
$\mathcal{A}=C^{\infty}_{c}(\mathfrak{g})\subseteq C^{*}G$. If
$P:\mathfrak{g}\to\mathbb{C}$ is a polynomial with no constant term, then

$\delta_{P}:\mathcal{A}\to\mathcal{A},\quad\delta_{P}(f)=Pf$

is a derivation of order $n$ for some $n\in\mathbb{N}$.

###### Proof.

By Proposition 1.3 we can suppose that $P$ is linear. Let
$\mathfrak{g}^{1}=\mathfrak{g}$ and
$\mathfrak{g}^{r+1}=[\mathfrak{g}^{r},\mathfrak{g}]$. We choose a linear
decomposition $\mathfrak{g}=V^{1}\oplus\cdots\oplus V^{k}$ such that
$\mathfrak{g}^{r}=V^{r}\oplus\cdots\oplus V^{k}$. It is enough to consider
$P\in V_{i}^{*}$ for some $i$. We proceed by induction on $i$. For $i=1$,
$\delta_{P}$ is a derivation of order $1$. The induction case follows by using
the Baker-Campell-Hausdorff formula like in (1). ∎

It is convenient to enlarge the set of derivations by adding multipliers as
well. This leads us to the following definition.

######  Definition 1.5.

Let $\delta:\mathcal{A}\to A$ be a $\mathbb{C}$-linear map.

  * •

We say that $\delta$ is a generalised derivation of order $0$ if there exists
a map $\delta^{t}:\mathcal{A}\to A$ (necessarily unique by density of
$\mathcal{A}$) such that

$\displaystyle\delta(ab)=\delta(a)b,\quad
a\delta(b)=\delta^{t}(a)b,\quad\forall a,b\in\mathcal{A}$ (4)

We say that $\delta$ leaves $\mathcal{A}$ invariant if
$\delta(\mathcal{A})\subseteq\mathcal{A}$ and
$\delta^{t}(\mathcal{A})\subseteq\mathcal{A}$.

  * •

For $n\in\mathbb{N}$, we define generalised derivations of order $n$
inductively as follows. We say that $\delta$ is a derivation of order $n$ if
there exists a finite family of generalised derivations
$\delta_{01},\cdots,\delta_{0k}:\mathcal{A}\to A$ of order $0$ and a finite
family of derivations $\delta_{11},\cdots,\delta_{1k}:\mathcal{A}\to A$ of
order $n$ and a finite family of generalised derivations
$\delta_{21},\cdots,\delta_{2l},\delta_{31},\cdots,\delta_{3l}:\mathcal{A}\to
A$ of order $<n$ such that

$\delta(ab)=\delta(a)b+\sum_{i=1}^{k}\delta_{0i}(a)\delta_{1i}(b)+\sum_{i=1}^{l}\delta_{2i}(a)\delta_{3i}(b),\quad\forall
a,b\in\mathcal{A}.$ (5)

We say that $\delta$ leaves $\mathcal{A}$ invariant if
$\delta(\mathcal{A})\subseteq\mathcal{A}$ and each
$\delta_{0i},\delta_{1i},\delta_{2j},\delta_{3j}$ leaves $\mathcal{A}$
invariant.

We denote by $\operatorname{\mathrm{gDer}}(\mathcal{A})_{n}$ the set of
generalised derivations of order $n$ that leave $\mathcal{A}$ invariant. Let
$\operatorname{\mathrm{gDer}}(\mathcal{A}):=\bigcup_{n\in\mathbb{N}}\operatorname{\mathrm{gDer}}_{n}(\mathcal{A})$.

######  Proposition 1.6.

The space $\operatorname{\mathrm{gDer}}(\mathcal{A})$ is a unital filtered
algebra, i.e., if $\delta\in\operatorname{\mathrm{gDer}}_{n}(\mathcal{A})$ and
$\delta^{\prime}\in\operatorname{\mathrm{gDer}}_{m}(\mathcal{A})$, then

$\displaystyle\mathrm{Id}\in\operatorname{\mathrm{gDer}}_{0}(\mathcal{A}),\quad\delta+\delta^{\prime}\in\operatorname{\mathrm{gDer}}_{\max(n,m)}(\mathcal{A}),\quad\delta\circ\delta^{\prime}\in\operatorname{\mathrm{gDer}}_{n+m}(\mathcal{A}).$

###### Proof.

The statements $\mathrm{Id}\in\operatorname{\mathrm{gDer}}_{0}(\mathcal{A})$
and
$\delta+\delta^{\prime}\in\operatorname{\mathrm{gDer}}_{\max(n,m)}(\mathcal{A})$
are straightforward. We prove that $\delta\circ\delta^{\prime}$ is a
generalised derivation by induction on the order. The case $n=m=0$ is
straightforward. We now prove the proposition for $n=0$ and for $m$ arbitrary
by induction on $m$. Let $\delta$ be a generalised derivation of order $0$,
$\delta^{\prime}$ a generalised derivation of order $m$. Then we have

$\displaystyle\delta\circ\delta^{\prime}(ab)$
$\displaystyle=\delta\left(\delta^{\prime}(a)b+\sum_{j=1}^{k^{\prime}}\delta_{0j}^{\prime}(a)\delta_{1j}^{\prime}(b)+\sum_{j=1}^{l^{\prime}}\delta_{2j}^{\prime}(a)\delta_{3j}^{\prime}(b)\right)$
$\displaystyle=\delta\circ\delta^{\prime}(a)b+\sum_{j=1}^{k^{\prime}}\delta\circ\delta_{0j}^{\prime}(a)\delta_{1j}^{\prime}(b)+\sum_{j=1}^{l^{\prime}}\delta\circ\delta_{2j}^{\prime}(a)\delta_{3j}^{\prime}(b)$

Hence by induction applied to $\delta\circ\delta_{1i}^{\prime}$ and using that
$\delta\circ\delta_{0i}^{\prime}$ is of order $0$, it follows that
$\delta\circ\delta^{\prime}$ is a derivation of order $m$. Similarly one
proves the proposition for $n$ arbitrary and $m=0$. We now proceed to the case
$n,m\geq 1$ assuming by induction that the proposition is true for
$(n^{\prime},m^{\prime})$ with $n^{\prime}+m^{\prime}<n+m$ and $n^{\prime}\leq
n$ and $m^{\prime}\leq m$. We have

$\displaystyle\delta\circ\delta^{\prime}(ab)$
$\displaystyle=\delta\left(\delta^{\prime}(a)b+\sum_{j=1}^{k^{\prime}}\delta_{0j}^{\prime}(a)\delta_{1j}^{\prime}(b)+\sum_{j=1}^{l^{\prime}}\delta_{2j}^{\prime}(a)\delta_{3j}^{\prime}(b)\right)$
$\displaystyle=\delta\circ\delta^{\prime}(a)b+\sum_{i=1}^{k}\delta_{0i}\circ\delta^{\prime}(a)\delta_{1i}(b)+\sum_{i=1}^{l}\delta_{2i}\circ\delta^{\prime}(a)\delta_{3i}(b)$
$\displaystyle+\sum_{j=1}^{k^{\prime}}\delta\circ\delta_{0j}^{\prime}(a)\delta_{1j}^{\prime}(b)+\sum_{i=1}^{k}\sum_{j=1}^{k^{\prime}}\delta_{0i}\circ\delta_{0j}^{\prime}(a)\delta_{1i}\circ\delta_{1j}^{\prime}(b)+\sum_{i=1}^{k}\sum_{j=1}^{k^{\prime}}\delta_{2i}\circ\delta_{0j}^{\prime}(a)\delta_{3i}\circ\delta_{1j}^{\prime}(b)$
$\displaystyle+\sum_{j=1}^{l^{\prime}}\delta\circ\delta_{2j}^{\prime}(a)\delta_{3j}^{\prime}(b)+\sum_{i=1}^{k}\sum_{j=1}^{k^{\prime}}\delta_{0i}\circ\delta_{2j}^{\prime}(a)\delta_{1i}\circ\delta_{3j}^{\prime}(b)+\sum_{i=1}^{k}\sum_{j=1}^{k^{\prime}}\delta_{2i}\circ\delta_{2j}^{\prime}(a)\delta_{3i}\circ\delta_{3j}^{\prime}(b)$

By Proposition 1.3, $\delta_{1i}\circ\delta_{1j}^{\prime}$ is a derivation of
order $n+m$. It follows that $\delta\circ\delta^{\prime}$ is a generalised
derivation of order $n+m$.∎

######  Proposition 1.7.

Let $\mathfrak{g}$, $G$, $A$, $\mathcal{A}$ as in Proposition 1.4. If $D$ is a
differential operator on $\mathfrak{g}$ with polynomial coefficients, then

$\delta_{D}:\mathcal{A}\to\mathcal{A},\quad\delta_{D}(f)=D(f)$

is a generalised derivation of order $n$ for some $n\in\mathbb{Z}^{+}$.

###### Proof.

Let $X\in\mathfrak{g}$, $X_{R},X_{L}$ the associated right and left invariant
vector fields respectively. One can easily see that $\delta_{X_{R}}$ is a
generalised derivation of order $0$ with $\delta_{X_{R}}^{t}=\delta_{X_{L}}$.
By writing $D$ as a sum of products of polynomials and monomials of the form
$X_{1R}\cdots X_{kR}$ for various $X_{1},\cdots,X_{k}\in\mathfrak{g}$, the
result follows from Proposition 1.4 and Proposition 1.6. ∎

Let $\delta\in\operatorname{\mathrm{gDer}}(\mathcal{A})$. Since $\delta$ is an
unbounded operator, it is desirable to take the closure of $\delta$. Some care
should be taken, as there exists derivations of order $1$ that aren’t closable
(the closure of the graph is no longer the graph of a linear map), see [BR75a,
BR76a] for examples and further discussion of this issue. This issue doesn’t
concern us as we only care about the domain and not the linear map on the
closure. This leads us to the following definition

######  Definition 1.8.

Let $\mathcal{D}(\mathcal{A})$ be the set of $x\in A$ such that there exists a
sequence $x_{n}\in\mathcal{A}$ such that $x_{n}\to x$ and for every
$\delta\in\operatorname{\mathrm{gDer}}(\mathcal{A})$, $\delta(x_{n})$ and
$\delta(x_{n}^{*})$ converge in $A$.

It is clear from the definition that
$\mathcal{A}\subseteq\mathcal{D}(\mathcal{A})$ and if a generalised derivation
$\delta$ is closable, then
$\mathcal{D}(\mathcal{A})\subseteq\operatorname{Dom}(\bar{\delta})$.

######  Theorem 1.9.

The set $\mathcal{D}(\mathcal{A})$ is a $*$-subalgebra of $A$ which is closed
under smooth functional calculus, i.e.,

  1. 1.

if $a\in\mathcal{D}(\mathcal{A})$ and $f$ is a holomorphic function defined on
some neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$, then
$f(a)\in\mathcal{D}(\mathcal{A})$.

  2. 2.

if $a\in\mathcal{D}(\mathcal{A})$ is normal ($a^{*}a=aa^{*}$) and $f$ is a
smooth function defined on some neighbourhood of $\mathrm{spec}(a)$ with
$f(0)=0$, then $f(a)\in\mathcal{D}(\mathcal{A})$.

Here $\mathrm{spec}(a)$ is the spectrum of $a$ as an element of $A$.

###### Proof.

It is straightforward to prove that if $x,y\in\mathcal{D}(\mathcal{A})$ and
$x_{n}\to x$ and $y_{n}\to y$ as in Definition 1.8, then by induction on the
order of $\delta$, $\delta(x_{n}y_{n})$ and $\delta(y_{n}^{*}x_{n}^{*})$
converge. Hence $\mathcal{D}(\mathcal{A})$ is a $*$-algebra.

We first prove Theorem 1.9.2 if $a=a^{*}\in\mathcal{D}(\mathcal{A})$. Let
$x_{n}\in\mathcal{A}$ a sequence such that $x_{n}\to a$ and $\delta(x_{n})$
and $\delta(x_{n}^{*})$ converge for every
$\delta\in\operatorname{\mathrm{gDer}}(\mathcal{A})$. By replacing $x_{n}$
with $\frac{x_{n}+x_{n}^{*}}{2}$, we can suppose that $x_{n}=x_{n}^{*}$. By
abuse of notation, for $m\in\mathbb{N}$, we define
$\delta(a^{m}):=\lim_{n}\delta(x_{n}^{m})$. Without loss of generality, we
suppose $a\neq 0$ and $x_{n}\neq 0$ for all $n\in\mathbb{N}$. We start with
the following lemma.

######  Lemma 1.10.

For every $l\in\mathbb{N}$ and
$\delta\in\operatorname{\mathrm{gDer}}_{l}(\mathcal{A})$, there exists $C>0$
such that for all $m,n\in\mathbb{N}$,

$\left\lVert\delta(x_{n}^{m})\right\rVert\leq Cm^{l}\left\lVert
x_{n}\right\rVert^{m},\quad\text{and}\quad\left\lVert\delta(a^{m})\right\rVert\leq
Cm^{l}\left\lVert a\right\rVert^{m}$ (6)

###### Proof.

It is clear that the second inequality follows from the first by taking the
limit in $n$. We first prove the lemma for derivations by induction on $l$.
The case $l=1$ is left to the reader (in this case we can take
$C=\sup_{n}\frac{\left\lVert\delta(x_{n})\right\rVert}{\left\lVert
x_{n}\right\rVert}$). Suppose the lemma holds for derivations of order $<l$.
By induction hypothesis applied to $\delta_{1i}$, there exists $C_{1i}$ such
that $\left\lVert\delta_{1i}(x_{n}^{m})\right\rVert\leq
C_{1i}m^{l-1}\left\lVert x_{n}\right\rVert^{m}$ for all $m,n\in\mathbb{N}$. We
take

$C=\sup_{n}\max\left(\frac{\left\lVert\delta(x_{n})\right\rVert}{\left\lVert
x_{n}\right\rVert},\sum_{i=1}^{k}\frac{C_{1i}\left\lVert\delta_{2i}(x_{n})\right\rVert}{\left\lVert
x_{n}\right\rVert}\right).$

We prove (6) by induction on $m$. We have

$\delta(x_{n}^{m})=\delta(x_{n}^{m-1})x_{n}+x_{n}^{m-1}\delta(x_{n})+\sum_{i=1}^{k}\delta_{1i}(x_{n}^{m-1})\delta_{2i}(x_{n}).$

Hence

$\displaystyle\left\lVert\delta(x_{n}^{m})\right\rVert$ $\displaystyle\leq
C(m-1)^{l}\left\lVert x_{n}\right\rVert^{m}+\left\lVert
x_{n}\right\rVert^{m-1}\left\lVert\delta(x_{n})\right\rVert+\sum_{i=1}^{k}C_{1i}(m-1)^{l-1}\left\lVert
x_{n}\right\rVert^{m-1}\left\lVert\delta_{2i}(x_{n})\right\rVert$
$\displaystyle\leq C(m-1)^{l}\left\lVert x_{n}\right\rVert^{m}+C\left\lVert
x_{n}\right\rVert^{m}+C(m-1)^{l-1}\left\lVert x_{n}\right\rVert^{m}\leq
Cm^{l}\left\lVert x_{n}\right\rVert^{m}$

We have thus proved the lemma for derivations. One can then prove it for
generalised derivations again by induction. The proof is very similar and is
left to the reader. ∎

Let $\xi\in\mathbb{R}$. By Lemma 1.10, the sums

$\sum_{m=0}^{\infty}(i\xi)^{m}\frac{\delta(x_{n}^{m+1})}{m!},\quad\sum_{m=0}^{\infty}(i\xi)^{m}\frac{\delta(a^{m+1})}{m!}$

converges in norm uniformly in $n$. By abuse of notation, we denote the limit
by $\delta(x_{n}e^{i\xi x_{n}})$ and $\delta(ae^{i\xi a})$. Since convergence
is uniform, $\lim_{n\to+\infty}\delta(x_{n}e^{i\xi x_{n}})=\delta(ae^{i\xi
a})$.

######  Lemma 1.11.

For every $l\in\mathbb{N}$,
$\delta\in\operatorname{\mathrm{gDer}}_{l}(\mathcal{A})$, there exists
constant $C^{\prime}$ such that for all $n\in\mathbb{N}$ and
$\xi\in\mathbb{R}$,

$\left\lVert\delta(x_{n}e^{i\xi x_{n}})\right\rVert\leq
C^{\prime}(|\xi|^{l}+1)\quad\text{and}\quad\left\lVert\delta(ae^{i\xi
a})\right\rVert\leq C^{\prime}(|\xi|^{l}+1).$

###### Proof.

Clearly the second inequality follows from the first by taking limit in $n$.
We prove the lemma for derivations by induction on the order. For order $1$,
by (2), we have

$\delta(x_{n}^{m+1})=\sum_{j=0}^{m}x_{n}^{j}\delta(x_{n})x_{n}^{m-j}.$

Hence by expanding $e^{i\xi sx_{n}}=\sum_{m=0}^{+\infty}\frac{(i\xi
sx_{n})^{m}}{m!}$, we deduce

$\delta(x_{n}e^{i\xi x_{n}})=\delta(x_{n})e^{i\xi
x_{n}}+i\xi\int_{0}^{1}x_{n}e^{i\xi sx_{n}}\delta(x_{n})e^{i\xi(1-s)x_{n}}ds.$

The lemma follows for derivations of order $1$ because $x_{n}=x_{n}^{*}$ which
implies $\left\lVert e^{i\xi sx_{n}}\right\rVert=\left\lVert
e^{i\xi(1-s)x_{n}}\right\rVert=1$. Suppose that $\delta$ is a derivation of
order $l$ and the lemma holds for derivations of order $<l$. By (3), it
follows that

$\delta(x_{n}^{m+1})=\sum_{j=0}^{m}x_{n}^{j}\delta(x_{n})x_{n}^{m-j}+\sum_{r=1}^{k}\sum_{j=1}^{m}\delta_{1r}(x_{n}^{j})\delta_{2r}(x_{n})x_{n}^{m-j}.$

Hence

$\delta(x_{n}e^{i\xi x_{n}})=\delta(x_{n})e^{i\xi
x_{n}}+i\xi\int_{0}^{1}x_{n}e^{i\xi
x_{n}}\delta(x_{n})e^{i\xi(1-s)x_{n}}ds+\sum_{r=1}^{k}i\xi\int_{0}^{1}\delta_{1r}(x_{n}e^{i\xi
sx_{n}})\delta_{2r}(x_{n})e^{i\xi(1-s)x_{n}}ds.$

Hence the result follows by the induction hypothesis on
$\delta_{1r}(x_{n}e^{i\xi sx_{n}})$. The argument for generalised derivations
is similar. If $n=0$, then $\delta(x_{n}e^{i\xi x_{n}})=\delta(x_{n})e^{i\xi
x_{n}}$ and the result follow. Let $\delta$ be a generalised derivation of
order $l$, and suppose the lemma holds for generalised derivations of order
$<l$. Then by (5), for $m\geq 1$

$\displaystyle\delta(x_{n}^{m+1})=\delta(x_{n}^{m})x_{n}+\sum_{r=1}^{k}\delta_{0r}(x_{n})x_{n}^{m-1}\delta_{1r}(x_{n})+\sum_{r=1}^{k^{\prime}}\delta_{2r}(x_{n}^{m})\delta_{3r}(x_{n}).$

which implies

$\delta(x_{n}^{m+1})=\delta(x_{n})x_{n}^{m}+\sum_{r=1}^{k}\sum_{j=0}^{m-1}\delta_{0r}(x_{n})x_{n}^{j}\delta_{1r}(x_{n})x_{n}^{m-1-j}+\sum_{r=1}^{k^{\prime}}\sum_{j=1}^{m}\delta_{2r}(x_{n}^{j})\delta_{3r}(x_{n})x_{n}^{m-j}.$

It follows that

$\displaystyle\delta(x_{n}e^{i\xi x_{n}})=\delta(x_{n})e^{i\xi x_{n}}$
$\displaystyle+\sum_{r=1}^{k}i\xi\int_{0}^{1}\delta_{0r}(x_{n})e^{i\xi
sx_{n}}\delta_{1r}(x_{n})e^{i\xi(1-s)x_{n}}ds$
$\displaystyle+\sum_{r=1}^{k^{\prime}}i\xi\int_{0}^{1}\delta_{2r}(x_{n}e^{i\xi
sx_{n}})\delta_{3r}(x_{n})e^{i\xi(1-s)x_{n}}ds.$

The result follows. ∎

Let $f$ be a $C^{\infty}$ function on a neighbourhood of
$\operatorname{Spec}(a)\subseteq\mathbb{R}$. We extend $f$ to a $C^{\infty}$
function on $\mathbb{R}$ with compact support. Since $f(0)=0$, we can write
$f(x)=xg(x)$ where $g$ is a $C^{\infty}$ function with compact support. It
follows that the Fourier transform $\hat{g}$ of $g$ is a Schwartz function. By
the Fourier inversion formula

$f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}xe^{ix\xi}\hat{g}(\xi)d\xi.$

Hence

$f(a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}ae^{ia\xi}\hat{g}(\xi)d\xi$

and the integral is absolutely convergent. We approximate the integral by
Riemann sums, and then replace each term $ae^{ia\xi}$ with a sum
$\sum_{m=0}^{r}\frac{x_{n}^{m}}{m!}$ for $r$ and $n$ big enough. This way, we
construct a sequence $s_{n}\in\mathcal{A}$ such that $s_{n}\to f(a)$ and
$s_{n}^{*}\to\bar{f}(a)$ and
$\delta(s_{n})\to\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(ae^{ia\xi})\hat{g}(\xi)d\xi$
and
$\delta(s_{n}^{*})\to\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(ae^{-ia\xi})\overline{\hat{g}}(\xi)d\xi$
for every $\delta\in\operatorname{\mathrm{gDer}}(\mathcal{A})$. This shows
that $f(a)\in\mathcal{D}(\mathcal{A})$.

We now prove Theorem 1.9.2 in the general case where $a$ is normal. Like
before, we extend $f$ to a $C^{\infty}$ compactly supported function on
$\mathbb{C}$. We write $z=x+iy$ for coordinates in $\mathbb{C}$. Consider the
function $g(z)=f(x)$. Then $g(a)=f_{|\mathbb{R}}(\frac{a+a^{*}}{2})$. It
follows from what we proved earlier that $g(a)\in\mathcal{D}(\mathcal{A})$.
Hence by replacing $f$ with $f-g$, we can suppose that $f=0$ on $\mathbb{R}$.
By a similar argument we can suppose that $f=0$ on both $\mathbb{R}$ and
$i\mathbb{R}$. Hence there exists a $C^{\infty}$ compactly supported function
$g:\mathbb{C}\to\mathbb{C}$ such that $f(z)=xyg(z)$. Since $g$ is smooth
compactly supported, it follows that $\hat{g}$ is a Schwartz function on
$\mathbb{C}$. Let $Re(a)=\frac{a+a^{*}}{2}$ and $Im(a)=\frac{a-a^{*}}{2i}$. By
the Fourier inversion formula

$f(a)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}Re(a)e^{i\xi
Re(a)}Im(a)e^{i\eta Im(a)}\hat{g}(\xi+i\eta)d\xi d\eta.$

We then write the integral as a Riemann sum like we did in the self adjoint
case, and Theorem 1.9.2 follows.

For Theorem 1.9.1, since $f(0)=0$, $f(z)=zg(z)$ for some holomorphic function
$g$. Let $\gamma$ be a contour around $\mathrm{spec}(a)$ such that $\gamma$ is
in domain of $f$ and the Cauchy integral formula applies

$f(z)=\frac{z}{2\pi i}\int_{\gamma}\frac{g(w)}{w-z}dw.$

We thus have

$f(a)=\frac{1}{2\pi i}\int_{\gamma}g(w)\frac{a}{w-a}dw.$

######  Lemma 1.12.

The element $\frac{a}{w-a}\in\mathcal{D}(\mathcal{A})$ for
$w\notin\mathrm{spec(a)}$. Furthermore we can choose a sequence
$x_{n}(w)\in\mathcal{A}$ such that for any
$K\subseteq\mathbb{C}\backslash\mathrm{spec(a)}$ compact,

  * •

$w\mapsto x_{n}(w)$ is uniformly continuous on $K$

  * •

$x_{n}(w)\to\frac{a}{w-a}$ uniformly in $K$

  * •

for every $\delta\in\operatorname{\mathrm{gDer}}(\mathcal{A})$,
$\delta(x_{n}(w))$ and $\delta(x_{n}(w)^{*})$ converge uniformly on $K$. (the
convergence isn’t uniform in the choice of $\delta$)

###### Proof.

The element $b=(w-a)^{*}(w-a)-|w|^{2}\in\mathcal{D}(\mathcal{A})$ is self
adjoint whose spectrum is contained in $]-|w|^{2},+\infty[$. So by Theorem
1.9.2, applied to $f(x)=\frac{1}{x+|w|^{2}}-\frac{1}{|w|^{2}}$, we deduce that
$f(b)\in\mathcal{D}(\mathcal{A})$. Since
$(w-a)^{*}a\in\mathcal{D}(\mathcal{A})$, it follows that
$\frac{a}{w-a}=(f(a)+\frac{1}{|w|^{2}})(w-a)^{*}a\in\mathcal{D}(\mathcal{A})$.
The rest of the lemma follows from the proof of Theorem 1.9.2, by remarking
that $f$ as well as all its derivatives vary uniformly in $w$ on compact
subsets of $\mathbb{C}\backslash\mathrm{spec(a)}$. ∎

It follows from Lemma 1.12 that $\frac{1}{2\pi
i}\int_{\gamma}g(w)x_{n}(w)dw\to f(a)$ and that $\frac{1}{2\pi
i}\int_{\gamma}g(w)\delta(x_{n}(w))dw$ converges for every
$\delta\in\operatorname{\mathrm{gDer}}(\mathcal{A})$. Same for the adjoint. By
approximating the integral by Riemann sums, the result follows. ∎

######  Corollary 1.13.

Let $\mathfrak{g},G$ be as in Proposition 1.4. Let
$\mathcal{S}(\mathfrak{g})\subseteq C^{*}G$ be $*$-subalgebra of Schwartz
functions. Then

  1. 1.

if $a\in\mathcal{S}(\mathfrak{g})$ and $f$ is a holomorphic function defined
on some neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$, then
$f(a)\in\mathcal{S}(\mathfrak{g})$.

  2. 2.

if $a\in\mathcal{S}(\mathfrak{g})$ is normal and $f$ is a smooth function
defined on some neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$, then
$f(a)\in\mathcal{S}(\mathfrak{g})$.

Corollary 1.13 is well known, see for example [HJ84a].

######  Remark 1.14.

For $l\in\mathbb{N}$, one can also define $\mathcal{D}_{l}(\mathcal{A})$ which
consists of all $a\in A$ such that there exists $x_{n}\in\mathcal{A}$ such
that $x_{n}\to a$ and for every
$\delta\in\operatorname{\mathrm{gDer}}_{l}(\mathcal{A})$, $\delta(x_{n})$ and
$\delta(x_{n}^{*})$ converge. The proof of theorem 1.9 proves as well the
following

######  Theorem 1.15.

For every $l\in\mathbb{N}$, there exists $l^{\prime}\in\mathbb{N}$ such that
the set $\mathcal{D}_{l}(\mathcal{A})$ is a $*$-subalgebra of $A$ which is
closed under $C^{l^{\prime}}$ functional calculus, i.e.,

  1. 1.

if $a\in\mathcal{D}_{l}(\mathcal{A})$ and $f$ is a holomorphic function
defined on some neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$, then
$f(a)\in\mathcal{D}_{l}(\mathcal{A})$.

  2. 2.

if $a\in\mathcal{D}_{l}(\mathcal{A})$ is normal ($a^{*}a=aa^{*}$) and $f$ is a
$C^{l^{\prime}}$ function defined on some neighbourhood of $\mathrm{spec}(a)$
with $f(0)=0$, then $f(a)\in\mathcal{D}_{l}(\mathcal{A})$.

##  2 Schwartz functions on Connes’s tangent groupoid

Let $M$ be a compact smooth manifold. The set

$\mathbb{T}M=M\times M\times\mathbb{R}_{+}^{\times}\sqcup TM\times\\{0\\}$

can be equipped with the structure of a smooth manifold with boundary
$TM\times\\{0\\}$ as follows. The subset $M\times
M\times\mathbb{R}_{+}^{\times}$ is declared an open subset of $\mathbb{T}M$
with its usual smooth structure. If $U\subseteq M$ is an open set,
$\phi:U\to\mathbb{R}^{\dim(M)}$ is a diffeomorphism, then $U\times
U\times\mathbb{R}_{+}^{\times}\sqcup TU\times\\{0\\}$ is declared an open
subset of $\mathbb{T}M$ and the following bijection is declared a
diffeomorphism

$\displaystyle U\times U\times\mathbb{R}_{+}^{\times}\sqcup TU\times\\{0\\}$
$\displaystyle\to\mathbb{R}^{\dim(M)}\times\mathbb{R}^{\dim(M)}\times\mathbb{R}_{+}$
$\displaystyle(y,x,t)$
$\displaystyle\mapsto(\frac{\phi(y)-\phi(x)}{t},\phi(x),t),\quad x,y\in
U,t\in\mathbb{R}_{+}^{\times}$ $\displaystyle(v,x,0)$
$\displaystyle\mapsto(d\phi_{x}(v),\phi(x),0),\quad x\in U,v\in T_{x}M$

One can check that this defines a smooth structure on $\mathbb{T}M$. A more
conceptual way to define the smooth structure is to notice that the smooth
structure (and the topology) is uniquely determined by requiring that the maps

$\displaystyle\mathbb{T}M\to M\times
M\times\mathbb{R}_{+},\quad(y,x,t)\mapsto(y,x,t),\quad(v,x,0)\mapsto(x,x,0)$
(7)

and

$\displaystyle\operatorname{DNC}(f):\mathbb{T}M\to\mathbb{C},\quad(y,x,t)\mapsto\frac{f(y)-f(x)}{t},\quad(v,x,0)\mapsto
df_{x}(v),$ (8)

be smooth, where $f\in C^{\infty}(M)$.

### Differential operators.

Let $X\in\mathcal{X}(M)$ be a vector field. We can define a vector field
$\mathbb{X}$ on $\mathbb{T}M$ as follows. If $f\in C^{\infty}(\mathbb{T}M)$ is
a smooth function, then we define $\mathbb{X}\cdot f\in
C^{\infty}_{c}(\mathbb{T}M)$ by

$\mathbb{X}\cdot f(y,x,t)=tX_{y}f(y,x,t),\quad\mathbb{X}\cdot
f(v,x,0)=X(x)\cdot f(v,x,0),$

where $X_{y}$ means that $X$ acts on the $y$ variable, and $X(x)\cdot
f(v,x,0)$ means that we view $X(x)$ as a constant vector field on $T_{x}M$
which acts on the smooth function $f(\cdot,x,0)$. By the description of charts
above, one checks that $\mathbb{X}$ is indeed a vector field.

More generally, let $D$ be a differential operator of order $d$ acting on $M$.
Then we can define a differential operator denoted $\mathbb{D}$ of order $d$
acting on $\mathbb{T}M$ by the formula

$\mathbb{D}\cdot f(y,x,t)=t^{d}D_{y}f(y,x,t),\quad\mathbb{D}\cdot
f(v,x,0)=D(x)\cdot f(v,x,0),$ (9)

where $D_{y}$ means that $D$ acts on the $y$ variable, and $D(x)\cdot
f(v,x,0)$ means that we view the principal part of $D$ as a constant
coefficient differential operator on $T_{x}M$ acting on the smooth function
$f(\cdot,x,0)$. Locally by the principal part, we mean that if
$D=\sum_{|I|\leq d}g_{I}\frac{\partial}{\partial x_{I}}$, then
$D(x)=\sum_{|I|=d}g_{I}(x)\frac{\partial}{\partial x_{I}}$ seen as a constant
coefficient differential operator on $T_{x}M$. Using the local charts of
$\mathbb{T}M$, one can check that $\mathbb{D}$ is indeed a differential
operator.

### Convolution and adjoint.

Starting from now we suppose that $M$ is equipped with a Riemannian metric. If
$f,g\in\mathbb{C}^{\infty}_{c}(\mathbb{T}M)$, then we define their convolution
$f\star g\in C^{\infty}_{c}(\mathbb{T}M)$ by the formula

$\displaystyle f\star g(y,x,t)$
$\displaystyle=t^{-\dim(M)}\int_{M}f(y,z,t)g(z,x,t)dz,\quad t\neq 0$ (10)
$\displaystyle f\star g(v,x,0)$
$\displaystyle=\int_{T_{x}M}f(v-w,x,0)g(w,x,0)dw,$

where the integral over $T_{x}M$ is with respect to the constant Riemannian
metric on $T_{x}M$ induced from the Riemannian metric on $M$. We also define
the adjoint by

$f^{*}(y,x,t)=\bar{f}(x,y,t),\quad f^{*}(v,x,0)=\bar{f}(-v,x,0)$

We leave it to the reader to check that $C^{\infty}_{c}(\mathbb{T}M)$ with the
operations defined above is a $*$-algebra.

### Adjoint of Differential operators on $\mathbb{T}M$.

Let $D$ be a differential operator on $M$ of order $d$, $\mathbb{D}$ the
associated differential operator on $\mathbb{T}M$. We sometimes denote
$\mathbb{D}(f)$ by $\mathbb{D}\star f$. We also define $f\star\mathbb{D}$ by
the formula

$f\star\mathbb{D}:=(\mathbb{D}^{*}\star f^{*})^{*},$

where $\mathbb{D}^{*}$ is the differential operator on $\mathbb{T}M$
associated to the formal adjoint $D^{*}$ of $D$. Equivalently

$f\star\mathbb{D}(y,x,t)=t^{d}D^{\prime}_{x}f(y,x,t),\quad
f\star\mathbb{D}(v,x,0)=D(x)\cdot f(v,x,0),$

where $D^{\prime}$ is the formal transpose given by

$\int_{M}Df(x)g(x)dx=\int_{M}f(x)D^{\prime}g(x)dx,\quad f,g\in C^{\infty}(M).$

One can check that

$(\mathbb{D}\star f)\star g=\mathbb{D}\star(f\star
g),\quad(f\star\mathbb{D})\star g=f\star(\mathbb{D}\star g)\quad f,g\in
C^{\infty}_{c}(\mathbb{T}M).$ (11)

This justifies our notation $f\star\mathbb{D}$ and $\mathbb{D}\star f$.

### Derivations in direction of $M$.

The functions $\mathbb{D}\star f$ and $f\star\mathbb{D}$ agree at $t=0$. It
follows that the map

$\displaystyle\delta_{D}:C^{\infty}_{c}(\mathbb{T}M)\to
C^{\infty}_{c}(\mathbb{T}M),\quad f\mapsto\frac{1}{t}(\mathbb{D}\star
f-f\star\mathbb{D})$

is a well defined differential operator on $\mathbb{T}M$ of order $d$. If $D$
is an order $0$ operator given by multiplication by $g\in C^{\infty}(M)$, then
$\delta_{g}$ is an order $0$ differential operator given by pointwise
multiplication by $\operatorname{DNC}(g)$, see (8). For $X\in\mathcal{X}(M)$,
one can describe $\delta_{X}$ as follows. The flow of $X$ gives an
$\mathbb{R}_{+}^{\times}$ action on $\mathbb{T}M$ by

$\beta_{\lambda}(y,x,t)=(\exp(\lambda X)\cdot y,\exp(\lambda X)\cdot
x,t),\quad\beta_{\lambda}(v,x,0)=(d\exp(\lambda X)(x)(v),\exp(\lambda X)\cdot
x,0).$

The differential operator $\delta_{X}$ is the derivative of the action $\beta$
(plus multiplication by the divergence of $X$). Since $\delta_{D}$ is given by
a commutator, it satisfies the Leibniz rule

$\delta_{D}(f\star g)=\delta_{D}(f)\star g+f\star\delta_{D}(g),\quad f,g\in
C^{\infty}_{c}(\mathbb{T}M)$ (12)

and

$\delta_{D_{1}D_{2}}(f)=\delta_{D_{1}}(f)\star\mathbb{D}_{2}+\mathbb{D}_{1}\star\delta_{D_{2}}(f),\quad
f\in C^{\infty}_{c}(\mathbb{T}M)$ (13)

### Derivation in direction of $t$.

The group $\mathbb{R}_{+}^{\times}$ acts smoothly on the left on $\mathbb{T}M$
by the formula

$\alpha_{\lambda}:\mathbb{T}M\to\mathbb{T}M,\quad\alpha_{\lambda}(y,x,t)=(y,x,\lambda^{-1}t),\quad\alpha_{\lambda}(v,x,0)=(\lambda
v,x,0),\quad\lambda\in\mathbb{R}_{+}^{\times}.$ (14)

We also let $\mathbb{R}_{+}^{\times}$ acts on $C^{\infty}$ functions by the
formula

$\displaystyle\alpha_{\lambda}(f)=\lambda^{-\dim(M)}f\circ\alpha_{\lambda^{-1}},\quad
f\in C^{\infty}_{c}(\mathbb{T}M),\lambda\in\mathbb{R}_{+}^{\times}.$

One has

$\alpha_{\lambda}(f\star
g)=\alpha_{\lambda}(f)\star\alpha_{\lambda}(g),\quad\alpha_{\lambda}(f)^{*}=\alpha_{\lambda}(f^{*}).$
(15)

We define $\delta_{\alpha}$ to be the derivative of the
$\mathbb{R}_{+}^{\times}$ given by

$\displaystyle\delta_{\alpha}(f)(y,x,t)$
$\displaystyle=-\dim(M)f(y,x,t)+t\frac{\partial}{\partial t}f(y,x,t)$
$\displaystyle\delta_{\alpha}(f)(v,x,0)$
$\displaystyle=-\dim(M)f(v,x,0)-\sum_{i}v_{i}\frac{\partial}{\partial
v_{i}}f(v,x,0),$

where $v_{i}$ are any local coordinates for $T_{x}M$. It follows from (15)
that $\delta_{\alpha}$ satisfies the Leibniz rule

$\delta_{\alpha}(f\star g)=\delta_{\alpha}(f)\star
g+f\star\delta_{\alpha}(g),\quad f,g\in C^{\infty}_{c}(\mathbb{T}M).$ (16)

######  Lemma 2.1.

One can find a finite family of smooth functions $f_{1},\cdots,f_{k}\in
C^{\infty}(M)$ and vector fields $X_{1},\cdots,X_{k}\in\mathcal{X}(M)$ such
that for all $x\in M$, $\sum_{i=1}^{k}df_{i}(x)X_{i}(x)$ is equal to the
identity map $T_{x}M\to T_{x}M$.

###### Proof.

Let $f_{1},\cdots,f_{k}:M\to\mathbb{R}$ be smooth functions such that for all
$x\in M$, $df_{1,x},\cdots,df_{k,x}$ span $T_{x}^{*}M$. Let
$\theta=M\times\mathbb{R}^{k}$ be the trivial vector bundle on $M$ of rank
$k$. The forms $df_{i}$ define a surjective bundle map $\theta\to T^{*}M$. Its
dual is an injective bundle map $i:TM\to\theta$. Let $p:\theta\to TM$ be any
bundle map such that $p\circ i=\mathrm{Id}$. The map $p$ determines the vector
fields $X_{1},\cdots,X_{k}$. ∎

We fix a choice of $f_{i},X_{i}$ for the rest of this section. By taking the
trace of $\sum_{i=1}^{k}df_{i}(x)X_{i}(x)$, we deduce that

$\sum_{i=1}^{k}X_{i}(f_{i})(x)=\dim(M),\quad\forall x\in M$ (17)

For $f\in C^{\infty}_{c}(\mathbb{T}M)$, the functions $\delta_{\alpha}(f)$ and
$-\dim(M)f-\sum_{i=1}^{k}\delta_{f_{i}}(\mathbb{X}_{i}\star f)$ agree at
$t=0$. Hence we can define

$\displaystyle\hat{\delta}(f):$
$\displaystyle=\frac{1}{t}\left(\delta_{\alpha}(f)+\dim(M)f+\sum_{i=1}^{k}\delta_{f_{i}}(\mathbb{X}_{i}\star
f)\right)$ (18)
$\displaystyle=\frac{1}{t}\left(\delta_{\alpha}(f)+\sum_{i=1}^{k}\mathbb{X}_{i}\star\delta_{f_{i}}(f)\right),$

where in the second equality we used (17).

######  Lemma 2.2.

The map $\hat{\delta}$ satisfies

$\hat{\delta}(f\star g)=\hat{\delta}(f)\star
g+f\star\hat{\delta}(g)+\sum_{i=1}^{k}\delta_{X_{i}}(f)\star\delta_{f_{i}}(g),\quad
f,g\in C^{\infty}_{c}(\mathbb{T}M)$

and hence a derivation of order $2$ on $C^{\infty}_{c}(\mathbb{T}M)$.

###### Proof.

By (12) and (16)

$\hat{\delta}(f\star g)-\hat{\delta}(f)\star
g-f\star\hat{\delta}(g)=\frac{1}{t}\left(\sum_{i=1}^{k}\mathbb{X}_{i}\star
f\star\delta_{f_{i}}(g)-f\star\mathbb{X}_{i}\star\delta_{f_{i}}(g)\right)=\sum_{i=1}^{k}\delta_{X_{i}}(f)\star\delta_{f_{i}}(g)\qed$

### Schwartz functions.

######  Definition 2.3.

We define $\mathcal{S}(\mathbb{T}M)$ to be the space of smooth function $f\in
C^{\infty}(\mathbb{T}M)$ such that $f$ is bounded and all iterated
applications of the following differential operators give bounded functions on
$\mathbb{T}M$

  * •

the operator $f\mapsto tf$, where $t:\mathbb{T}M\to\mathbb{R}_{+}$ is the
natural projection.

  * •

the operator $f\mapsto\mathbb{D}\star f$, where $D$ is any differential
operator on $M$

  * •

the operator $f\mapsto\delta_{D}(f)$, where $D$ is any differential operator
on $M$

  * •

the operator $f\mapsto\hat{\delta}(f)$.

We can add the application $f\mapsto\delta_{\alpha}(f)$ to the above list, but
this is redundant as it follows from the others using (18).

######  Proposition 2.4.

Let $f\in\mathcal{S}(\mathbb{T}M)$. Then for every $x\in M$, the function
$v\mapsto f(v,x,0)$ is a Schwartz function on the vector space $T_{x}M$ in the
classical sense.

###### Proof.

If $X\in\mathcal{X}(M)$ is a vector field, then $v\mapsto(\mathbb{X}\star
f)(v,x,0)$ is the application of the constant vector field $X(x)$ to $v\mapsto
f(v,x,0)$. While if $g\in C^{\infty}(M)$, then the map
$v\mapsto\delta_{g}(f)(v,x,0)$ is the pointwise product of the map $v\mapsto
f(v,x,0)$ with the linear map $dg_{x}:T_{x}M\to\mathbb{C}$. By iterated
application of the previous two operations, it follows that $v\mapsto
f(v,x,0)$ is Schwartz. ∎

######  Example 2.5.

Let $g_{1},\cdots,g_{l}:M\to\mathbb{R}$ be smooth functions such that the map

$M\mapsto\mathbb{R}^{l},\quad x\mapsto(g_{1}(x),\cdots,g_{l}(x))$

is an embedding. Then the map

$e^{-\sum_{i=1}^{l}\operatorname{DNC}(g_{i})^{2}-t^{2}}\in
C^{\infty}(\mathbb{T}M)$

is Schwartz, where $\operatorname{DNC}(g_{i})$ is defined in (8).

The following proposition summarizes the main properties of Schwartz
functions. In its proof, we remark that by (13), in Definition 2.3 the second
and third conditions can be replaced by $\mathbb{D}\star f$ and
$\delta_{D}(f)$ are bounded where $D$ is either a vector field or a smooth
function on $M$.

######  Proposition 2.6.

The following holds

  1. 1.

The definition of Schwartz functions doesn’t depend on the choice of
$f_{i},X_{i}$ in Lemma 2.1.

  2. 2.

If $f\in\mathcal{S}(\mathbb{T}M)$, then $f\in C_{0}(\mathbb{T}M)$, i.e. $f$
vanishes at infinity.

  3. 3.

If $f,g\in\mathcal{S}(\mathbb{T}M)$, then the integral in $f\star g$ is
absolutely convergent and $f\star g\in\mathcal{S}(\mathbb{T}M)$

  4. 4.

If $f\in\mathcal{S}(\mathbb{T}M)$, then $f^{*}\in\mathcal{S}(\mathbb{T}M)$

###### Proof.

For the first part, we first need some lemmas.

######  Lemma 2.7.

If $f\in C^{\infty}_{c}(M\times M\times\mathbb{R}_{+})$ and
$g\in\mathcal{S}(\mathbb{T}M)$, then the pointwise product
$(f\circ\pi)g\in\mathcal{S}(\mathbb{T}M)$, where $\pi:\mathbb{T}M\to M\times
M\times\mathbb{R}_{+}$ is the map in (7)

###### Proof.

$(f\circ\pi)g$ is bounded because $g$ and $f$ are bounded. We argue that each
of the applications in Definition 2.3 gives functions of the form
$(f^{\prime}\circ\pi)g^{\prime}$ for some $f^{\prime}\in
C^{\infty}_{c}(M\times M\times\mathbb{R}_{+})$ and
$g^{\prime}\in\mathcal{S}(\mathbb{T}M)$. The result then follows by induction.
We thus have

  * •

the function $t(f\circ\pi)g=(f\circ\pi)(tg)$

  * •

If $D$ is a function on $M$, then
$\mathbb{D}\star((f\circ\pi)g)=(f\circ\pi)(\mathbb{D}\star g)$. If $D$ is a
vector field, then we have

$\mathbb{D}\star((f\circ\pi)g)=(D_{y}(f)\circ\pi)(tg)+(f\circ\pi)(\mathbb{D}\star
g),$ (19)

where $D_{y}(f)$ is the action of $D$ on the $y$ variable in $f(y,x,t)$. The
equality can be checked directly on $M\times M\times\mathbb{R}_{+}^{\times}$
and by density equality follows on $\mathbb{T}M$.

  * •

If $D$ is a function, then
$\delta_{D}((f\circ\pi)g)=(f\circ\pi)\delta_{D}(g)$. If $D$ is a vector field,
then one can as in (19) show that

$\delta_{D}((f\circ\pi)g)=(D_{y}(f)\circ\pi-
D_{x}^{\prime}(f)\circ\pi)(g)+(f\circ\pi)\delta_{D}(g)$

  * •

one has

$\delta_{\alpha}((f\circ\pi)g)=(f\circ\pi)\delta_{\alpha}(g)+(\frac{\partial}{\partial
t}(f)\circ\pi)(tg)$

and so

$\hat{\delta}((f\circ\pi)g)=(f\circ\pi)\hat{\delta}(g)+(\frac{\partial}{\partial
t}(f)\circ\pi)(g)+\sum_{i=1}^{k}(X_{iy}(f)\circ\pi)(\delta_{f_{i}}(g))$

This finishes the proof. ∎

######  Lemma 2.8.

Let $f\in C^{\infty}(M\times M)$ be a smooth function that vanishes to the
order $1$ on the diagonal, and let $\tilde{f}\in C^{\infty}(\mathbb{T}M)$ be
the smooth function given by

$(y,x,t)\mapsto\frac{f(y,x)}{t^{2}},\quad(v,x,0)\mapsto\frac{1}{2}d^{2}f_{x}(v),$

where $d^{2}f$ is the Hessian of $f$. Then if $g\in\mathcal{S}(\mathbb{T}M)$
then the pointwise product $\tilde{f}g\in\mathcal{S}(\mathbb{T}M)$.

###### Proof.

Any function $f$ which vanishes to the order $1$ can be written as a finite
sum of functions of the form $h(y,x)(h_{1}(y)-h_{1}(x))(h_{2}(y)-h_{2}(x))$
where $h\in C^{\infty}(M\times M),h_{1},h_{2}\in C^{\infty}(M)$. It follows
that the pointwise product $\tilde{f}g$ is sum of pointwise product of
$h\circ\pi$ with $\delta_{h_{1}}(\delta_{h_{2}}(g))$ where $\pi:\mathbb{T}M\to
M\times M$ is the natural map. The result then follows from Lemma 2.7. ∎

######  Lemma 2.9.

Let $g_{1},\cdots,g_{k^{\prime}}$ and $Y_{1},\cdots,Y_{k^{\prime}}$ be another
family satisfying Lemma 2.1. Then there exists a family of smooth functions
$h_{ij}:M\to\mathbb{R}$ for $1\leq i\leq k$ and $1\leq j\leq k^{\prime}$ such
that

$df_{i}=\sum_{j=1}^{k^{\prime}}h_{ij}dg_{j},\quad
Y_{j}=\sum_{i=1}^{k}h_{ij}X_{i}.$ (20)

###### Proof.

Let $x_{1},\cdots,x_{\dim(M)}$ be local coordinates in $M$. Then
$df_{i}=\sum_{l=1}^{\dim(M)}\phi_{il}dx_{l}$ and
$X_{i}=\sum_{l=1}^{\dim(M)}\phi^{\prime}_{il}\frac{\partial}{\partial x_{l}}$
for some functions $\phi_{il},\phi_{il}^{\prime}$. Similarly
$dg_{j}=\sum_{l=1}^{{\dim(M)}}\psi_{jl}dx_{l}$ and
$Y_{j}=\sum_{l=1}^{{\dim(M)}}\psi^{\prime}_{jl}\frac{\partial}{\partial
x_{l}}$. Then one can take
$h_{ij}=\sum_{l=1}^{{\dim(M)}}\phi_{il}\psi_{jl}^{\prime}$. To define $h$
globally one takes a partition of unity. ∎

We can now prove Proposition 2.6.1. Let $\hat{\delta}^{\prime}$ be the
operator associated to $g_{1},\cdots,g_{k^{\prime}}$ and
$Y_{1},\cdots,Y_{k^{\prime}}$. We have

$\displaystyle\hat{\delta}(f)-\hat{\delta}^{\prime}(f)$
$\displaystyle=\frac{1}{t}\left(\sum_{i=1}^{k}\delta_{f_{i}}(\mathbb{X}_{i}\star
f)-\sum_{j=1}^{k^{\prime}}\delta_{g_{j}}(\mathbb{Y}_{j}\star f)\right)$
$\displaystyle=\sum_{i=1}^{k}\frac{1}{t^{2}}\left(\left(f_{i}(y)-f_{i}(x)-\sum_{j=1}^{k^{\prime}}(g_{j}(y)-g_{j}(x))h_{ij}(y)\right)(\mathbb{X}_{i}\star
f)\right).$

By (20), the function
$f_{i}(y)-f_{i}(x)-\sum_{j=1}^{k^{\prime}}(g_{j}(y)-g_{j}(x))h_{ij}(y)$
vanishes to order $1$ on the diagonal. By Lemma 2.8,
$\hat{\delta}^{\prime}(f)\in\mathcal{S}(\mathbb{T}M)$. Proposition 2.6.1
follows.

For Proposition 2.6.2, suppose that $f\notin C_{0}(\mathbb{T}M)$. Then there
exists $\epsilon>0$ a sequence in $\mathbb{T}M$ and which goes to infinity yet
$|f|\geq\epsilon$ for every element in the sequence. By passing to a
subsequence we can suppose that the sequence is either of the form
$(y_{n},x_{n},t_{n})$ or of the form $(v_{n},x_{n},0)$. Suppose we have the
first case. Then by taking a subsequence, we can suppose that $x_{n}\to x$ and
$y_{n}\to y$ and $t_{n}\to t\in[0,+\infty]$. If $t=+\infty$, we get a
contradiction to the fact that $tf$ is bounded. If $t\in]0,+\infty[$, then the
sequence $(y_{n},x_{n},t_{n})$ converges in $\mathbb{T}M$ to $(y,x,t)$, again
a contradiction. If $x\neq y$ then we get a contradiction to the fact that
$\delta_{g}(f)$ is bounded where $g\in C^{\infty}(M)$ is any smooth function
with $g(x)=g(y)$. So we have $t=0$ and $x=y$. The sequence
$(y_{n},x_{n},t_{n})$ converging to infinity implies that there exists $g\in
C^{\infty}(M)$ such that
$\left|\frac{g(y_{n})-g(x_{n})}{t_{n}}\right|\to+\infty$. We get then a
contradiction to the fact that $\delta_{g}(f)$ is bounded. The case of a
sequence $(v_{n},x_{n},0)$ is similar.

For Proposition 2.6.3, first the integral in $f\star g$ is absolutely
convergent by Proposition 2.4. Hence $f\star g$ is a well defined function on
$\mathbb{T}M$. We now show its continuity. Let $g_{i}$ be as in Example 2.5.
Then consider the function

$\phi=(1+\operatorname{DNC}(g_{1})^{2}+\cdots+\operatorname{DNC}(g_{l})^{2})^{\frac{\dim(M)}{2}+1}\in
C^{\infty}(\mathbb{T}M).$ (21)

The function $f$ being Schwartz implies that $\phi f$ is bounded. One can
easily show by looking at local coordinates of $\mathbb{T}M$ that there exists
$C>0$ (only depends on $\phi$) such that

$\left\lVert f\star g\right\rVert_{\infty}\leq C\left\lVert\phi
f\right\rVert_{\infty}\left\lVert g\right\rVert_{\infty}.$

Since $\phi f$ and $g$ are Schwartz functions, $\phi f,g\in
C_{0}(\mathbb{T}M)$. We can thus approximate them uniformly with compactly
supported functions. It follows that $f\star g\in C_{0}(\mathbb{T}M)$.
Smoothness of $f\star g$ as well as the fact that $f\star
g\in\mathcal{S}(\mathbb{T}M)$ follow easily from (11), (12), (15) and Lemma
2.2.

For Proposition 2.6.4, since $f^{*}$ is bounded if $f$ is bounded, the result
follows from the following identities

$\displaystyle tf^{*}=(tf)^{*},\quad\mathbb{D}\star
f^{*}=(-t\delta_{D^{*}}(f)+\mathbb{D}^{*}\star
f)^{*},\quad\delta_{D}(f^{*})=-\delta_{D^{*}}(f)^{*},\quad\delta_{\alpha}(f^{*})=\delta_{\alpha}(f)^{*},$

and the identity

$\displaystyle\hat{\delta}(f^{*})^{*}-\hat{\delta}(f)$
$\displaystyle=\frac{1}{t}\left(\sum_{i=1}^{k}-\delta_{f_{i}}(f\star\mathbb{X}_{i}^{*})-\delta_{f_{i}}(\mathbb{X}_{i}\star
f)\right)$
$\displaystyle=\frac{1}{t}\left(\sum_{i=1}^{k}\delta_{f_{i}}(f\star\mathbb{X}_{i})-\delta_{f_{i}}(\mathbb{X}_{i}\star
f)\right)+\sum_{i=1}^{k}\delta_{f_{i}}(f\star\mathrm{div}(X_{i}))$
$\displaystyle=\sum_{i=1}^{k}-\delta_{f_{i}}(\delta_{X_{i}}(f))+\delta_{f_{i}}(f\star\mathrm{div}(X_{i}))\qed$

###  $C^{*}$-algebra of the tangent groupoid.

Let $f\in C^{\infty}_{c}(\mathbb{T}M)$, $t\neq 0$. We define the operator

$\displaystyle\pi_{t}(f):L^{2}M$ $\displaystyle\to L^{2}M$ $\displaystyle g$
$\displaystyle\mapsto(y\mapsto t^{-\dim(M)}\int_{M}f(y,x,t)g(x)dx),\quad g\in
L^{2}(M)$

For each $x\in M$, we also define the operator

$\displaystyle\pi_{x}(f):L^{2}T_{x}M$ $\displaystyle\to L^{2}T_{x}M$
$\displaystyle g$
$\displaystyle\mapsto(v\mapsto\int_{T_{x}M}f(v-w,x,0)g(w)dw),\quad g\in
L^{2}(T_{x}M)$

We then define

$\left\lVert
f\right\rVert:=\max{\left(\sup_{t\in\mathbb{R}_{+}^{\times}}\left\lVert\pi_{t}(f)\right\rVert,\sup_{x\in
M}\left\lVert\pi_{x}(f)\right\rVert\right)}.$

We define the $C^{*}$-algebra $C^{*}\mathbb{T}M$ to be the completion of
$C^{\infty}_{c}(\mathbb{T}M)$ with respect to $\left\lVert\cdot\right\rVert$.
We remark the the $C^{*}$-algebra $C^{*}\mathbb{T}M$ lies in a short exact
sequence

$0\to\mathcal{K}(L^{2}M)\otimes C_{0}(\mathbb{R}_{+}^{\times})\to
C^{*}\mathbb{T}M\to C_{0}(T^{*}M)\to 0,$

where $\mathcal{K}(L^{2}M)$ is the $C^{*}$-algebra of compact operators on
$L^{2}M$, see [Con94a, Proposition 5 Page 108]

######  Proposition 2.10.

The space $\mathcal{S}(\mathbb{T}M)$ is a $*$-subalgebra of
$C^{*}\mathbb{T}M$.

###### Proof.

It is well known that

$\left\lVert\pi_{t}(f)\right\rVert\leq\sup_{x\in
M}\max\left(t^{-\dim(M)}\int_{M}|f(x,y,t)|dy,t^{-\dim(M)}\int_{M}|f(y,x,t)|dy\right),\quad
t\in\mathbb{R}_{+}^{\times},f\in C^{\infty}_{c}(\mathbb{T}M)$

and

$\left\lVert\pi_{x}(f)\right\rVert\leq\sup_{x\in
M}\left(\int_{T_{x}M}|f(v,x,0)|dv\right),\quad x\in M,f\in
C^{\infty}_{c}(\mathbb{T}M).$

Hence

$\left\lVert f\right\rVert\leq\sup_{x\in
M,t\in\mathbb{R}_{+}^{\times}}\max\left(t^{-\dim(M)}\int_{M}|f(x,y,t)|dy,t^{-\dim(M)}\int|f(y,x,t)|dy,\int_{T_{x}M}|f(v,x,0)|dv\right)$
(22)

It follows that $C^{*}\mathbb{T}M$ contains measurable functions on
$\mathbb{T}M$ for which the right hand side of (22) is finite (usually denoted
$L^{1}(\mathbb{T}M)$). Let $\phi$ be as in (21). If
$f\in\mathcal{S}(\mathbb{T}M)$, then the right hand side of (22) is bounded by
$\left\lVert\phi f\right\rVert_{\infty}$. Hence $f\in C^{*}\mathbb{T}M$.
Finally $\mathcal{S}(\mathbb{T}M)$ is a $*$-subalgebra by Proposition 2.6. ∎

### Relation between uniform norm and $C^{*}$-norm.

The following theorem which is essentially just the Sobolev embedding theorem
will be very useful in allowing us to replace the uniform norm with the
$C^{*}$-norm which is more convenient to use.

######  Theorem 2.11.

Let $\Delta$ be the positive Laplace operator on $M$, $\mathbbl{\Delta}$ the
corresponding differential operator on $\mathbb{T}M$ as in (9), and
$k\in\mathbb{N}$ with $2k>\frac{\dim(M)}{2}$. Then there exists $C>0$,

$\left\lVert f\right\rVert_{\infty}\leq
C\left\lVert(1+t^{\dim(M)})(1+\mathbbl{\Delta})^{k}\star
f\star(1+\mathbbl{\Delta})^{k}\right\rVert,\quad\forall f\in
C^{\infty}_{c}(\mathbb{T}M)$

######  Lemma 2.12.

Let $k\in\mathbb{N}$ with $2k>\frac{\dim(M)}{2}$. There exists a constant
$C>0$ such that for all $t>0$ and $x\in M$, if $u_{x}$ denotes the Dirac delta
distribution on $M$ at $x$, then

$\left\lVert(1+t^{2}\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}\leq
C\max(t^{-\frac{\dim(M)}{2}},1).$

###### Proof.

The Sobolev embedding lemma implies that $u_{x}\in H^{-2k}(M)$ where
$H^{s}(M)$ denotes the $s$ Sobolev space. Hence
$(1+t^{2}\Delta)^{-k}(u_{x})\in L^{2}M$. For $t>1$ we have and use the
inequality

$\displaystyle\left\lVert(1+t^{2}\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}$
$\displaystyle=\left\lVert(1+t^{2}\Delta)^{-k}(1+\Delta)^{k}(1+\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}$
$\displaystyle\leq\left\lVert(1+t^{2}\Delta)^{-k}(1+\Delta)^{k}\right\rVert_{B(L^{2}M)}\left\lVert(1+\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}$
$\displaystyle\leq\left\lVert(1+\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M},$

where in the last inequality we used the fact that
$\frac{(1+x)^{k}}{(1+t^{2}x)^{k}}\leq 1$ for all $x\in[0,+\infty[$. For $t<1$
we proceed differently. For $g\in L^{2}M$ notice that

$\langle(1+t^{2}\Delta)^{-k}(u_{x}),g\rangle_{L^{2}M}=|(1+t^{2}\Delta)^{-k}(g)(x)|.$

We need to maximize $\frac{|(1+t^{2}\Delta)^{-k}(g)(x)|}{\left\lVert
g\right\rVert_{L^{2}M}}$ as $g$ varies in $L^{2}M$. The function $g$ needs
vanish outside a neighbourhood of $x$ to reach the supremum. We can thus
assume that $M=\mathbb{R}^{\dim(M)}$ and $x=0$ and the metric on $M$ is
Euclidean outside the unit ball. The inequality follows from a change of
variable $x\to\frac{x}{t}$. ∎

###### Proof of Theorem 2.11.

Let $t>0$. We apply

$\pi_{t}((1+\mathbbl{\Delta})^{k}\star
f\star(1+\mathbbl{\Delta})^{k})=t^{-\dim(M)}(1+t^{2}\Delta)^{k}\star
f\star(1+t^{2}\Delta)^{k}$

to $(1+t^{2}\Delta)^{-k}(u_{x})$ to deduce that

$\displaystyle
t^{-\dim(M)}\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$
$\displaystyle\leq
C\max(t^{-\frac{\dim(M)}{2}},1)\left\lVert(1+\mathbbl{\Delta})^{k}\star
f\star(1+\mathbbl{\Delta})^{k}\right\rVert$

One has

$\displaystyle|f(y,x,t)|$ $\displaystyle=|\left\langle
u_{y},f(\cdot,x,t)\right\rangle_{H^{-2k}(M)\times H^{2k}(M)}|$
$\displaystyle=|\langle(1+t^{2}\Delta)^{-k}(u_{y}),(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\rangle_{L^{2}(M)\times
L^{2}(M)}|$
$\displaystyle\leq\left\lVert(1+t^{2}\Delta)^{-k}(u_{y})\right\rVert_{L^{2}M}\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$
$\displaystyle\leq
C\max(t^{-\frac{\dim(M)}{2}},1)\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$

It follows that

$\displaystyle|f(y,x,t)|$ $\displaystyle\leq
C^{2}t^{\dim(M)}\max(t^{-\dim(M)},1)\left\lVert(1+\mathbbl{\Delta})^{k}\star
f\star(1+\mathbbl{\Delta})^{k}\right\rVert$
$\displaystyle=C^{2}\max(1,t^{\dim(M)})\left\lVert(1+\mathbbl{\Delta})^{k}\star
f\star(1+\mathbbl{\Delta})^{k}\right\rVert.$

By taking the limit as $t\to 0^{+}$, one obtains a bound of $f$ on
$TM\times\\{0\\}$. The theorem follows. ∎

### Schwartz functions are closed under smooth calculus.

###### Proof of Theorem A.

We will prove that
$\mathcal{D}(\mathcal{S}(\mathbb{T}M))=\mathcal{S}(\mathbb{T}M)$. Theorem A
would then follow from Theorem 1.9. Let
$a\in\mathcal{D}(\mathcal{S}(\mathbb{T}M))$,
$x_{n}\in\mathcal{S}(\mathbb{T}M)$ as in Definition 1.8. Since

$x\mapsto(1+t^{\dim(M)})(1+\mathbbl{\Delta})^{k}\star
x\star(1+\mathbbl{\Delta})^{k}$

is a generalised derivation on $\mathcal{S}(\mathbb{T}M)$, it follows that
$(1+t^{\dim(M)})(1+\mathbbl{\Delta})^{k}\star
x_{n}\star(1+\mathbbl{\Delta})^{k}$ converge in $C^{*}\mathbb{T}M$. By Theorem
2.11, it follows that $a$ is a bounded continuous function and $x_{n}\to a$ in
uniform norm. All the maps in Definition 2.3 are generalised derivations on
$\mathcal{S}(\mathbb{T}M)$. It follows that any iterated application of them
is also a generalised derivation on $\mathcal{S}(\mathbb{T}M)$. Let $\delta$
be an iterated application of them. Since
$x\mapsto(1+t^{\dim(M)})(1+\mathbbl{\Delta})^{k}\star\delta(x)\star(1+\mathbbl{\Delta})^{k}$
is a generalised derivation on $\mathcal{S}(\mathbb{T}M)$, again by Theorem
2.11 $\delta(x_{n})$ converges in the uniform norm to a bounded continuous
functions on $\mathbb{T}M$. It follows that $a$ is smooth and $\delta(a)$
bounded for every $\delta$. Hence $a\in\mathcal{S}(\mathbb{T}M)$. ∎

By the above
$\mathcal{D}(C^{\infty}_{c}(\mathbb{T}M))\subseteq\mathcal{S}(\mathbb{T}M)$. I
don’t know if this inclusion is strict or not. [sorting=nyt]

## References

  * [BR75] O. Bratteli and D.. Robinson  “Unbounded derivations of $C^{\ast}$-algebras”  In _Comm. Math. Phys._ 42, 1975, pp. 253–268  URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899048
  * [BR76] O. Bratteli and D.. Robinson  “Unbounded derivations of $C^{\ast}$-algebras. II”  In _Comm. Math. Phys._ 46.1, 1976, pp. 11–30  URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899543
  * [Car08] P. Carrillo Rouse  “A Schwartz type algebra for the tangent groupoid”  In _$K$ -theory and noncommutative geometry_, EMS Ser. Congr. Rep.  Eur. Math. Soc., Zürich, 2008, pp. 181–199  DOI: 10.4171/060-1/7
  * [Con94] A. Connes  “Noncommutative geometry”  Academic Press, Inc., San Diego, CA, 1994, pp. xiv+661
  * [Ful84] W. Fulton  “Intersection theory” 2, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]  Springer-Verlag, Berlin, 1984, pp. xi+470  DOI: 10.1007/978-3-662-02421-8
  * [HJ84] A. Hulanicki and J.. Jenkins  “Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators”  In _Studia Math._ 80.3, 1984, pp. 235–244  DOI: 10.4064/sm-80-3-235-244
  * [Ped00] G.. Pedersen  “Operator differentiable functions”  In _Publ. Res. Inst. Math. Sci._ 36.1, 2000, pp. 139–157  DOI: 10.2977/prims/1195143229

## References

  * [BR75a] O. Bratteli and D.. Robinson  “Unbounded derivations of $C^{\ast}$-algebras”  In _Comm. Math. Phys._ 42, 1975, pp. 253–268  URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899048
  * [BR76a] O. Bratteli and D.. Robinson  “Unbounded derivations of $C^{\ast}$-algebras. II”  In _Comm. Math. Phys._ 46.1, 1976, pp. 11–30  URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899543
  * [Ful84a] W. Fulton  “Intersection theory” 2, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]  Springer-Verlag, Berlin, 1984, pp. xi+470  DOI: 10.1007/978-3-662-02421-8
  * [HJ84a] A. Hulanicki and J.. Jenkins  “Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators”  In _Studia Math._ 80.3, 1984, pp. 235–244  DOI: 10.4064/sm-80-3-235-244
  * [Con94a] A. Connes  “Noncommutative geometry”  Academic Press, Inc., San Diego, CA, 1994, pp. xiv+661
  * [Ped00a] G.. Pedersen  “Operator differentiable functions”  In _Publ. Res. Inst. Math. Sci._ 36.1, 2000, pp. 139–157  DOI: 10.2977/prims/1195143229
  * [Car08a] P. Carrillo Rouse  “A Schwartz type algebra for the tangent groupoid”  In _$K$ -theory and noncommutative geometry_, EMS Ser. Congr. Rep.  Eur. Math. Soc., Zürich, 2008, pp. 181–199  DOI: 10.4171/060-1/7

(Omar Mohsen) Paris-Saclay University, Paris, France e-mail:
<EMAIL_ADDRESS>


# Manifestation of heavy 3/2-spin lepton in large-angle
$e^{+}e^{-}\rightarrow\gamma\gamma$ reaction at high energies

G. I. Gakh National Science Centre, Kharkov Institute of Physics and
Technology, Akademicheskaya 1, and V. N. Karazin Kharkov National University,
Dept. of Physics and Technology, 31 Kurchatov, 61108 Kharkov, Ukraine M.I.
Konchatnij<EMAIL_ADDRESS>National Science Centre, Kharkov
Institute of Physics and Technology, Akademicheskaya 1, and V. N. Karazin
Kharkov National University, Dept. of Physics and Technology, 31 Kurchatov,
61108 Kharkov, Ukraine N.P. Merenkov  merenkov@@kipt.kharkov.ua National
Science Centre, Kharkov Institute of Physics and Technology, Akademicheskaya
1, and V. N. Karazin Kharkov National University, Dept. of Physics and
Technology, 31 Kurchatov, 61108 Kharkov, Ukraine A. G. Gakh<EMAIL_ADDRESS>V. N. Karazin Kharkov National University, Dept. of Physics and Technology, 31
Kurchatov, 61108 Kharkov, Ukraine

###### Abstract

Manifestation of the heavy 3/2-spin lepton $(h^{\pm})$, as possible virtual
intermediate state in Feynmann diagrams, have been searched in the
$e^{+}\,e^{-}\to\gamma\,\gamma$ reaction at high energies and large photon
angles. The spin-vector field $\Psi_{\alpha}$ of the 3/2-lepton is described
by the Rarita-Schwinger one and phenomenological Lagrangian of
$h^{\pm}\,e^{\pm}\,\gamma$-interaction is chosen similarly to interaction of
$\Delta$ isobar with nucleon and $\gamma$ quant. It is described by two
constants with dimension $[M^{-1}]$ and $[M^{-2}].$ The differential cross
section and polarization asymmetries have been calculated for the case when
both beams are polarized longitudinally along their directions, as well
transversally, in the reaction plane, and normally, perpendicularly to it.
Numerical estimations are performed in wide diapason of the collision energy
and parameters entering phenomenological Lagrangian.

PACS: 12.20.-m, 13.40.-f, 13.60.-Hb, 13.88.+e

##  I Introduction

Recent analysis of experimental data collected from 1989 up to 2003, to search
the non-pointness of the electron in $e^{+}\,e^{-}$ annihilation reactions
(see Chen et al. (2023) and references therein) showed a significant signal at
a confidence level 5 standard deviations. To explain this finding authors
suggest the existence of an exited electron with mass about 310 GeV and some
contact interaction with a cutoff scale about 1250 GeV. Although the more
recent experimental data obtained in electron-proton Aaron et al. (2008) and
proton-proton Abazov et al. (2008); Aad et al. (2013); Khachatryan et al.
(2016) collisions do not see an evidence for the exited lepton, this result is
awaiting confirmation at future circular and linear electron-positron
colliders such as FCC Abada et al. (2019), CEPC Dong et al. (2018), ILC Beh
(2013) and ClIC Aic (2012).

In present paper we consider one more mechanism that can influence such
deviation (it may be more exotic one): effects due to existence of a heavy
lepton $h^{\pm}$ with spin 3/2 which, like a heavy exited electron, appears as
a virtual intermediate state in Feynman diarams. We assume that 3/2-spin heavy
lepton is described by the Rarita-Schwinger spin-vector field Rarita and
Schwinger (1941) and it interacts with at least the QED sector of the Standard
Model and can decay into electron and $\gamma$ quant just similarly to decay
of the $\Delta$ isobar into nucleon and photon (see, for example, Jones and
Scadron (1973); Pascalutsa et al. (2007)). Therefore it is quite natural to
look for the manifestation of this lepton in the large-angle high energy
annihilation reaction

$e^{-}(p_{1})+e^{+}(p_{2})\to\gamma(k_{1})+\gamma(k_{2}).$ (1)

The possibility of the existence 3/2-spin heavy lepton within composite models
Harari (1984) and searching for a signal associated with it have been
discussed early in a number works Leite Lopes et al. (1980); Fleury et al.
(1983); Choudhury et al. (1985); Almeida et al. (1996); Cakir and Ozansoy
(2008). Here we mainly focus on the effects due to arbitrary beam
polarizations and consider the influence of the virtual intermediate heavy
lepton on various polarization asymmetries in the angular distribution of
photons.

##  II Formalism

To estimate effects due to 3/2-spin heavy lepton in the reaction (1) we
restrict ourselves with standard QED diagrams (Fig. 1 a)) and diagrams with
intermediate heavy lepton $h^{\pm}$ (Fig. 1 b)) instead of electron (HL
diagrams). We consider the case when both, electron and positron, can be
arbitrary polarized, namely: longitudinally (along their 3-momentum direction)
with 4-vector $S_{1\mu}^{l}$, transversally (in the reaction plane) with
$S_{1\mu}^{t}$ and normal (perpendicularly to the reaction plane) with
$S_{1\mu}^{n}$. Our aim is to search manifestation of the HL diagrams on the
level differential cross section and different polarization asymmetries at
high beam energies achievable at future linear colliders ILC and CLIC.

###  II.1 Normalization and polarization 4-vectors

We choose a normalization such that, being averaged over the electron and
positron polarizations, the differential angular cross section reads

$\frac{d\,\sigma}{d\,\Omega}=\frac{\alpha^{2}}{16\,s}|M^{2}|,\
s=(p_{1}+p_{2})^{2},\ t=(p_{1}-k_{1})^{2},\ u=(p_{1}-k_{2})^{2},$ (2)

where $\alpha=e^{2}/(4\pi)=1/137$ fine structure constant and the matrix
element squared is defined by a sum of the QED, QED-HL interference and HL
contributions. The same expression is valid when electron and positron are
polarized. When calculating traces in $|M^{2}|$ at chosen normalization we
have to use the electron and positron projections as (we omit the spin
indecies)

$u(p_{1})\,\bar{u}(p_{1})=(\hat{p}_{1}+m)(1+\gamma_{5}\,\hat{S}_{1}),\ \
v(p_{2})\,\bar{v}(p_{2})=(\hat{p}_{2}-m)(1+\gamma_{5}\,\hat{S}_{2}),$ (3)

where $u(p_{1})$ and $v(p_{2})$ are the electron and positron bispinors,
respectively.

At high energies, neglecting the electron mass, we can express 4-vectors
$S_{1,2\mu}^{l,t,n}$ in terms of particle 4-momenta, namely

$S_{1\mu}^{l}=\lambda_{1}\frac{p_{1\mu}}{m},\
S_{2\mu}^{l}=\lambda_{2}\frac{p_{2\mu}}{m},\
S_{1\mu}^{t}=\eta_{1}\sqrt{\frac{s}{u\,t}}\Big{(}\frac{u}{s}p_{1\mu}+\frac{t}{s}p_{2\mu}+k_{1\mu}\Big{)},\
S_{2\mu}^{t}=\frac{\eta_{2}}{\eta_{1}}S_{1\mu}^{t},$ (4)
$S_{1\mu}^{n}=\xi_{1}\frac{\varepsilon_{\mu\nu\lambda\rho}\,p_{1}^{\nu}\,p_{2}^{\lambda}\,k_{1}^{\rho}}{K},\
K=\frac{1}{2}\sqrt{stu},\ S_{2\mu}^{n}=\frac{\xi_{2}}{\xi_{1}}\,S_{1\mu},$

where $\lambda_{1,2},\,\eta_{1,2}$ and $\xi_{1,2}$ are corresponding
polarization degrees. In the case of longitudinal polarization one must first
perform multiplication in (3) and then go to the limit $m\to 0.$ Take
attention that part of the differential cross section which does not depend on
polarizations is the same for unpolarized and polarized beams.

Figure 1: Feynman diagrams for the process
$e^{+}\,e^{-}\rightarrow\gamma\,\gamma$. a) QED diagrams, b) diagrams with the
heavy 3/2-spin lepton $h$ in $t-$ and $u-$ channels.

Let $\big{(}d\sigma/d\Omega\big{)}_{un}$ is the cross section in unpolarized
case and $\big{(}d\sigma/d\Omega\big{)}_{i\,j},\,i,j=l,\,t,\,n$ is the cross
section in the case of positron polarization $S_{2}^{i}$ and electron one
$S_{1}^{j}$. Then we define the polarization asymmetries
$A_{ll},\,\,A_{tt},\,\,A_{nn},\,\,A_{lt}$ and $A_{tl}$ as follows

$\frac{d\sigma_{ll}}{d\Omega}=\frac{d\sigma_{un}}{d\,\Omega}\big{(}1\pm\lambda_{1}\,\lambda_{2}A_{ll}\big{)},\
\frac{d\sigma_{tt}}{d\Omega}=\frac{d\sigma_{un}}{d\,\Omega}\big{(}1\pm\eta_{1}\,\eta_{2}A_{tt}\big{)},\
\frac{d\sigma_{nn}}{d\Omega}=\frac{d\sigma_{un}}{d\,\Omega}\big{(}1\pm\xi_{1}\,\xi_{2}A_{nn}\big{)},$
(5)
$\frac{d\sigma_{lt}}{d\Omega}=\frac{d\sigma_{un}}{d\,\Omega}\big{(}1\pm\eta_{1}\,\lambda_{2}A_{lt}\big{)},\
\frac{d\sigma_{tl}}{d\Omega}=\frac{d\sigma_{un}}{d\,\Omega}\big{(}1\pm\lambda_{1}\,\eta_{2}A_{tl}\big{)}.$

Signs in front $A_{ij}$ can be changed by changing the sign of one of the
4-vector polarization to the opposite; sign ”+” corresponds to choice given by
definitions (4).

Note that $A_{nt}=A_{nl}=A_{tn}=A_{ln}=0$ for obvious reason. In the limit
$m\to 0$ the Born approximation gives $A_{lt}^{B}=A_{tl}^{B}=0$ and

$\left(\frac{d\sigma}{d\Omega}\right)_{un}^{B}=\frac{\alpha^{2}}{s}\Big{(}\frac{1+\cos^{2}{\theta}}{1-\cos^{2}{\theta}}\Big{)},\
\ A_{ll}^{B}=-1,\
A_{tt}^{B}=A_{nn}^{B}=\frac{1-\cos^{2}{\theta}}{1+\cos^{2}{\theta}}.$ (6)

###  II.2 About 3/2-spin heavy lepton

We suggest that even-parity 3/2-spin lepton $h^{\pm}$ is described by the
Rarita-Schwinger spin-vector field
$\Psi^{\alpha}_{p}(x)=u^{\alpha}(p)\exp(-ip\cdot x)$ and Lagrangian its
interaction with electron and photon is very similar to the Lagrangian
describing electromagnetic decay of $\Delta$ isobar into nucleon and $\gamma$
quant: $\Delta\to N+\gamma.$ There are a lot possibilities to choose this
Lagrangian, see for example, Pascalutsa (1998); Pascalutsa and Timmermans
(1999); Kristiano et al. (2017) but we will use well known one from Jones and
Scadron (1973)

$L_{\Psi^{\alpha}\,e\,\gamma}=e\,\bar{\psi}\,\gamma_{5}\Big{(}\frac{ig_{1}}{M}\gamma^{\mu}-\frac{g_{2}}{M^{2}}\partial^{\mu}\,\Big{)}\Psi^{\alpha}\,F_{\alpha\mu}+h.\,c.,\
F_{\alpha\mu}=\partial_{\mu}\,A_{\alpha}-\partial_{\alpha}\,A_{\mu},$ (7)

where dimensionless constants $g_{1}$ and $g_{2}$ are real (C P parity is
conserved), parameter $M$ has the dimension of mass, and $F_{\alpha\mu}$ is
tensor of an electromagnetic field. Below we take mass of the $h^{\pm}$ as the
parameter $M$ and assume $M^{2}\approx s.$

Lagrangian (7) defines an amplitude $A_{D}$ of the decay $h^{-}(p_{1})\to
e^{-}(p_{2})+\gamma(k)$ in terms of constants $g_{1}$ and $g_{2}$ as follows

$A_{D}=e\,\bar{u}(p_{2})\,\gamma_{5}\,\Gamma^{\alpha\mu}\,u_{\alpha}(p_{1})\,\varepsilon^{*}_{\mu}(k),$
(8)

where

$\Gamma^{\alpha\mu}=\frac{g_{1}}{M}(k^{\alpha}\,\gamma^{\mu}-\hat{k}\,g^{\alpha\mu})+\frac{g_{2}}{M^{2}}[k^{\alpha}\,p_{1}^{\mu}-(k\cdot
p_{1})g^{\alpha\mu}].$

The square of this amplitude reads

$|A_{D}|^{2}=e^{2}\,Sp\big{\\{}(\hat{p}_{2}-m)\,\Gamma^{\alpha\mu}\,(\hat{p}_{1}+M)\,P_{\alpha\beta}(p_{1})\Gamma^{\beta}_{\mu}\big{\\}},$
(9)
$P_{\alpha\beta}(p_{1})=g_{\alpha\beta}-\frac{1}{3}\gamma_{\alpha}\,\gamma_{\beta}-\frac{1}{3\,p_{1}^{2}}\big{(}\hat{p}_{1}\gamma_{\alpha}p_{1\beta}+p_{1\alpha}\gamma_{\beta}\hat{p}_{1}\big{)},\
\ \Gamma^{\beta}_{\mu}=\Gamma^{\beta\lambda}\,g_{\lambda\mu},$

where we taken into account that
$\sum_{pol}\varepsilon^{*}_{\mu}\varepsilon_{\rho}=-g_{\mu\rho}.$ Using
conservation low one obtains the partial decay width neglecting the electron
mass

$\Gamma(h^{\pm}\to
e^{\pm}\,\gamma)=\frac{\alpha\,M}{24}\big{(}3g_{1}^{2}+g_{2}^{2}+3g_{1}\,g_{2}\big{)}.$
(10)

Note that in this limit the analytic form of the corresponding negative-parity
HL decay width coincides with (10).

###  II.3 Unpolarized cross section

In accordance with Feynmann rules, contribution HL diagrams to the reaction
amplitude can be written as follows (we extracted factor $\alpha^{2}$ in
accordance with the definition (2) of the differential cross section)

$M_{h}=\bar{v}(p_{2})\gamma_{5}\big{[}\Gamma^{\beta\nu}_{22}\Delta_{\beta\alpha}(p)\Gamma\,^{\alpha\mu}_{11}+\Gamma^{\beta\mu}_{21}\Delta_{\beta\alpha}(q)\Gamma^{\alpha\nu}_{12}\big{]}\gamma_{5}u(p_{1})\epsilon^{*}_{\mu}(k_{1})\,\epsilon^{*}_{\nu}(k_{2}),$
(11)

where $\epsilon_{\mu}(k_{1})$ and $\epsilon_{\nu}(k_{2})$ are 4-vectors of
photon polarizations and

$p=p_{1}-k_{1}=k_{2}-p_{2},\ q=p_{1}-k_{2}=k_{1}-p_{2}.$

The sort notation for all $\Gamma$’s in (11) is

$\Gamma^{\beta\nu}_{ij}=\frac{g_{1}}{M}(k_{j}^{\beta}\gamma^{\nu}-\hat{k}_{j}\,g^{\beta\nu})+\frac{g_{2}}{M^{2}}[k_{j}^{\beta}p_{i}^{\nu}-(p_{i}\cdot
k_{j})\,g^{\beta\nu}],\,(i,j=1,2)$ (12)

Propagator $\Delta_{\alpha\beta}$ of the particle with spin 3/2 reads (see
Eqs. (13), (14) in Ref. PT1999)

$\Delta_{\alpha\beta}(p)=-\frac{\hat{p}+M}{p^{2}-M^{2}}P_{\alpha\beta}(p),\
P_{\alpha\beta}(p)\,p^{\beta}=p^{\alpha}\,P_{\alpha\beta}(p)=0,\
P_{\alpha\beta}(p)\,\gamma^{\beta}=\gamma^{\alpha}\,P_{\alpha\beta}(p)=0.$
(13)

To obtain $|M_{h}|^{2}$ one has to compute rather long trace

$|M_{h}|^{2}=Spur\big{\\{}(\hat{p}_{2}-m)\gamma_{5}\big{[}\Gamma^{\beta\nu}_{22}\Delta_{\beta\alpha}(p)\Gamma\,^{\alpha\mu}_{11}+\Gamma^{\beta\mu}_{21}\Delta_{\beta\alpha}(q)\Gamma^{\alpha\nu}_{12}\big{]}\gamma_{5}\times$
(14)
$(\hat{p_{1}}+m)\gamma_{5}\big{[}\Gamma_{11\,\mu}^{\delta}\widetilde{\Delta}_{\delta\sigma}(p)\Gamma_{22\,\nu}^{\sigma}+\Gamma_{12\,\nu}^{\delta}\widetilde{\Delta}_{\delta\sigma}(q)\Gamma_{21\,\mu}^{\sigma}\big{]}\gamma_{5}\big{\\}},\
\widetilde{\Delta}_{\delta\sigma}(p)=-P_{\delta\sigma}(p)\frac{\hat{p}+M}{p^{2}-M^{2}}.$

As concerns the interference of the pure QED and HL diagrams, it can be
written as

$2\,M_{h}\,M_{e}^{*}=2\,Spur\big{\\{}(\hat{p}_{2}-m)\gamma_{5}\big{[}\Gamma^{\beta\nu}_{22}\Delta_{\beta\alpha}(p)\Gamma\,^{\alpha\mu}_{11}+\Gamma^{\beta\mu}_{21}\Delta_{\beta\alpha}(q)\Gamma^{\alpha\nu}_{12}\big{]}\gamma_{5}\times$
(15)
$(\hat{p}_{1}+m)\big{[}\gamma_{\mu}\frac{\hat{p}+m}{p^{2}-m^{2}}\gamma_{\nu}+\gamma_{\nu}\frac{\hat{q}+m}{q^{2}-m^{2}}\gamma_{\mu}\big{]}\big{\\}}.$

In what follows we neglect the electron mass and introduce the dimensionless
invariant variables

$s=\frac{2(p_{1}\cdot p_{2})}{M^{2}},\ t=-\frac{2(p_{1}\cdot k_{1})}{M^{2}},\
u=-\frac{2(p_{1}\cdot k_{2})}{M^{2}},\ s+t+u=0,\ \cos{\theta}=1+\frac{2t}{s}.$
(16)

These variables are very convenient especially in the limit $m\to 0.$ In terms
of these variables we obtain for HL diagrams contribution into matrix element
squared

$\left|M_{h}\right|^{2}=2g_{1}^{4}\,A_{11}(s,t)+\frac{1}{3}g_{1}^{2}\,g_{2}^{2}\,A_{12}(s,t)+\frac{1}{36}g_{2}^{4}\,A_{22}(s,t),$
(17)

where

$A_{11}(s,t)=-2[3+2s^{2}+t^{2}+s(5+t)]-(1+3s+3s^{2})\Big{[}\frac{1}{(1-t)^{2}}+\frac{1}{(1+s+t)^{2}}\Big{]}+$
$\frac{8+22s+21s^{2}+10s^{3}+s^{4}}{2+s}\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)},$
$A_{12}(s,t)=4s^{3}+3s^{2}(11+6t)+12(2+t)^{2}+6s(11+2t+3t^{2})+3(1+2s)^{2}\Big{[}\frac{1}{(1-t)^{2}}+\frac{1}{(1+s+t)^{2}}\Big{]}$
$-\frac{30+114s+126s^{2}+46s^{3}+s^{4}}{2+s}\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)},$
$A_{22}(s,t)=s^{3}(23-5t)-s^{2}(74+44t+21t^{2})-4s(21+9t+11t^{2}+8t^{3})-4(14+9t^{2}+4t^{4})$
$-(5+6s+6s^{2})\Big{[}\frac{1}{(1-t)^{2}}+\frac{1}{(1+s+t)^{2}}\Big{]}$
$+\frac{66+152s+175s^{2}+52s^{3}-18s^{4}}{2+s}\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)}.$

The QED-HL interference contributes as follows

$2\,Re[M_{h}\,M_{e}^{*}]=8\,g_{1}^{2}\Big{[}\frac{(1+s)^{2}}{1-t}+\frac{(1+s)^{2}}{1+s+t}-2-3s\Big{]}+$
(18)
$\frac{2\,g_{2}^{2}}{3}\Big{[}8+14s+s^{2}+8st+8t^{2}-(4+9s+3s^{2})\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)}\Big{]}.$

To calculate the angular distribution in high energy limit one can use

$A_{11}(s,{\theta})=\frac{4(8+22s+21s^{2}+10s^{3}+s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}-6-10s-\frac{s^{2}(7+\cos^{2}{\theta})}{2}$
$-4(1+3s+3s^{2})\Big{[}\frac{1}{(2+s-s\cos{\theta})^{2}}+\frac{1}{(2+s+s\cos{\theta})^{2}}\Big{]},$
$A_{12}(s,{\theta})=24+66s+30s^{2}-\frac{s^{3}}{2}+3s^{2}\cos^{2}{\theta}\Big{(}1+\frac{3s}{2}\Big{)}$
$+12(1+2s)^{2}\Big{[}\frac{1}{(2+s-s\cos{\theta})^{2}}+\frac{1}{(2+s+s\cos{\theta})^{2}}\Big{]}-\frac{4[30+114s+126s^{2}+46s^{3}+s^{4}]}{(2+s)^{2}-s^{2}\cos^{2}{\theta}},$
$A_{22}(s,{\theta})=-56-84s-65s^{2}+34s^{3}+\frac{s^{4}}{4}-\Big{(}9+11s-\frac{3s^{2}}{4}\Big{)}s^{2}\cos^{2}{\theta}-s^{4}\cos^{4}{\theta}+$
$\frac{4(66+152s+175s^{2}+52s^{3}-18s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}-(5+6s+6s^{2})\Big{[}\frac{4}{(2+s-s\cos{\theta})^{2}}+\frac{4}{(2+s+s\cos{\theta})^{2}}\Big{]}.$

Regards the QED-HL interference we have

$2\,Re[M_{3/2}^{+}M^{e*}]=8e^{2}\,g_{1}^{2}\Big{[}-2-3s+\frac{4(1+s)^{2}(2+s))}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}\Big{]}+$
(19)
$\frac{2e^{2}\,g_{2}^{2}}{3}\Big{[}8+14s+s^{2}-2s^{2}\sin^{2}{\theta}-\frac{4(4+9s+3s^{2})(2+s)}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}\Big{]}.$

Note that in unpolarized case neither $|M_{h}|^{2}$ nor $Re[M_{h}\,M_{e}^{*}]$
contains product of the odd powers $g_{1}$ and $g_{2},$ so they do not depend
on relative sign of these constants. With accounting for all the contribution,
we can write down the the differential over the photon solid angle cross
section in unpolarized case as

$\frac{d\sigma_{un}}{d\,\Omega}=\frac{\alpha^{2}}{16M^{2}}\Big{[}\frac{16(1+\cos^{2}\theta)}{s(1-\cos^{2}\theta)}+\frac{2\,Re[M_{h}\,M_{e}^{*}]}{s}+\frac{|M_{h}|^{2}}{s}\Big{]}.$
(20)

In Fig. 2 we plot the differential cross section (20) (in units $M^{-2})$ at
different values of constant $g_{1}:1,\,0.35,\,0.1$ and two angles
$\theta=45^{o},\,90^{o}$ as a function of the ratio $g=g_{2}/g_{1}$ and
dimensionless invariant variable $s$.

Figure 2: Unpolarized differential cross section defined by Eq. (20) (in units
$M^{-2}$) at two values of the photon angle: $\theta=45^{o},$ the upper row;
$\theta=\,90^{o}$, the lower row; and three values of constants $g_{1}$:
1$-$left panel; 0.35$-$middle panel; 0.1$-$ right panel. $g$ is the ratio
$g_{2}/g_{1}$.

##  III Polarization asymmetries

To estimate different polarization asymmetries one has to compute
corresponding polarization dependent parts of the cross section, using the
electron and positron projections as given by Eq.(3) and expressions for
different polarization 4-vectors defined by relations (4).

###  III.1 Case of longitudinal polarization

The direct calculation in the case of longitudinal polarization of the
electron and positron along beam directions results

$\left|M_{h}^{+ll}\right|^{2}=2g_{1}^{4}L_{11}(s,t)+\frac{1}{3}L_{12}(s,t)+\frac{1}{36}L_{22}(s,t),$
(21)
$L_{11}(s,t)=2[3+2s^{2}+t^{2}+s(5+t)]+(1+3s+3s^{2}+2s^{3})\Big{[}\frac{1}{(1-t)^{2}}+\frac{1}{(1+s+t)^{2}}\Big{]}$
$-\,\frac{8+22s+21s^{2}+6s^{3}+s^{4}}{2+s}\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)},$
$L_{12}(s,t)=2s^{3}+3s^{2}(5+2t)-12(2+t^{2})-6s(3+2t-t^{2})-3(1+2s)\Big{[}\frac{1}{(1-t)^{2}}+\frac{1}{(1+s+t)^{2}}\Big{]}+$
$\frac{30+54s+6s^{2}-14s^{3}+s^{4}}{2+s}\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)},$
$L_{22}(s,t)=s^{3}(47+5t)+s^{2}(178+44t+21t^{2})+4s(43+9t+11t^{2}+8t^{3})+4(14+9t^{2}+4t^{4})+$
$(5+28s+54s^{2}+36s^{3})\Big{[}\frac{1}{(1-t)^{2}}+\frac{1}{(1+s+t)^{2}}\Big{]}$
$-\,\frac{66+284s+463s^{2}+320s^{3}+78s^{4}}{2+s}\Big{(}\frac{1}{1-t}+\frac{1}{1+s+t}\Big{)}.$

For the corresponding contribution into QED HL interference we have

$2\,Re[M_{h}\,M_{e}^{*}]^{ll}=-8\,g_{1}^{2}\frac{[s^{3}+2t^{2}+s^{2}(1+3t)+st(2+3t)]}{(1-t)(1+s+t)}+$
(22)
$2\,g_{2}^{2}\frac{8t^{4}+s^{3}(2+t)+3s^{2}t(2+3t)+2st^{2}(3+8t)}{3(1-t)(1+s+t}.$

As one can see, the odd powers of constant $g_{1}$ and $g_{2}$ are also absent
as in the unpolarized case.

To study the $(s,\,\theta)$ distribution one can use for contribution into
$\left|M_{h}^{+ll}\right|^{2}$

$L_{11}(s,\theta)=6+10s+4s^{2}-\frac{s^{2}}{2}\,\sin^{2}{\theta}-\frac{4(8+22s+21s^{2}+6s^{3}+s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}+$
$4(1+3s+3s^{2}+2s^{3})\big{[}\frac{1}{(2+s-s\cos{\theta})^{2}}+\frac{1}{(2+s-s\cos{\theta})^{2}}\big{]},$
$L_{12}(s,\theta)=-24-18s+15s^{2}+23s^{3}+3s^{2}\Big{(}1-\frac{s}{2}\Big{)}\sin^{2}{\theta}+\frac{4(30+54s+6s^{2}-14s^{3}+s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}$
$-12(1+2s)\big{[}\frac{1}{(2+s-s\cos{\theta})^{2}}+\frac{1}{(2+s-s\cos{\theta})^{2}}\big{]},$
$L_{22}(s,\theta)=56+172s+178s^{2}+47s^{3}+\Big{(}s^{2}\sin^{2}{\theta}-9-11s-\frac{5s^{2}}{4}\Big{)}s^{2}\sin^{2}{\theta}+$
$4(5+28s+54s^{2}+36s^{3})\big{[}\frac{1}{(2+s-s\cos{\theta})^{2}}+\frac{1}{(2+s-s\cos{\theta})^{2}}\big{]}$
$-\,\frac{4(66+284s+463s^{2}+320s^{3}+78s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}.$

The corresponding result for the QED-HL interference reads

$2\,Re[M_{h}\,M_{e}^{*}]^{ll}=-8\,g_{1}^{2}s^{2}\frac{4(1+s)-(2+3s)\sin^{2}{\theta}}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}+\frac{2}{3}\,g_{2}^{2}\,s^{3}\frac{8+(2s\sin^{2}{\theta}-6-s)\sin^{2}{\theta}}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}.$
(23)

Asymmetry $A_{ll}$, in the case of longitudinally polarized electron and
positron along their beam directions, is plotted In Fig. 3 at the same values
of the angle $\theta$ and constant $g_{1}$ as in Fig. 2.

Figure 3: Asymmetry $A_{ll}$ as defined by Eq. (5) at the same conditions as
in Fig. 2 for unpolarized cross section.

###  III.2 Transverse polarization of beams in the reaction plane

As one can see from (4), polarization 4-vector $S_{1}^{t}$ in the rest frame
of the electron has only space component which (for $\lambda_{1}=1)$ is
$S_{1}=(0,{\bf n}_{1t})$ and in this system ${\bf n}_{1t}\cdot{\bf p}_{2}=0.$
Transition to c.m.s. does not change $S_{1}^{t}$, and because the same is
valid for $S_{2}$ (with the change ${\bf n}_{1t}\rightleftarrows{\bf n}_{2t},\
{\bf p}_{2}\rightleftarrows{\bf p}_{1}$), it is possible ${\bf n}_{1t}=\pm{\bf
n}_{2t}$ only. The same is valid for the normal 4-vectors $S_{1}^{n}$ and
$S_{2}^{n}$.

The $|M_{h}^{tt}|^{2}$, when both, the electron and positron are transversely
polarized in the reaction plane along the same direction can be written as

$\left|M_{3/2}^{+tt}\right|^{2}=e^{4}\,\big{[}2\,g_{1}^{4}\,T_{11}(s,t)+\frac{1}{3}\,g_{1}^{2}\,g_{2}^{2}\,T_{12}(s,t)+\frac{1}{36}\,g_{2}^{4}\,T_{22}(s,t)\big{]}$
(24) $T_{11}(s,t)=\frac{2t^{2}(s+t)^{2}}{(1-t)(1+s+t)},$
$T_{12}(s,t)=\frac{(s+t)[s^{3}-4st^{3}+t^{4}-5s^{2}t(1+t)]}{(1+s+t)^{2}}-\frac{t\,[8st^{3}+t^{4}+6s^{3}(1+t)+s^{2}t(5+13t)]}{(1-t)^{2}}$
$+\frac{s\,[s^{3}+9t^{4}+2st^{2}(1+9t)+s^{2}t(2+9t)]}{(1-t)(1+s+t)},$
$T_{22}(s,t)=\frac{3\,(s+t)^{2}\,[s^{2}t(t-2)+st^{2}(t-3)+t^{4}+s^{3}(1+3t)]}{(1+s+t)^{2}}$
$+\frac{3\,t^{2}\,[t^{4}+6s^{3}(1+t)+st^{2}(3+5t)+2s^{2}t(5t+4)]}{(1-t)^{2}}$
$-\frac{2\,t(s+t)[s^{3}(6-12t)+5t^{4}-s^{2}t(9+7t)+st^{2}(10t-9)]}{(1-t)(1+s+t)}.$

The QED-HL interference in this case reads

$2\,Re\big{[}M_{h}\,M_{e}^{*}\big{]}=\frac{2\,t(s+t}{3(1-t)(1+s+t)}[-12\,g_{1}^{2}\,(2+s)+g_{2}^{2}[3s^{2}t+8t^{2}+s(2+8t)].$
(25)

The corresponding expressions for angular distribution read

$T_{11}(s,\theta)=\frac{s^{4}\,\sin^{4}{\theta}}{2[(2+s)^{2}-s^{2}\,\cos^{2}{\theta}]},$
$T_{12}(s,\theta)=\frac{s^{4}(1+\cos{\theta})[56-11s+4(3s-10)\cos{\theta}+s(10\cos^{2}{\theta}-12\cos^{3}{\theta}+\cos^{4}{\theta})]}{8\,(2+s+s\cos{\theta})^{2}}$
$+\frac{s^{4}[8+9s+(8-18s)\,\cos^{2}{\theta}+9s\,\cos^{4}{\theta}]}{4[(2+s)^{2}-s^{2}\cos^{2}{\theta}]}$
$+\frac{s^{4}(1-\cos{\theta})[56-11s+(40-12s)\cos{\theta}+s(10\,\cos^{2}{\theta}+12\,\cos^{3}{\theta}+\cos^{4}{\theta})]}{8(2+s-s\cos{\theta})^{2}},$
$T_{22}(s,\theta)=\frac{s^{5}\,\sin^{2}{\theta}\,[132+53s-36\cos^{2}{\theta}-58\cos^{2}{\theta}+5s\cos^{4}{\theta}]}{8[(2+s)^{2}-s^{2}\cos^{2}{\theta}}$
$+\frac{3s^{5}(1+\cos{\theta})^{2}\,[44-17s+(6s-40)\cos{\theta}+4(3+4s)\cos^{2}{\theta}-6s\cos^{3}{\theta}+s\cos^{4}{\theta}]}{16(2+s+s\cos{\theta})^{2}}$
$+\frac{3s^{5}(1-\cos{\theta})^{2}[44-17s+40\cos{\theta}+12\cos^{2}{\theta}+s(-6\cos{\theta}+16\cos^{2}{\theta}+6\cos^{3}{\theta}+\cos^{\theta})]}{16(2+s-s\cos{\theta})^{2}}.$

The interference with the QED amplitude is

$2\,Re\,\big{[}M_{h}\,M_{e}^{*}\big{]}^{tt}=\frac{2s^{2}\sin^{2}{\theta}}{3[(2+s)^{2}-s^{2}\cos^{2}{\theta}]}\big{[}12\,g_{1}^{2}(2+s)-g_{2}^{2}s(2+s+2s\cos^{2}{\theta})\big{]}.$
(26)

Asymmetry caused by the transverse polarizations of beams along the same
direction is shown in Fig. 4. It changes sign if directions of polarizations
are opposite.

Figure 4: The same as in Fig. 3 but for asymmetry $A_{tt}.$

###  III.3 longitudinal-transverse correlation

If the positron beam is polarized longitudinally and the electron beam $-$
transversally (or vise versa), the corresponding spin-dependent contribution
is absent in the Born approximation. But contribution caused by the
intermediate 3/2-spin heavy lepton is nonzero in this case and has a form

$\left|M_{h}^{+lt}\right|^{2}=(-st)\sqrt{\frac{s+t}{-st}}\Big{(}2g_{1}^{3}\,g_{2}\,P^{lt}_{31}(s,t)+g_{1}^{2}\,g_{2}^{2}\,P^{lt}_{22}(s,t)+\frac{1}{3}g_{1}\,g_{2}^{3}\,P^{lt}_{13}(s,t)+\frac{1}{3}g_{2}^{4}\,P^{lt}_{04}(s,t)\Big{)},$
(27)
$P^{lt}_{31}(s,t)=(1+2s+2s^{2})\Big{[}\frac{1}{(1+s+t)^{2}}-\frac{1}{(1-t)^{2}}\Big{]}-(4+5s+s^{2})\Big{[}\frac{1}{1+s+t}-\frac{1}{1-t}\Big{]}-2(s+2t),$
$P^{lt}_{22}(s,t)=-(1+2s+2s^{2})\Big{[}\frac{1}{(1+s+t)^{2}}-\frac{1}{(1-t)^{2}}\Big{]}+(2+3s+s^{2})\Big{[}\frac{1}{1+s+t}-\frac{1}{1-t}\Big{]},$
$P^{lt}_{13}(s,t)=(2-s)(s+2t)-\Big{[}\frac{1}{(1+s+t)^{2}}-\frac{1}{(1-t)^{2}}\Big{]}+4\Big{[}\frac{1}{1+s+t}-\frac{1}{1-t}\Big{]},$
$P^{lt}_{04}(s,t)=-(2+s)(s+2t)+(1+3s+3s^{2})\Big{[}\frac{1}{(1+s+t)^{2}}-\frac{1}{(1-t)^{2}}\Big{]}+(4+8s+5s^{2})\Big{[}\frac{1}{1+s+t}-\frac{1}{1-t}\Big{]}.$

As one can see, term with $g_{1}^{4}$ is absent and terms appear with the odd
powers of $g_{1}$ and $g_{2}.$ If contrary, the electron beam is polarized
longitudinally and the positron beam transversally, we have

$P^{tl}_{31}(s,t)=P^{lt}_{31}(s,t),\ P^{tl}_{13}(s,t)=P^{lt}_{13}(s,t),\
P^{tl}_{22}(s,t)=-P^{lt}_{22}(s,t),\ P^{tl}_{04}(s,t)=-P^{lt}_{04}(s,t)$

(terms with even powers $g_{1}$ and 2 change sign).

The QED-HL interference is also nonzero and reads

$2Re\big{[}M_{h}\,M_{e}^{*}\big{]}^{lt}=2e^{4}s^{2}\sqrt{\frac{-t(s+t)}{s}}\Big{[}2g_{1}g_{2}\Big{(}\frac{1}{1+s+t}-\frac{1}{1-t}\Big{)}+g_{2}^{2}\Big{(}\frac{s+t}{1+s+t}+\frac{s+t}{1-t}\Big{)}\Big{]}.$
(28)

In the case when the positron beam is polarized transversally and the electron
one longitudinally, term with $g_{1}\,g_{2}$ change sign.

Taking into account that

$(-st)\sqrt{\frac{s+t}{-st}}=\frac{s^{3/2}}{2}\sin{\theta},$

one can rewrite corresponding the $(s,\,\theta)-$distribution

$P^{lt}_{31}(s,\,\theta)=2s\cos{\theta}\Big{[}-1+\frac{2(4+5s+s^{2})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}-\frac{8(2+s)(1+2s+2s^{2})}{[(2+s)^{2}-s^{2}\cos^{2}{\theta}]^{2}}\Big{]},$
$P^{lt}_{22}(s,\,\theta)=4s\cos{\theta}\Big{[}-\frac{2+3s+s^{2}}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}+\frac{4(2+s)(1+2s+2s^{2})}{[(2+s)^{2}-s^{2}\cos^{2}{\theta}]^{2}}\Big{]},$
$P^{lt}_{13}(s,\,\theta)=s\cos{\theta}\Big{[}2-s-\frac{16}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}+\frac{16(2+s)}{[(2+s)^{2}-s^{2}\cos^{2}{\theta}]^{2}}\Big{]},$
$P^{lt}_{04}(s,\,\theta)=s\cos{\theta}\Big{[}-2-s-\frac{4(4+8s+5s^{2})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}-\frac{16(2+s)(1+3s+3s^{2})}{[(2+s)^{2}-s^{2}\cos^{2}{\theta}]^{2}}\Big{]}.$

The QED-HL interference for this distribution is

$2\,Re\,\big{[}M_{3/2}^{+}\,M^{e*}\big{]}^{lt}=4\,s^{5/2}\sin{\theta}\,\cos{\theta}\,\frac{-2g_{1}g_{2}\,s+g_{2}^{2}(2+s)}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}.$
(29)

It is clear that corresponding asymmetries go to zero if $\theta$ go to
90${}^{o}.$ They are shown in Figs. 5, 6.

Figure 5: Asymmetry $A_{lt}$ at photon angle $\theta=45^{o}$ (the upper row);
$\theta=60^{0}$ (lower row) and the values of constants $g_{1}$ as in Fig. 2.
It arises due to intermediate 3/2-spin heavy lepton and goes to zero at
$\theta\to\pi/2.$

Figure 6: The same as in Fig. 5 but for asymmetry $A_{tl}.$

###  III.4 normal polarizations

As it noted above, at $\eta_{1}=\eta_{2}=1,$ the 4-vectors of normal
polarization are $S_{1}^{n}=(0,\,{\bf n_{1n}})$ and $S_{2}^{n}=(0,\,{\bf
n_{2n}})$ with ${\bf n_{1n}}=\pm{\bf n_{2n}}.$ If both beams are polarized
perpendicularly to the reaction plane in the same direction we have

$|M_{h}|^{2}=2g_{1}^{4}\,N_{11}(s,t)+\frac{1}{3}g_{1}^{2}\,g_{2}^{2}\,N_{12}(s,t)+\frac{1}{36}N_{22}(s,t),$
(30) $N_{11}(s,t)=\frac{2t^{2}(s+t)^{2}}{(1-t)(1+s+t)},$
$N_{12}(s,t)=\frac{[s^{3}-4st^{3}+t^{4}-5s^{2}t(1+t)](s+t)}{(1+s+t)^{2}}-\frac{t[8st^{3}+t^{4}+6s^{3}(1+t)+s^{2}t(5+13t)]}{(1-t)^{2}}$
$+\frac{s\,[s^{3}+9t^{4}+2st^{2}(1+9t)+s^{2}t(2+9t)]}{(1-t)(1+s+t)},$
$N_{22}(s,t)=\frac{3(s+t)^{2}[s^{2}t(t-2)+st^{2}(t-3)+t^{4}+s^{3}(1+3t)]}{(1+s+t)^{2}}$
$+\frac{3t^{2}[t^{4}+6s^{3}(1+t)+st^{2}(3+5t)+2s^{2}t(4+5t)]}{(1-t)^{2}}$
$-\frac{2t[s^{3}(6-12t)+5t^{4}-s^{2}t(9+7t)+st^{2}(10t-9)]}{(1-t)(1+s+t)}.$

The QED-HL interference in this case coincides with (25).

$N_{11}(s,\theta)=-2(1+s)+\frac{s^{2}\,\sin^{2}{\theta}}{2}+\frac{8(1+s)}{(2+s)^{2}-s^{2}\cos^{2}{\theta}},$
$N_{12}(s,\theta)=-8-6s-10s^{2}-\frac{s^{2}}{2}+\frac{s^{2}(s-2)\cos^{2}{\theta}}{2}+\frac{4(10+18s+22s^{2}+12s^{3}-s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}$
$-\frac{8\,(1+2s+2s^{2})[(2+s)^{2}+s^{2}\cos^{2}{\theta}]}{[(2+s)^{2}-s^{2}\cos^{2}{\theta}]^{2}},$
$N_{22}(s,\theta)=40+92s+63s^{2}-34s^{3}+\frac{s^{4}}{4}+\frac{s^{2}(28+52s-5s^{2})\cos^{2}{\theta}}{4}+s^{4}\cos^{4}{\theta}$
$-\frac{4(46+144s+177s^{2}+52s^{3}-18s^{4})}{(2+s)^{2}-s^{2}\cos^{2}{\theta}}+\frac{24(1+2s+2s^{2})((2+s)^{2}+s^{2}\cos^{2}{\theta})}{[(2+s)^{2}-s^{2}\cos^{2}{\theta}]^{2}}.$

As well as longitudinal and transverse asymmetries the normal one does not
contain the odd powers of $g_{1}$ and $g_{2}.$ It plotted in Fig. 7.

Figure 7: the same asin Fig. 3 but for asymmetry $A_{nn}$.

##  IV Discussion and conclusion

Manifestation of the 3/2-spin heavy lepton $h^{\pm}$ has been considered in
large-angle $e^{+}\,e^{-}\to\gamma\,\gamma$ reaction at high energies. We
suggest that lepton $h^{\pm}$ can interact with QED sector of the SM and
appears as a virtual intermediate state in Feynmann diagrams, changing the
intermediate electrons in $t$ and $u$ channels of this reaction.
Phenomenological Lagrangian of the $h^{\pm}\,e^{\pm}\,\gamma$ interaction
chosen similarly to one describing decay of the $\Delta$ isobar into electron
and photon. It contains two unknown real constants which are dimensionless by
dividing by the parameter with the dimension of mass, and we take this
parameter as the heavy lepton mass $M$.

At energies $\sqrt{s}\approx$ 100 GeV the born differential section (6) is
about 2 pb and decreases with increasing energy as $s^{-1}$. Electroweak
radiative corrections (EWRC) to the total cross section at such energies
consist of about few percent and the positive QED share partially compensated
by a negative weak correction Carloni Calame et al. (2019); Bondarenko et al.
(2023). Contribution due to 3/2-spin heavy lepton increases very quickly with
the energy and therefore can, in principle, be competitive with EWRC at ILC
and CLIC colliders.

Our finding is that at $g_{1}=1$ and $M=$ 100 GeV contribution due to HL and
QED-HL interference dominates already at $\sqrt{s}=$ 50 GeV and such situation
seems implausible. At $g_{1}=0.35$ it is of the order born cross section what
is unrealistic too, and if $g_{1}=0.1$ it can be on the level of EWRC. Very
interesting is the possibility to probe effects due to heavy lepton by
measuring the longitudinal-transverse asymmetries $A_{lt}$ and $A_{tl}$ Figs.
5, 6 where the pure QED mechanism does not contribute.

Acknowledgment

The authors acknowledge support by the National Academy of Sciences of Ukraine
via the program “Participation in the international projects in high energy
and nuclear physics” (project no. 0121U111693).

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