This article with minor changes was accepted for publication in

Journal of Fourier Analysis and Applications

The Heyde characterization theorem on compact totally

disconnected and connected Abelian groups

Gennadiy Feldman

Abstract

By the well-known Heyde theorem, the Gaussian distribution on the real line is
characterized by the symmetry of the conditional distribution of one linear
form of independent random variables given another. In the case of two
independent random variables we give a complete description of compact totally
disconnected Abelian groups $X$, where an analogue of this theorem is valid.
We also prove that even a weak analogue of the Heyde theorem fails on compact
connected Abelian groups $X$. Coefficients of considered linear forms are
topological automorphisms of $X$. The proofs are based on the study of
solutions of a functional equation on the character group of the group $X$ in
the class of Fourier transforms of probability distributions.

Keywords Compact totally disconnected Abelian group $\cdot$ Compact connected
Abelian group $\cdot$ Topological automorphism $\cdot$ Haar distribution

Mathematics Subject Classification Primary 43A35 Secondary 60B15 $\cdot$ 62E10

##  1  Introduction

Let $(\Omega,\mathfrak{A},P)$ be a probability space, $X$ be a topological
Abelian group, $\mathfrak{B}$ be the $\sigma$-algebra of Borel sets in $X$.
Consider random variables $\xi$ and $\eta$ defined on
$(\Omega,\mathfrak{A},P)$ with values in $(X,\mathfrak{B})$. The definition of
the conditional distribution of the random variable $\eta$ given $\xi$ see
e.g. in [1, Appendix, §5]. We note that the conditional distribution of $\eta$
given $\xi$ is symmetric if and only if the random vectors $(\xi,\eta)$ and
$(\xi,-\eta)$ are identically distributed.

According to the classical Skitovich–Darmois theorem the independence of two
linear forms of independent random variables is a characteristic property of
the Gaussian distribution on the real line. Another important characterization
result was proved by C.C. Heyde, where instead of the independence the
symmetry of the conditional distribution of one linear form given another is
considered ([19], see also [20, § 13.4.1]). For two independent random
variables the Heyde theorem can be formulated as follows.

######  Theorem A

Let $\xi_{1}$ and $\xi_{2}$ be independent real random variables with
distributions $\mu_{1}$ and $\mu_{2}$. Let $\alpha_{j},\beta_{j}$ be non zero
real numbers such that
$\alpha_{1}\beta_{1}^{-1}+\alpha_{2}\beta_{2}^{-1}\not=0$. If the conditional
distribution of the linear form $L_{2}=\beta_{1}\xi_{1}+\beta_{2}\xi_{2}$
given $L_{1}=\alpha_{1}\xi_{1}+\alpha_{2}\xi_{2}$ is symmetric, then $\mu_{j}$
are Gaussian distributions.

Some generalisations of the Heyde theorem, where independent random variables
take values in a locally compact Abelian group $X$, and coefficients of linear
forms are topological automorphisms of $X$, were studied in [4, 5, 6, 8, 9,
10, 11, 12, 13, 14, 21, 22, 23], see also [7, Chapter VI]. We continue this
research. The main result of the article is a complete description of compact
totally disconnected Abelian groups for which a natural analogue of the Heyde
theorem for two independent random variables holds true. It is well known that
Gaussian distributions on a locally compact totally disconnected Abelian group
$X$ are degenerated ([24, Chapter IV, §6, Remark 2]). The role of Gaussian
distributions on such groups is played by shifts of the Haar distributions of
compact subgroups of $X$. Thus, we will be interested in the description of
compact totally disconnected Abelian groups $X$ for which shifts of the Haar
distributions of compact subgroups of $X$ are characterized by the symmetry of
the conditional distribution of one linear form of two independent random
variables with values in $X$ given another. In §2 we give a complete
description of such groups. In §3 we prove a generalization of this result. In
§4 we prove that even a weak analogue of the Heyde theorem fails on compact
connected Abelian groups. The proofs are based on the study of solutions of a
functional equation on the character group of the group $X$ in the class of
Fourier transforms of probability distributions.

In the article we use standard results of abstract harmonic analysis, in
particular the duality theory for locally compact Abelian groups (see [17]).
Let $X$ be a second countable locally compact Abelian group. We consider only
such groups without mentioning it specifically. Denote by $Y$ the character
group of the group $X$, and by $(x,y)$ the value of a character $y\in Y$ at an
element $x\in X$. If $K$ is a closed subgroup of $X$, denote by
$A(Y,K)=\\{y\in Y:(x,y)=1$ for all  $x\in K\\}$ its annihilator. Let $X_{1}$
and $X_{2}$ be locally compact Abelian groups with character groups $Y_{1}$
and $Y_{2}$ respectively. Let $\alpha:X_{1}\rightarrow X_{2}$ be a continuous
homomorphism. The adjoint homomorphism $\tilde{\alpha}:Y_{2}\rightarrow Y_{1}$
is defined by the formula $(\alpha x_{1},y_{2})=(x_{1},\tilde{\alpha}y_{2})$
for all $x_{1}\in X_{1},\ y_{2}\in Y_{2}$. Denote by ${\rm Aut}(X)$ the group
of topological automorphisms of $X$ and by $I$ the identity automorphism of a
group.

Let $\\{G_{\iota}:\iota\in{\cal I}\\}$ be a nonempty set of compact Abelian
groups. Denote by $\mathop{\mbox{\rm\bf P}}\limits_{\iota\in{\cal
I}}G_{\iota}$ the direct product of the groups $G_{\iota}$ equipped with the
product topology. If $\\{H_{\iota}:\iota\in{\cal I}\\}$ is a nonempty set of
discrete Abelian groups, then denote by $\mathop{\mbox{\rm\bf
P}^{*}}\limits_{\iota\in{\cal I}}H_{\iota}$ the weak direct product of the
groups $H_{\iota}$ equipped with the discrete topology. We will always
consider a finite Abelian group in the discrete topology. Denote by
$\mathbb{Z}(n)=\\{0,1,\dots,n-1\\}$ the group of residue classes modulo $n$.
Let $p$ be a prime number. Denote by $\Delta_{p}$ the group of $p$-adic
integers. A $p$-group is defined to be a group whose elements have order equal
to a non-negative power of $p$. In particular, for power $0$, it is possible
that $X_{p}=\\{0\\}$. Denote by $\cal P$ the set of prime numbers.

Denote by ${\rm M}^{1}(X)$ the convolution semigroup of probability
distributions on the group $X$. Let $\mu\in{\rm M}^{1}(X)$. Denote by

$\hat{\mu}(y)=\int_{X}(x,y)d\mu(x),\ \ y\in Y,$

the characteristic function (Fourier transform) of the distribution $\mu$, and
by $\sigma(\mu)$ the support of $\mu$. Define the distribution
$\bar{\mu}\in{\rm M}^{1}(X)$ by the formula $\bar{\mu}(B)=\mu(-B)$ for any
Borel subset $B$ of $X$. Then $\hat{\bar{\mu}}(y)=\overline{\hat{\mu}(y)}$.

Denote by $m_{K}$ the Haar distribution of a compact subgroup $K$ of the group
$X$, and by $I(X)$ the set of shifts of Haar distributions $m_{K}$ of compact
subgroups $K$ of the group $X.$ We note that the characteristic function of a
distribution $m_{K}$ is of the form

$\hat{m}_{K}(y)=\begin{cases}1,&\text{\ if\ }\ y\in A(Y,K),\\\ 0,&\text{\ if\
}\ y\not\in A(Y,K).\end{cases}$ (1)

A distribution $\gamma\in{\rm M}^{1}(X)$ is called Gaussian ([24, Chapter
IV, §6], see also [2]) if its characteristic function is represented in the
form

$\hat{\gamma}(y)=(x,y)\exp\\{-\varphi(y)\\},$

where $x\in X$, and $\varphi(y)$ is a continuous nonnegative function on the
group $Y$ satisfying the equation

$\varphi(u+v)+\varphi(u-v)=2[\varphi(u)+\varphi(v)],\ \ u,v\in Y.$

Denote by $E_{x}$ the degenerate distribution concentrated at an element $x\in
X$. Observe that degenerate distributions are Gaussian. Denote by $\Gamma(X)$
the set of Gaussian distributions on $X$.

##  2  The Heyde theorem on compact totally disconnected Abelian groups

Let $\xi_{j}$, $j=1,2$, be independent random variables with values in a
locally compact Abelian group $X$ and distributions $\mu_{j}$, and let
$\alpha_{j},\beta_{j}\in{\rm Aut}(X)$. Assume that the conditional
distribution of the linear form $L_{2}=\beta_{1}\xi_{1}+\beta_{2}\xi_{2}$
given $L_{1}=\alpha_{1}\xi_{1}+\alpha_{2}\xi_{2}$ is symmetric. Obviously, if
we are interested in the description of $\mu_{j}$, then we can suppose without
loss of generality that $L_{1}=\xi_{1}+\xi_{2}$ and
$L_{2}=\xi_{1}+\alpha\xi_{2}$, where $\alpha\in{\rm Aut}(X)$.

Denote by $G$ the subgroup of $X$ generated by all elements of $X$ of order 2.
Then $\alpha(G)=G$ and $g=-g$ for all $g\in G$. Assume that $\xi_{1}$ and
$\xi_{2}$ take values in $G$. Taking into account that the conditional
distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given
$L_{1}=\xi_{1}+\xi_{2}$ is symmetric if and only if the random vectors
$(L_{1},L_{2})$ and $(L_{1},-L_{2})$ are identically distributed, we see that
for any $\mu_{j}$ and any topological automorphism $\alpha$ the conditional
distribution of $L_{2}$ given $L_{1}$ is symmetric.

Put $K={\rm Ker}(I+\alpha)$ and suppose that $K\neq\\{0\\}$. Then
$\alpha(k)=-k$ for all $k\in K$. Assume that $\xi_{1}$ and $\xi_{2}$ are
arbitrary independent identically distributed random variables with values in
$K$. Consider the linear form $L_{1}=\xi_{1}+\xi_{2}$ and
$L_{2}=\xi_{1}+\alpha\xi_{2}$. Then $L_{2}=\xi_{1}-\xi_{2}$, and the
characteristic functions of the random vectors $(L_{1},L_{2})$ and
$(L_{1},-L_{2})$ are equal. Hence the random vectors $(L_{1},L_{2})$ and
$(L_{1},-L_{2})$ are identically distributed. So, the conditional distribution
of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$
is symmetric.

Let $X$ be a compact totally disconnected Abelian group. Our goal is to
describe all groups $X$ for which shifts of the Haar distributions of compact
subgroups of $X$ are characterized by the symmetry of the conditional
distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given
$L_{1}=\xi_{1}+\xi_{2}$. From what has been said above it follows that we
should suppose that $X$ contains no elements of order 2, and condition

${\rm Ker}(I+\alpha)=\\{0\\}$ (2)

is satisfied.

The main result of this section is the following theorem.

######  Theorem 2.1

Let $X$ be a compact totally disconnected Abelian group containing no elements
of order $2$, and let $\alpha$ be a topological automorphism of the group $X$
satisfying condition $(\ref{1})$. Let $\xi_{1}$ and $\xi_{2}$ be independent
random variables with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$.
The symmetry of the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ implies that
$\mu_{j}\in I(X)$, $j=1,2$, if and only if the group $X$ is topologically
isomorphic to a group of the form

$\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ p>2}{X}_{p},$ (3)

where $X_{p}$ is a finite $p$-group.

Since shifts of the Haar distributions of compact subgroups of the group $X$
play the role of Gaussian distributions on $X$, Theorem 2.1 gives a complete
description of compact totally disconnected Abelian groups where an analogue
of the Heyde theorem for two independent random variables holds. We note that
an analogue of the Heyde theorem for two independent random variables is valid
on an arbitrary discrete Abelian group (see [12, Theorem 1]).

To prove Theorem 2.1 we need some lemmas.

######  Lemma 2.2 ([7, Lemma 16.1])

Let $X$ be a locally compact Abelian group, and let $\alpha$ be a topological
automorphism of the group $X$. Let $\xi_{1}$ and $\xi_{2}$ be independent
random variables with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$.
The conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric if and only if the characteristic
functions $\hat{\mu}_{j}(y)$ satisfy the equation

$\hat{\mu}_{1}(u+v)\hat{\mu}_{2}(u+\tilde{\alpha}v)=\hat{\mu}_{1}(u-v)\hat{\mu}_{2}(u-\tilde{\alpha}v),\
\ u,v\in Y.$ (4)

######  Lemma 2.3 ([4], see also [7, Corollary 17.2 and Remark 17.5])

Let $X$ be a finite Abelian group containing no elements of order $2$. Let
$\alpha$ be an automorphism of the group $X$ satisfying condition $(\ref{1})$.
Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$. If the conditional distribution of
the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then $\mu_{j}=m_{K}*E_{x_{j}}$, where $K$ is a subgroup of $X$,
$x_{j}\in X$, $j=1,2$. Moreover, $\alpha(K)=K$.

######  Lemma 2.4 ([10, Theorem 2])

Let $X=\Delta_{p}$ be the group of $p$-adic integers. Then there exist a
topological automorphism $\alpha$ of the group $X$ satisfying condition
$(\ref{1})$, and independent random variables $\xi_{1}$ and $\xi_{2}$ with
values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$ such that the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, while $\mu_{j}\notin I(X)$,
$j=1,2$.

The following lemma plays a key role in the proof of Theorem 2.1.

######  Lemma 2.5

Let $p$ be a prime number, and let $p>2$. Consider the group

$X=\mathop{\mbox{\rm\bf P}}\limits_{n=1}^{\infty}\mathbb{Z}(p^{k_{n}}),\ \
k_{n}\leq k_{n+1},\ \ n=1,2,\dots$ (5)

Then there exist a topological automorphism $\alpha$ of the group $X$
satisfying condition $(\ref{1})$, and independent identically distributed
random variables $\xi_{1}$ and $\xi_{2}$ with values in $X$ and distribution
$\mu$ such that the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
while $\mu\notin I(X)$.

Proof  The group $Y$ is topologically isomorphic to the weak direct product of
the groups $\mathbb{Z}(p^{k_{n}})$. In order not to complicate the notation we
assume that

$Y={\mathop{\mbox{\rm\bf P}^{*}}\limits_{n=1}^{\infty}}\mathbb{Z}(p^{k_{n}}).$
(6)

We denote by $t=\\{t_{n}\\}_{n=1}^{\infty}$, $t_{n}\in\mathbb{Z}(p^{k_{n}})$,
elements of the group $X$ and by $s=\\{s_{n}\\}_{n=1}^{\infty}$,
$s_{n}\in\mathbb{Z}(p^{k_{n}})$, elements of the group $Y$. Let
$s=\\{s_{n}\\}_{n=1}^{\infty}\in Y$. We say that $s\in\mathbb{Z}(p^{k_{1}})$
if $s_{2}=s_{3}=\dots=0$.

Let $i\leq j$. By $\alpha_{i,j}$ denote the homomorphism
$\alpha_{i,j}:\mathbb{Z}(p^{k_{i}})\rightarrow\mathbb{Z}(p^{k_{j}})$ of the
form

$\alpha_{i,j}t_{i}=p^{k_{j}-k_{i}}t_{i},\ \ t_{i}\in\mathbb{Z}(p^{k_{i}}).$

It is easy to verify that the adjoint homomorphism
$\tilde{\alpha}_{i,j}:\mathbb{Z}(p^{k_{j}})\rightarrow\mathbb{Z}(p^{k_{i}})$
is of the form

$\tilde{\alpha}_{i,j}s_{j}=s_{j}({\rm mod}\ p^{k_{i}}),\ \
s_{j}\in\mathbb{Z}(p^{k_{j}}).$

Define a continuous homomorphism $\alpha:X\rightarrow X$ by the formula
$\alpha\\{t_{n}\\}_{n=1}^{\infty}=\\{h_{n}\\}_{n=1}^{\infty}$, where

$h_{n}=\begin{cases}(p^{k_{1}}-t_{1})({\rm mod}\ p^{k_{1}}),&\text{\ if\ }\
n=1,\\\ (\alpha_{n-1,n}t_{n-1}+p^{k_{n}}-t_{n})({\rm mod}\ p^{k_{n}}),&\text{\
if\ }\ n\geq 2.\end{cases}$ (7)

It is easy to see that $\alpha$ is a topological automorphism of the group $X$
satisfying condition (2). We find from (7) that the adjoint automorphism
$\tilde{\alpha}:Y\rightarrow Y$ is of the form
$\tilde{\alpha}\\{s_{n}\\}_{n=1}^{\infty}=\\{u_{n}\\}_{n=1}^{\infty}$, where

$u_{n}=(\tilde{\alpha}_{n,n+1}s_{n+1}+p^{k_{n}}-s_{n})({\rm mod}\ p^{k_{n}}),\
\ n\geq 1.$ (8)

It follows from (5) and (6) that $X=\mathbb{Z}(p^{k_{1}})\times G$, where
$G=\mathop{\mbox{\rm\bf P}}\limits_{n=2}^{\infty}\mathbb{Z}(p^{k_{n}})$ and
$Y=\mathbb{Z}(p^{k_{1}})\times H$, where $H=\mathop{\mbox{\rm\bf
P}^{*}}\limits_{n=2}^{\infty}\mathbb{Z}(p^{k_{n}})$. Obviously,
$A(Y,G)=\mathbb{Z}(p^{k_{1}})$.

Take $0<a<1$. Consider on the group $X$ the distribution
$\mu=am_{X}+(1-a)m_{G}.$ It follows from (1) that the characteristic function
$\hat{\mu}(y)$ is of the form

$\hat{\mu}(y)=\begin{cases}1,&\text{\ if\ }\ y=0,\\\ 1-a,&\text{\ if\ }\
y\in{\mathbb{Z}}(p^{k_{1}})\backslash\\{0\\},\\\ 0,&\text{\ if\ }\
y\notin{\mathbb{Z}}(p^{k_{1}}).\end{cases}$ (9)

Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random
variables with values in $X$ and distribution $\mu$. Let us verify that the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric. By Lemma 2.2, it suffices to
verify that the characteristic function $\hat{\mu}(y)$ satisfies the equation

$\hat{\mu}(u+v)\hat{\mu}(u+\tilde{\alpha}v)=\hat{\mu}(u-v)\hat{\mu}(u-\tilde{\alpha}v),\
\ u,v\in Y.$ (10)

Let $u,v\in Y$. Consider 3 cases.

1\. $u,v\in{\mathbb{Z}}(p^{k_{1}})$. It follows from (8) that
$\tilde{\alpha}y=-y$ when $y\in{\mathbb{Z}}(p^{k_{1}})$. Hence, the
restriction of equation (10) to the subgroup ${\mathbb{Z}}(p^{k_{1}})$ takes
the form

$\hat{\mu}(u+v)\hat{\mu}(u-v)=\hat{\mu}(u-v)\hat{\mu}(u+v),\ \
u,v\in{\mathbb{Z}}(p^{k_{1}}).$

It is obvious that (10) is fulfilled.

2\. Either $u\in{\mathbb{Z}}(p^{k_{1}}),$ $v\notin{\mathbb{Z}}(p^{k_{1}})$ or
$u\notin{\mathbb{Z}}(p^{k_{1}}),$ $v\in{\mathbb{Z}}(p^{k_{1}}).$ Then $u\pm
v\notin{\mathbb{Z}}(p^{k_{1}})$. It follows from (9) that $\hat{\mu}(u\pm
v)=0$. Then both sides of equation (10) are equal to zero.

3\. $u,v\notin{\mathbb{Z}}(p^{k_{1}}).$ Assume that the left-hand side in (10)
is not equal to zero. Then (9) implies that
$u+v,u+\tilde{\alpha}v\in{\mathbb{Z}}(p^{k_{1}})$. It follows from this that

$(I-\tilde{\alpha})v\in{\mathbb{Z}}(p^{k_{1}}).$ (11)

Let $v=\\{v_{n}\\}_{n=1}^{\infty}$, $v_{n}\in\mathbb{Z}(p^{k_{n}})$. We find
from (8) and (11) that

$(2v_{n}+p^{k_{n}}-\tilde{\alpha}_{n,n+1}v_{n+1})({\rm mod}\ p^{k_{n}})=0,\
n\geq 2.$ (12)

It follows from $p>2$ that if $v_{n}\neq 0$, then $2v_{n}({\rm mod}\
p^{k_{n}})\neq 0$, $n\geq 1$. Assume that $v_{2}\neq 0$. Then we find from
(12) that $v_{n}\neq 0$ for each $n\geq 2$. But this contradicts the fact that
$v_{n}=0$ for all but a finite set of indices $n$. Hence, $v_{2}=0$. Reasoning
by induction we prove that $v_{3}=0$, then $v_{4}=0$, and so on. Thus, we got
that $v=\\{v_{n}\\}_{n=1}^{\infty}\in{\mathbb{Z}}(p^{k_{1}})$, contrary to
assumption. The obtained contradiction shows that the left-hand side of (10)
is equal to zero. Similarly, we prove that when
$u,v\notin{\mathbb{Z}}(p^{k_{1}})$ the right-hand side of (10) is also equal
to zero. So, both sides of equation (10) are equal to zero. $\blacksquare$

The following lemma follows directly from Lemma 13.24 in [7].

######  Lemma 2.6

Let $X$ be a compact totally disconnected Abelian group containing no elements
of order $2$. Then either $X$ is topologically isomorphic to a group of the
form $(\ref{n9})$ or for some prime number $p$ there exists a compact subgroup
$K$ of $X$ such that $K$ is a topological direct factor of the group $X$, and
$K$ is topologically isomorphic to either the group of $p$-adic integers
$\Delta_{p}$ or a group of the form $(\ref{n1})$.

Proof of Theorem 2.1 Necessity. Assume that a compact totally disconnected
Abelian group $X$ contains no elements of order $2$, and $X$ is not
topologically isomorphic to a group of the form $(\ref{n9})$. By Lemma 2.6,
for some prime number $p$, $p>2$, there exists a compact subgroup $K$ of the
group $X$ such that $K$ is a topological direct factor of $X$, and $K$ is
topologically isomorphic to either a group of $p$-adic integers $\Delta_{p}$
or a group of the form $(\ref{n1})$. We have $X=K\times G$. Denote elements of
the group $X$ by $x=(k,g)$, where $k\in K$, $g\in G$. Applying either Lemma
2.4 or Lemma 2.5 we find a topological automorphism $\alpha_{K}$ of the group
$K$ satisfying condition $(\ref{1})$, and independent random variables
$\xi_{1}$ and $\xi_{2}$ with values in $K$ and distributions $\mu_{1}$ and
$\mu_{2}$ such that the conditional distribution of the linear form
$M_{2}=\xi_{1}+\alpha_{K}\xi_{2}$ given $M_{1}=\xi_{1}+\xi_{2}$ is symmetric,
whereas $\mu_{j}\notin I(K)$, $j=1,2$. Put $\alpha(k,g)=(\alpha_{K}k,g)$.
Since the group $X$ contains no elements of order 2, $\alpha$ is a topological
automorphism of $X$ satisfying condition $(\ref{1})$. Obviously, if we
consider $\xi_{j}$ as independent random variables with values in the group
$X$, then the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
whereas $\mu_{j}\notin I(X)$, $j=1,2$. Necessity is proved.

Sufficiency. Assume that a compact totally disconnected Abelian group $X$ is
topologically isomorphic to a group of the form (3), and let $\alpha$ be a
topological automorphism of the group $X$ satisfying condition $(\ref{1})$.
Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$. By Lemma 2.2, the symmetry of the
conditional distribution of the linear form $L_{2}$ given $L_{1}$ implies that
the characteristic functions $\hat{\mu}_{j}(y)$ satisfy equation (4). Put
$\nu_{j}=\mu_{j}*\bar{\mu}_{j}$. Then
$\hat{\nu}_{j}(y)=|\hat{\mu}_{j}(y)|^{2}\geq 0$ for all $y\in Y$, and the
characteristic functions $\hat{\nu}_{j}(y)$ also satisfy equation (4).

It follows from (3) that

$Y=\mathop{\mbox{\rm\bf P}^{*}}\limits_{p\in{\cal P},\ p>2}{Y}_{p},$ (13)

where the group $Y_{p}$ is isomorphic to the character group of the group
$X_{p}$. Put

$A_{m}=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ 2<p\leq m}{X}_{p},\ \
B_{m}=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ 2<p\leq m}{Y}_{p},\ \
m=3,4,\dots$ (14)

Then $A_{m}$ and $B_{m}$ are finite Abelian groups, $B_{m}$ is isomorphic to
the character group of the group $A_{m}$, $B_{m}\subset B_{m+1}$ and
$Y=\bigcup\limits_{m=3}^{\infty}B_{m}$. Denote by $\alpha_{m}$ the restriction
of $\alpha$ to the subgroup $A_{m}$. It is obvious that $\alpha_{m}\in{\rm
Aut}(A_{m})$ and $\alpha_{m}$ satisfies the condition $(\ref{1})$. Note that
the restriction of $\tilde{\alpha}$ to subgroup $B_{m}$ coincides with
$\tilde{\alpha}_{m}$. Consider the restriction of equation (4) for the
characteristic functions $\hat{\nu}_{j}(y)$ to the subgroup $B_{m}$. We have

$\hat{\nu}_{1}(u+v)\hat{\nu}_{2}(u+\tilde{\alpha}_{m}v)=\hat{\nu}_{1}(u-v)\hat{\nu}_{2}(u-\tilde{\alpha}_{m}v),\
\ u,v\in B_{m}.$ (15)

Taking into account Lemma 2.2 and applying Lemma 2.3 to the finite group
$A_{m}$, we obtain from (15) that $\hat{\nu}_{1}(y)=\hat{\nu}_{2}(y)$ for all
$y\in B_{m}$, and the characteristic functions $\hat{\nu}_{j}(y)$ take only
values 0 and 1 on $B_{m}$. This implies that
$\hat{\nu}_{1}(y)=\hat{\nu}_{2}(y)$ for all $y\in Y$, and the characteristic
functions $\hat{\nu}_{j}(y)$ take only values 0 and 1 on $Y$. Put $E=\\{y\in
Y:\hat{\nu}_{j}(y)=1\\}$. Then $E$ is a subgroup of $Y$. Put $K=A(X,E)$. Since
$E=A(Y,K)$, it follows from (1) that $\nu_{1}=\nu_{2}=m_{K}$. This easily
implies that $\mu_{j}=m_{K}*E_{x_{j}}$, where $x_{j}\in X$, $j=1,2$.
Sufficiency is proved.

In fact, we have proved somewhat more, namely $\mu_{j}$ are shifts of the same
Haar distribution of a compact subgroup $K$ of the group $X$. Furthermore, we
can assert that $\alpha(K)=K$. Indeed, put $E_{m}=E\cap B_{m}$. It follows
from Lemma 2.3 that $\tilde{\alpha}_{m}(E_{m})=E_{m}$. Hence,
$\tilde{\alpha}(E)=E$, so that $\alpha(K)=K$. $\blacksquare$

######  Remark 2.7

Let us compare Theorem 2.1 with the Skitovich–Darmois theorem for compact
totally disconnected Abelian groups, which has the following statement. It can
be deduced from Theorem 1 in [15], see also [7, Theorem 13.25].

######  Theorem

Let $X$ be a compact totally disconnected Abelian group containing no elements
of order $2$, and let $\alpha$ be a topological automorphism of the group $X$.
Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$. The independence of the linear
forms $L_{1}=\xi_{1}+\xi_{2}$ and $L_{2}=\xi_{1}+\alpha\xi_{2}$ implies that
$\mu_{j}\in I(X)$, $j=1,2$, if and only if the group $X$ is topologically
isomorphic to a group of the form

$\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal
P}}(\Delta_{p}^{n_{p}}\times{X}_{p}),$

where $n_{p}$ is a nonnegative integer, and $X_{p}$ is a finite $p$-group,
$X_{2}=\\{0\\}$.

We see that the class of compact totally disconnected Abelian groups, for
which the Skitovich–Darmois theorem is true, is wider than the class of
compact totally disconnected Abelian groups for which the Heyde theorem is
valid.

At the same time, we note that both the Skitovich–Darmois theorem and the
Heyde theorem are valid for discrete Abelian groups containing no elements of
order 2.

We supplement Theorem 2.1 with the following statements.

######  Proposition 2.8

Let $X$ be a compact totally disconnected Abelian group of the form
$(\ref{n9})$, let $K$ be a compact subgroup of $X$, and let $\alpha$ be a
topological automorphism of the group $X$ satisfying condition $(\ref{1})$.
Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random
variables with values in $X$ and distribution $m_{K}$. Then the following
assertions are equivalent:

$(i)$ the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric;

$(ii)$ $(I-\alpha)(K)=K$.

Proof  $(i)\Rightarrow(ii)$. Note that every closed subgroup of a group of the
form $(\ref{n9})$ is topologically isomorphic to a group of the form
$(\ref{n9})$. By Theorem 2.1, $\alpha(K)=K$, i.e. the restriction of $\alpha$
to the subgroup $K$ is a topological automorphism of $K$. Thus, we can prove
the statement assuming that $K=X$. We note that $(I-\alpha)(X)=X$ if and only
if ${\rm Ker}(I-\tilde{\alpha})=\\{0\\}$. Denote by $\tilde{\alpha}_{m}$ the
same topological automorphism as in the proof of sufficiency in Theorem 2.1.
We have ${\rm Ker}(I-\tilde{\alpha})=\\{0\\}$ if and only if ${\rm
Ker}(I-\tilde{\alpha}_{m})=\\{0\\}$ for $m=3,4,\dots$. Taking into account
Lemma 2.2, the statement ${\rm Ker}(I-\tilde{\alpha}_{m})=\\{0\\}$ follows
from the fact that the proposition is valid in the case when $X$ is a finite
Abelian group containing no elements of order $2$ ([12, Proposition 1]).

$(ii)\Rightarrow(i)$. This statement holds for an arbitrary locally compact
Abelian group $X$ and a compact subgroup $K$ of $X$ ([12, Proposition 1]).
$\blacksquare$

######  Proposition 2.9

Let $X$ be a compact totally disconnected Abelian group of the form

$X=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P}}{X}_{p},$

where $X_{p}$ is a finite $p$-group. Put

$G=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ p>2}{X}_{p}.$

Let $\alpha$ be a topological automorphism of the group $X$ satisfying
condition $(\ref{1})$. Let $\xi_{1}$ and $\xi_{2}$ be independent random
variables with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$. If the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then
$\mu_{j}=\rho_{j}*m_{K}*E_{g_{j}}$, where $\rho_{j}$ are some distributions on
$X_{2}$, $K$ is a compact subgroup of $G$, $g_{j}\in G$, $j=1,2$.

To prove Proposition 2.9 we need the following lemma which is a special case
of Theorem 3 proved in [12].

######  Lemma 2.10

Let $X$ be a finite Abelian group, let $X_{2}$ be its $2$-component, and let
$G$ be the subgroup of $X$ generated by all elements of odd order. Let
$\alpha$ be an automorphism of the group $X$ satisfying condition $(\ref{1})$.
Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$. If the conditional distribution of
the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then $\mu_{j}=\rho_{j}*m_{K}*E_{g_{j}}$, where
$\sigma(\rho_{j})\subset X_{2}$, $K$ is a subgroup of $G$, $g_{j}\in G$,
$j=1,2$.

Proof of Proposition 2.9 We argue as in the proof of sufficiency in Theorem
2.1. We have $X=X_{2}\times G$. Then $Y=L\times H$, where $L$ and $H$ are
groups topologically isomorphic to the character groups of the groups $X_{2}$
and $G$ respectively. Denote by $y=(l,h)$, where $l\in L$, $h\in H$, elements
of the group $Y$. Taking into account Lemma 2.2, Theorem 2.1 and the fact that
the restriction of $\alpha$ to the subgroup $G$ is a topological automorphism
of $G$ satisfying condition (2), it is easy to verify that Proposition 2.9
will be proved if we prove that there exist functions $a_{j}(l)$, $l\in L$,
and $b_{j}(h)$, $h\in H$, such that

$\hat{\mu}_{j}(l,h)=a_{j}(l)b_{j}(h),\ \ l\in L,\ \ h\in H,\ \ j=1,2.$

This implies that $a_{j}(l)$ are the characteristic functions of some
distributions $\rho_{j}$ on $X_{2}$ and $b_{j}(h)$ are the characteristic
functions of some shifts of a distribution $m_{K}$, where $K$ is a subgroup of
$G$.

We have $Y=\mathop{\mbox{\rm\bf P}^{*}}\limits_{p\in{\cal P}}{Y}_{p}$, where
the group $Y_{p}$ is isomorphic to the character group of the group $X_{p}$.
Define the groups $A_{m}$ and $B_{m}$ by (14). Then $A_{m}$ and $B_{m}$ are
finite Abelian groups, the group $L\times B_{m}$ is isomorphic to the
character group of the group $X_{2}\times A_{m}$, $B_{m}\subset B_{m+1}$ and
$Y=L\times\bigcup\limits_{m=3}^{\infty}B_{m}$. Denote by $\alpha_{m}$ the
restriction of $\alpha$ to the subgroup $X_{2}\times A_{m}$. It is obvious
that $\alpha_{m}\in{\rm Aut}(X_{2}\times A_{m})$ and $\alpha_{m}$ satisfies
condition $(\ref{1})$. Note that the restriction of $\tilde{\alpha}$ to the
subgroup $L\times B_{m}$ coincides with $\tilde{\alpha}_{m}$. By Lemma 2.2,
the symmetry of the conditional distribution of the linear form $L_{2}$ given
$L_{1}$ implies that the characteristic functions $\hat{\mu}_{j}(l,h)$ satisfy
equation (4). Consider the restriction of equation (4) to the subgroup
$L\times B_{m}$. We have

$\hat{\mu}_{1}(u+v)\hat{\mu}_{2}(u+\tilde{\alpha}_{m}v)=\hat{\mu}_{1}(u-v)\hat{\mu}_{2}(u-\tilde{\alpha}_{m}v),\
\ u,v\in L\times B_{m}.$ (16)

Taking into account Lemma 2.2 and applying Lemma 2.10 to the finite group
$X_{2}\times A_{m}$, we obtain from (16) that there exist functions
$a_{j}(l)$, $l\in L$, and $b_{j,m}(h)$, $h\in B_{m}$ such that
$\hat{\mu}_{j}(l,h)=a_{j}(l)b_{j,m}(h)$ for all $(l,h)\in L\times B_{m}$.
Moreover, $b_{j,m+1}(h)=b_{j,m}(h)$ for all $h\in B_{m}$. Put
$b_{j}(h)=b_{j,m}(h)$ for all $h\in B_{m}$. Then
$\hat{\mu}_{j}(l,h)=a_{j}(l)b_{j}(h)$ for all $(l,h)\in Y$, $j=1,2$.
$\blacksquare$

##  3  A generalization of Theorem 2.1

The following theorem has been proved in [12, Theorem 2].

######  Theorem B

Let $X={\mathbb{R}}^{n}\times D$, where $n\geq 0,$ and $D$ is a discrete
Abelian group containing no elements of order $2$. Let $\alpha$ be a
topological automorphism of the group $X$ satisfying condition $(\ref{1})$.
Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$. If the conditional distribution of
the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then $\mu_{j}=\gamma_{j}*m_{K}*E_{x_{j}}$, where
$\gamma_{j}\in\Gamma({\mathbb{R}}^{n})$, $K$ is a finite subgroup of $D$,
$x_{j}\in D$, $j=1,2$. Moreover, $\alpha(K)=K$.

In this section we shall prove a theorem which generalizes both Theorem 2.1
and Theorem B and describe a wide class of locally compact Abelian groups
where an analogue of the Heyde theorem holds.

######  Theorem 3.1

Let

$X={\mathbb{R}}^{n}\times D\times G,$ (17)

where $n\geq 0,$ $D$ is a discrete Abelian group containing no elements of
order $2$, and $G$ is a compact totally disconnected Abelian group of the form
$(\ref{n9})$. Let $\alpha$ be a topological automorphism of the group $X$
satisfying condition $(\ref{1})$. Let $\xi_{1}$ and $\xi_{2}$ be independent
random variables with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$.
If the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then
$\mu_{j}=\gamma_{j}*m_{K}*E_{x_{j}}$, where
$\gamma_{j}\in\Gamma({\mathbb{R}}^{n})$, $K$ is a compact subgroup of $X$,
$x_{j}\in X$, $j=1,2$. Moreover, $\alpha(K)=K$.

To prove Theorem 3.1 we need two lemmas. One of them is the following well-
known statement (see e.g. [7, Proposition 2.13]).

######  Lemma 3.2

Let $X$ be a locally compact Abelian group, and let $\mu\in{\rm M}^{1}(X)$.
Then the set $E=\\{y\in Y:\ \hat{\mu}(y)=1\\}$ is a closed subgroup of the
group $Y$, and $\sigma(\mu)\subset A(X,E)$.

The following lemma generalizes Theorem 2.1 and plays a key role in the proof
of Theorem 3.1.

######  Lemma 3.3

Let $X={\mathbb{R}}^{n}\times G$, where $n\geq 0,$ and $G$ is a compact
totally disconnected Abelian group of the form $(\ref{n9})$. Let $\alpha$ be a
topological automorphism of the group $X$ satisfying condition $(\ref{1})$.
Let $\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$
and distributions $\mu_{1}$ and $\mu_{2}$. If the conditional distribution of
the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then $\mu_{j}=\gamma_{j}*m_{K}*E_{x_{j}}$, where
$\gamma_{j}\in\Gamma({\mathbb{R}}^{n})$, $K$ is a compact subgroup of $G$,
$x_{j}\in G$, $j=1,2$. Moreover, $\alpha(K)=K$.

Proof  Denote elements of the group $X$ by $x=(t,g)$, where
$t\in{\mathbb{R}}^{n}$, $g\in G$. We have $Y={\mathbb{R}}^{n}\times H$, where
$H$ is a group topologically isomorphic to the character group of the group
$G$. Denote elements of the group $Y$ by $y=(s,h)$, where
$s\in{\mathbb{R}}^{n}$, $h\in H$. Since ${\mathbb{R}}^{n}$ is the connected
component of zero of the group $X$, we have
$\alpha({\mathbb{R}}^{n})={\mathbb{R}}^{n}$. Since $G$ is the subgroup of $X$
consisting of all compact elements of the group $X$, we have $\alpha(G)=G$.
This implies that the restriction of $\alpha$ to each of the subgroups
${\mathbb{R}}^{n}$ and $G$ is a topological automorphism of the corresponding
subgroup. Denote by $\alpha_{{\mathbb{R}}^{n}}$ and $\alpha_{G}$ these
restrictions respectively. This means that $\alpha$ can be written in the form
$\alpha(t,g)=(\alpha_{{\mathbb{R}}^{n}}t,\alpha_{G}g)$, $(t,g)\in X$. Note
that the restrictions of $\tilde{\alpha}$ to the subgroups ${\mathbb{R}}^{n}$
and $H$ coincide with $\tilde{\alpha}_{{\mathbb{R}}^{n}}$ and
$\tilde{\alpha}_{G}$ respectively. Hence, $\tilde{\alpha}$ can be written in
the form
$\tilde{\alpha}(s,h)=(\tilde{\alpha}_{{\mathbb{R}}^{n}}s,\tilde{\alpha}_{G}h)$,
$(s,h)\in Y$.

By Lemma 2.2, the symmetry of the conditional distribution of the linear form
$L_{2}$ given $L_{1}$ implies that the characteristic functions
$\hat{\mu}_{j}(s,h)$ satisfy equation (4) which takes the form

$\hat{\mu}_{1}(s_{1}+s_{2},h_{1}+h_{2})\hat{\mu}_{2}(s_{1}+\tilde{\alpha}_{{\mathbb{R}}^{n}}s_{2},h_{1}+\tilde{\alpha}_{G}h_{2})$
$=\hat{\mu}_{1}(s_{1}-s_{2},h_{1}-h_{2})\hat{\mu}_{2}(s_{1}-\tilde{\alpha}_{{\mathbb{R}}^{n}}s_{2},h_{1}-\tilde{\alpha}_{G}h_{2}),\
\ s_{j}\in{\mathbb{R}}^{n},\ \ h_{j}\in H.$ (18)

It is obvious that the topological automorphism $\alpha_{{\mathbb{R}}^{n}}$
satisfies condition (2). Substituting $h_{1}=h_{2}=0$ in (3), taking into
account Lemma 2.2 and Theorem B for the group $X={\mathbb{R}}^{n}$, we obtain
from the resulting equation that

$\hat{\mu}_{j}(s,0)=\exp\\{i\langle t_{j},s\rangle-\langle
A_{j}s,s\rangle\\},\ \ s\in\mathbb{R}^{n},\ \ j=1,2,$ (19)

where $t_{j}\in\mathbb{R}^{n}$, $\langle.,.\rangle$ is the scalar product, and
$A_{j}$ are symmetric positive semi-definite $n\times n$-matrices. Similarly,
substituting $s_{1}=s_{2}=0$ in (3), taking into account Lemma 2.2 and
applying Theorem 2.1 to the group $X=G$, we obtain from the resulting equation
that

$\hat{\mu}_{j}(0,h)=\hat{m}_{K}(h)(g_{j},h),\ \ h\in H,\ \ j=1,2,$ (20)

where $K$ is a compact subgroup in $G$, and $g_{j}\in G$. Moreover,
$\alpha_{G}(K)=K$.

Substituting (20) into (3) and taking into account (1), we find that
$2(g_{1}+\alpha_{G}g_{2})\in K$. Since $G$ is a group of the form $(\ref{n9})$
and $K$ is a compact subgroup of $G$, multiplication by 2 is a topological
automorphism of $K$. Hence,

$g_{1}+\alpha_{G}g_{2}\in K.$ (21)

Put $x_{1}=-\alpha_{G}g_{2}$, $x_{2}=g_{2}$. Then $x_{1}+\alpha_{G}x_{2}=0$.
It follows from (1), (20), and (21) that

$\hat{\mu}_{j}(0,h)=\hat{m}_{K}(h)(x_{j},h),\ \ h\in H,\ \ j=1,2.$ (22)

Consider new independent random variables $\eta_{j}=\xi_{j}-x_{j}$, and denote
by $\lambda_{j}$ the distribution of the random variable $\eta_{j}$, $j=1,2$.
Since

$\lambda_{j}=\mu_{j}*E_{-x_{j}},\ \ j=1,2,$ (23)

and $x_{1}+\alpha x_{2}=0$, the characteristic functions
$\hat{\lambda}_{j}(y)$ also satisfy equation (4). By Lemma 2.2, the
conditional distribution of the linear form $N_{2}=\eta_{1}+\alpha\eta_{2}$
given $N_{1}=\eta_{1}+\eta_{2}$ is symmetric. It follows from (1), (22), and
(23) that $\hat{\lambda}_{j}(0,h)=1$ for all $h\in A(H,K)$. By Lemma 3.2, this
implies that $\sigma(\lambda_{j})\subset A(X,A(H,K))={\mathbb{R}}^{n}\times
K$. Since $\alpha({\mathbb{R}}^{n}\times K)={\mathbb{R}}^{n}\times K$, we can
assume that the independent random variables $\eta_{j}$ take values in the
group $X={\mathbb{R}}^{n}\times K$. Then $Y={\mathbb{R}}^{n}\times L$, where
$L$ is a group topologically isomorphic to the character group of the group
$K$. Denote elements of the group $Y$ by $y=(s,l)$, where
$s\in{\mathbb{R}}^{n}$, $l\in L$. In so doing,

$\hat{\lambda}_{j}(0,l)=\begin{cases}1,&\text{\ if\ }\ l=0,\\\ 0,&\text{\ if\
}\ l\neq 0,\ \ j=1,2.\end{cases}$ (24)

Denote by $\alpha_{K}$ the restriction of $\alpha$ to the subgroup $K$ and
consider equation (4) for the characteristic functions
$\hat{\lambda}_{j}(s,l)$ on the group $Y={\mathbb{R}}^{n}\times L$. We have

$\hat{\lambda}_{1}(s_{1}+s_{2},l_{1}+l_{2})\hat{\lambda}_{2}(s_{1}+\tilde{\alpha}_{{\mathbb{R}}^{n}}s_{2},l_{1}+\tilde{\alpha}_{K}l_{2})$
$=\hat{\lambda}_{1}(s_{1}-s_{2},l_{1}-l_{2})\hat{\lambda}_{2}((s_{1}-\tilde{\alpha}_{{\mathbb{R}}^{n}}s_{2},l_{1}-\tilde{\alpha}_{K}l_{2}),\
\ s_{j}\in{\mathbb{R}}^{n},\ \ l_{j}\in L.$ (25)

Substitute $s_{1}=s_{2}=s$, $l_{1}=l$, $l_{2}=-l$ in (3). We obtain

$\hat{\lambda}_{1}(2s,0)\hat{\lambda}_{2}((I+\tilde{\alpha}_{{\mathbb{R}}^{n}})s,(I-\tilde{\alpha}_{K})l)=\hat{\lambda}_{1}(0,2l)\hat{\lambda}_{2}((I-\tilde{\alpha}_{{\mathbb{R}}^{n}})s,(I+\tilde{\alpha}_{K})l),\
\ s\in{\mathbb{R}}^{n},\ \ l\in L.$ (26)

Assume that $l\neq 0$. Then $2l\neq 0$, and (24) implies that
$\hat{\lambda}_{1}(0,2l)=0$. Hence, it follows from (26) that
$\hat{\lambda}_{1}(2s,0)\hat{\lambda}_{2}((I+\tilde{\alpha}_{{\mathbb{R}}^{n}})s,(I-\tilde{\alpha}_{K})l)=0$.
Taking into account (19), this implies that

$\hat{\lambda}_{2}((I+\tilde{\alpha}_{{\mathbb{R}}^{n}})s,(I-\tilde{\alpha}_{K})l)=0.$

Since $\alpha_{{\mathbb{R}}^{n}}$ satisfies condition (2) and
$\alpha({\mathbb{R}}^{n})={\mathbb{R}}^{n}$, we have
$I+\alpha_{{\mathbb{R}}^{n}}\in{\rm Aut}({{\mathbb{R}}^{n}})$. Hence,
$I+\tilde{\alpha}_{{\mathbb{R}}^{n}}\in{\rm Aut}({{\mathbb{R}}^{n}})$ and

$\hat{\lambda}_{2}(s,(I-\tilde{\alpha}_{K})l)=0,\ \ s\in{{\mathbb{R}}^{n}},\ \
l\in L,\ \ l\neq 0.$ (27)

In particular,

$\hat{\lambda}_{2}(0,(I-\tilde{\alpha}_{K})l)=0,\ \ l\in L,\ \ l\neq 0.$ (28)

It follows from (24) and (28) that

${\rm Ker}(I-\tilde{\alpha}_{K})=\\{0\\}.$ (29)

Since $K$ is a compact subgroup of the group $G$, the group $K$ is
topologically isomorphic to a compact totally disconnected Abelian group of
the form $(\ref{n9})$. This implies that the group $L$ is topologically
isomorphic to a group of the form (13). Then, it follows from (29) that
$I-\tilde{\alpha}_{K}\in{\rm Aut}(L)$. Hence, we find from (27) that

$\hat{\lambda}_{2}(s,l)=0,\ \ s\in{{\mathbb{R}}^{n}},\ \ l\in L,\ \ l\neq 0.$
(30)

Similarly, we obtain

$\hat{\lambda}_{1}(s,l)=0,\ s\in{{\mathbb{R}}^{n}},\ \ l\in L,\ \ l\neq 0.$
(31)

It follows from (19), (30), and (31) that

$\hat{\lambda}_{j}(s,l)=\begin{cases}\exp\\{i\langle t_{j},s\rangle-\langle
A_{j}s,s\rangle\\},&\text{\ if\ }\ l=0,\\\ 0,&\text{\ if\ }\ l\neq 0,\ \
j=1,2.\end{cases}$ (32)

Taking into account (32), and returning from the distributions $\lambda_{j}$
to the distributions $\mu_{j}$, we prove the statement of the lemma.
$\blacksquare$

Proof of Theorem 3.1 We have $Y={\mathbb{R}}^{n}\times L\times H$, where $L$
and $H$ are groups topologically isomorphic to the character groups of the
groups $D$ and $G$ respectively. Let

$G=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ p>2}{G}_{p},$

where $G_{p}$ is a finite $p$-group. Then

$H=\mathop{\mbox{\rm\bf P}^{*}}\limits_{p\in{\cal P},\ p>2}{H}_{p},$

where $H_{p}$ is a group topologically isomorphic to the character group of
the group $G_{p}$. Put

$A_{m}=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ 2<p\leq m}{G}_{p},\ \
B_{m}=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ 2<p\leq m}{H}_{p},\ \
C_{m}=\mathop{\mbox{\rm\bf P}}\limits_{p\in{\cal P},\ p>m}{G}_{p},\ \
m=3,4,\dots$

Since ${\mathbb{R}}^{n}$ is the connected component of zero of the group $X$,
we have $\alpha({\mathbb{R}}^{n})={\mathbb{R}}^{n}$. It follows from the
definition of the product topology that $\alpha(C_{m})\subset G$ for some $m$.
Taking into account that $C_{m}$ is the only subgroup of $G$ which is
topologically isomorphic to $C_{m}$, we have $\alpha(C_{m})=C_{m}$, and hence
$\alpha({\mathbb{R}}^{n}\times C_{m})={\mathbb{R}}^{n}\times C_{m}$.

Represent the group $X$ in the form $X={\mathbb{R}}^{n}\times D\times
A_{m}\times C_{m}$. Consider the factor-group $X/({\mathbb{R}}^{n}\times
C_{m})$. Denote by $[x]$ its elements. Note that the factor-group
$X/({\mathbb{R}}^{n}\times C_{m})$ is topologically isomorphic to the group
$D\times A_{m}$. Since $\alpha({\mathbb{R}}^{n}\times
C_{m})={\mathbb{R}}^{n}\times C_{m}$, the topological automorphism $\alpha$
induces a topological automorphism $[\alpha]$ on the factor-group
$X/({\mathbb{R}}^{n}\times C_{m})$ by the formula $[\alpha][x]=[\alpha x]$. It
is easy to see that any continuous monomorphism of the group
${\mathbb{R}}^{n}\times C_{m}$ to itself is a topological automorphism. This
easily implies that

${\rm Ker}(I+[\alpha])=\\{0\\}.$ (33)

The character group of the factor-group $X/({\mathbb{R}}^{n}\times C_{m})$ is
topologically isomorphic to the annihilator $A(Y,{\mathbb{R}}^{n}\times
C_{m})=L\times B_{m}$. Since $\alpha({\mathbb{R}}^{n}\times
C_{m})={\mathbb{R}}^{n}\times C_{m}$, we have $\tilde{\alpha}(L\times
B_{m})=(L\times B_{m})$. Note also that the restriction of $\tilde{\alpha}$ to
the subgroup $L\times B_{m}$ is the adjoint automorphism to $[\alpha]$.

Let $\eta_{1}$ and $\eta_{2}$ be independent random variables with values in
the factor-group $X/({\mathbb{R}}^{n}\times C_{m})$ such that the
characteristic functions of their distributions coincide with the restrictions
to the subgroup $L\times B_{m}$ of the characteristic functions
$\hat{\mu}_{j}(y)$. By Lemma 2.2, the conditional distribution of the linear
form $M_{2}=\eta_{1}+[\alpha]\eta_{2}$ given $M_{1}=\eta_{1}+\eta_{2}$ is
symmetric. Obviously, the factor-group $X/({\mathbb{R}}^{n}\times C_{m})$ is
discrete and contains no elements of order 2. Hence, taking into account (33)
and applying Theorem B to the discrete group $X/({\mathbb{R}}^{n}\times
C_{m})$, we obtain that the restrictions of the characteristic functions
$\hat{\mu}_{j}(y)$ to the subgroup $L\times B_{m}$ are of the form

$\hat{\mu}_{j}(y)=\begin{cases}(g_{j},y),&\text{\ if\ }\ y\in E,\\\ 0,&\text{\
if\ }\ y\notin E,\ \ j=1,2,\end{cases}$

where $g_{j}\in D\times A_{m}$, $E$ is an open subgroup in $L\times B_{m}$,
which is the annihilator of a finite subgroup in the group
$X/({\mathbb{R}}^{n}\times C_{m})$. Moreover,

$\tilde{\alpha}(E)=E.$ (34)

Using the same reasoning as in the proof of Lemma 3.3, we can replace the
independent random variables $\xi_{j}$ by $\eta_{j}=\xi_{j}-x_{j}$ with
distributions $\lambda_{j}$ in such a way that the conditional distribution of
the linear form $N_{2}=\eta_{1}+\alpha\eta_{2}$ given
$N_{1}=\eta_{1}+\eta_{2}$ is symmetric, and the restrictions of the
characteristic functions $\hat{\lambda}_{j}(y)$ to the subgroup $L\times
B_{m}$ are of the form

$\hat{\lambda}_{j}(y)=\begin{cases}1,&\text{\ if\ }\ y\in E,\\\ 0,&\text{\ if\
}\ y\notin E,\ \ j=1,2.\end{cases}$ (35)

By Lemma 3.2, (35) implies that $\sigma(\lambda_{j})\subset A(X,E)$. It is
easy to see that $A(X,E)={\mathbb{R}}^{n}\times F$, where $F$ is a compact
totally disconnected Abelian group of the form $(\ref{n9})$. It follows from
(34) that $\alpha({\mathbb{R}}^{n}\times F)={\mathbb{R}}^{n}\times F$.
Applying Lemma 3.3 to the group ${\mathbb{R}}^{n}\times F$, we get the
required representation for the distributions $\lambda_{j}$, and hence, for
the distributions $\mu_{j}$ too. $\blacksquare$

An analogue of the Heyde theorem is valid not only for locally compact Abelian
groups of the form (17). The following statement holds true (compare with [16,
§3]).

######  Proposition 3.4

There exists a locally compact Abelian group $X$ containing no elements of
order $2$ such that $X$ is not topologically isomorphic to any group of the
form $(\ref{09.09.15.1})$, and the following statement holds. If $\alpha$ is a
topological automorphism of the group $X$ satisfying condition $(\ref{1})$,
and $\xi_{1}$ and $\xi_{2}$ are independent random variables with values in
$X$ and distributions $\mu_{1}$ and $\mu_{2}$, then the symmetry of the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ implies that $\mu_{j}=m_{K}*E_{x_{j}}$, where
$K$ is a compact subgroup of $X$, $x_{j}\in X$, $j=1,2$. Moreover,
$\alpha(K)=K$.

Proof Denote by $p_{n}$ the $n$th prime number. Put $T=\mathop{\mbox{\rm\bf
P}}\limits_{n=2}^{\infty}{\mathbb{Z}}(p_{n}^{2})$ and consider $T$ only as
"algebraic" group. We do not consider $T$ as a topological group equipped with
the product topology.

Consider a subgroup $K_{n}=\\{0,p_{n},2p_{n},\dots,(p_{n}-1)p_{n}\\}$ of
${\mathbb{Z}}(p^{2}_{n})$. It is obvious that $K_{n}$ is isomorphic to
${\mathbb{Z}}(p_{n})$. Denote by $Y$ a subgroup of $T$ consisting of all
elements $y=(y_{2},\dots,y_{n},\dots)$, $y_{n}\in{\mathbb{Z}}(p^{2}_{n})$,
such that $y_{n}\notin K_{n}$ only for a finite number indexes $n$. Put

$F=\mathop{\mbox{\rm\bf P}}\limits_{n=2}^{\infty}K_{n}.$

Then $F$ is a subgroup of $Y$. Consider the group $F$ in the product topology.
Obviously, $F$ is a compact Abelian group. Define a topology on $Y$ as
follows. A subset $U$ in $Y$ is called open if for each $y\in U$ there is a
neighborhood of zero $V_{y}$ in $F$ such that $y+V_{y}\subset U$. Then $Y$ is
a second countable locally compact Abelian group, and $F$ is an open subgroup
in $Y$. It is obvious that $Y$ is a non-discrete, non-compact and totally
disconnected group. Denote by $X$ the character group of the group $Y$.

Since multiplication by 2 is a topological isomorphism of the group $Y$, the
group $X$ contains no elements of order 2. To see that the group $X$ is not
topologically isomorphic to a group of the form (17), suppose the contrary. It
is obvious that then in (17) $n=0$, and then the group $Y$ can be represented
in the form $Y=L\times H$, where the group $L$ is compact, and the group $H$
is discrete. It is easy to see that there exists $m$, such that
$\mathop{\mbox{\rm\bf P}}\limits_{n=m}^{\infty}K_{n}\subset L$. This implies
that the group $H$ contains only a finite number different of subgroups
$K_{n}$, and hence, $H$ is a finite group. It follows from this that $Y$ is a
compact group. The obtained contradiction shows that the group $X$ is not
topologically isomorphic to a group of the form $(\ref{09.09.15.1})$.

We shall verify that

${(I+\tilde{\alpha})(F)}=F.$ (36)

Note that $\alpha$ satisfies condition $(\ref{1})$ if and only if

$\overline{(I+\tilde{\alpha})(Y)}=Y.$ (37)

Put $B=\mathop{\mbox{\rm\bf
P}^{*}}\limits_{n=2}^{\infty}{\mathbb{Z}}(p^{2}_{n})$, and consider $B$ as a
subgroup of the group $Y$ in the topology induced on $B$ by the topology of
$Y$. Denote by $\tilde{\alpha}_{B}$ the restriction of $\tilde{\alpha}$ to the
subgroup $B$. Assume that ${\rm Ker}(I+\tilde{\alpha}_{B})\neq\\{0\\}$. This
implies that $(I+\tilde{\alpha}_{B})(B)$ is a proper subgroup of $B$. It
follows from this that $(I+\tilde{\alpha})(B)$ is contained in a proper closed
subgroup of the group $Y$. Since $B$ is a dense subgroup in $Y$, this implies
that $(I+\tilde{\alpha})(Y)$ is also contained in a proper closed subgroup of
the group $Y$, contrary to $(\ref{10.09.15.1})$. Hence, ${\rm
Ker}(I+\tilde{\alpha}_{B})=\\{0\\}$, and then $I+\tilde{\alpha}_{B}$ is an
automorphism of the group $B$. Put $C=\mathop{\mbox{\rm\bf
P}^{*}}\limits_{n=2}^{\infty}K_{n}$, and consider $C$ as a subgroup of the
group $F$ in the topology induced on $C$ by the topology of $Y$. Then $C$ is a
dense subgroup in $F$. Since $I+\tilde{\alpha}_{B}$ is an automorphism of the
group $B$, we have

$(I+\tilde{\alpha})(C)=C.$ (38)

Take $y_{0}\in F$. The subgroup $C$ is dense in $F$. Hence, there exists a
sequence $y_{j}\in C$ such that $y_{j}\rightarrow y_{0}$. It follows from (38)
that $y_{j}=(I+\tilde{\alpha})z_{j}$, where $z_{j}\in C$. Since $F$ is a
compact group, there exist a sequence $\\{j_{k}\\}$ and an element $z\in F$
such that $z_{j_{k}}\rightarrow z$. Hence $y_{0}=(I+\tilde{\alpha})z$. Thus,
(36) is proved.

By Lemma 2.2, the characteristic functions $\hat{\mu}_{j}(y)$ satisfy equation
(4). Put $\nu_{j}=\mu_{j}*\bar{\mu}_{j}$. Then
$\hat{\nu}_{j}(y)=|\hat{\mu}_{j}(y)|^{2}\geq 0$ for all $y\in Y$, and the
characteristic functions $\hat{\nu}_{j}(y)$ also satisfy equation (4). It is
easy to see that Proposition 3.4 will be proved if we prove that
$\nu_{1}=\nu_{2}=m_{K}$, where $K$ is a compact subgroup of the group $X$, and
$\alpha(K)=K$. Therefore we can solve equation (4), assuming that
$\hat{\mu}_{j}(y)\geq 0$ for all $y\in Y$. Denote by $\tilde{\alpha}_{F}$ the
restriction of $\tilde{\alpha}$ to the subgroup $F$. It is obvious that
$\tilde{\alpha}_{F}\in{\rm Aut}(F)$. Consider the restriction of equation (4)
to the subgroup $F$. We have

$\hat{\mu}_{1}(u+v)\hat{\mu}_{2}(u+\tilde{\alpha}_{F}v)=\hat{\mu}_{1}(u-v)\hat{\mu}_{2}(u-\tilde{\alpha}_{F}v),\
\ u,v\in F.$ (39)

The group $F$ is topologically isomorphic to the character group of the
discrete group $\mathop{\mbox{\rm\bf P}^{*}}\limits_{n=2}^{\infty}K_{n},$ and
$\tilde{\alpha}_{F}$ is the adjoint automorphism to an automorphism
$\bar{\alpha}$ of the group $\mathop{\mbox{\rm\bf
P}^{*}}\limits_{n=2}^{\infty}K_{n}.$ Note that $(\ref{10.09.15.2})$ implies
that $\bar{\alpha}$ satisfies condition (2). Taking into account Lemma 2.2, by
applying Theorem B to the discrete group $\mathop{\mbox{\rm\bf
P}^{*}}\limits_{n=2}^{\infty}K_{n},$ we conclude from (39) that the
restrictions of the characteristic functions $\hat{\mu}_{j}(y)$ to the
subgroup $F$ are of the form

$\hat{\mu}_{j}(y)=\begin{cases}1,&\text{\ if\ }\ y\in E,\\\ 0,&\text{\ if\ }\
y\notin E,\ \ j=1,2,\end{cases}$

where $E$ is an open subgroup of $F$. In so doing,

$\tilde{\alpha}_{F}(E)=E.$ (40)

Put $M=A(X,E)$. It follows from (40) that $\alpha(M)=M$. By Lemma 3.2,
$\sigma(\mu_{j})\subset M$, $j=1,2$. Thus we reduced the proof of Proposition
3.4 to the case when the random variables $\xi_{j}$ take values in the group
$M$. The character group of the group $M$ is topologically isomorphic to the
factor-group $Y/E$. Since $E$ is an open subgroup of $F$, there is some
natural number $m$ such that the group $Y/E$ is topologically isomorphic to a
group of the form

$\mathop{\mbox{\rm\bf
P}}\limits_{n=2}^{m}{\mathbb{Z}}(p^{l_{n}}_{n})\times\mathop{\mbox{\rm\bf
P}^{*}}\limits_{n=m+1}^{\infty}{\mathbb{Z}}(p_{n}),$

where $l_{n}\in\\{0,1,2\\}$. Hence, the group $M$ is topologically isomorphic
to a group of the form (3). The proposition follows now from Theorem 2.1.
$\blacksquare$

##  4  The Heyde theorem fails on compact connected Abelian groups

We will prove in this section that there does not exist a compact connected
Abelian group on which even a weak analogue of the Heyde theorem holds (see
below Theorem 4.1).

Let $n$ be a natural number. Denote by $f_{n}:X\rightarrow X$ an endomorphism
of the group $X$, defined by the formula $f_{n}x=nx$ for all $x\in X$. Put
$f_{n}(X)=X^{(n)}$. Denote by $\mathbb{Q}$ the additive group of rational
numbers considering in the discrete topology. Let
$\text{\boldmath$a$}=(a_{0},a_{1},\dots,a_{n},\dots)$ be an arbitrary infinite
sequence of integers, where each of $a_{n}$ is greater than 1. Denote by
$\Sigma_{\text{\boldmath$a$}}$ the corresponding $a$-adic solenoid ([17,
(10.12)]). The group $\Sigma_{\text{\boldmath$a$}}$ is compact, connected and
has dimension 1 ([17, (10.13), (24.28)]). Denote by
$H_{\text{\boldmath$a$}}$ a subgroup of $\mathbb{Q}$ of the form

$\displaystyle{H_{\text{\boldmath$a$}}=\left\\{{m\over a_{0}a_{1}\cdots
a_{n}}:\ n=0,1,\dots;\ m=0,\pm 1,\pm 2,\dots\right\\}}.$

The character group of the group $\Sigma_{\text{\boldmath$a$}}$ is
topologically isomorphic to the group $H_{\text{\boldmath$a$}}$ ([17,
(10.12)]). Note that if $\text{\boldmath$a$}=(2,3,4,\dots)$, then
$H_{\text{\boldmath$a$}}=\mathbb{Q}$.

The main result of this section is the following statement.

######  Theorem 4.1

Let $X$ be a compact connected Abelian group containing no elements of order
$2$. Then there exist a topological automorphism $\alpha$ of the group $X$
satisfying condition $(\ref{1})$ and independent random variables $\xi_{1}$
and $\xi_{2}$ with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$
such that the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
whereas $\mu_{j}\notin\Gamma(X)*I(X)$, $j=1,2$.

We note that if $\mu\in\Gamma(X)*I(X)$, then $\mu$ is invariant with respect
to a compact subgroup $K$ of the group $X$ and, under the natural homomorphism
$X\rightarrow X/K$, induces a Gaussian distribution on the factor-group $X/K$.
Thus, Theorem 4.1 shows that even a weak analogue the Heyde theorem is not
valid for compact connected Abelian groups.

Proof of Theorem 4.1 Since $X$ is a compact connected Abelian group, $Y$ is a
discrete torsion free Abelian group. Two cases are possible: either the group
$X$ contains elements of finite order or $X$ is a torsion-free group.

1\. Assume that the group $X$ contains elements of finite order. Denote by $p$
the minimum of orders of nonzero elements of $X$. Then $p$ is a prime and
taking into account that $X$ contains no elements of order 2, $p\geq 3$. Since
$X$ is a connected Abelian group, we have $X^{(n)}=X$ for all natural $n$.
This implies that if ${\rm Ker}f_{n}=\\{0\\}$, then $f_{n}\in{\rm Aut}(X)$.
Set $\alpha=-f_{p-1}$. Then $\alpha\in{\rm Aut}(X)$. Since
$I+\alpha=-f_{p-2}$, we have $I+\alpha\in{\rm Aut}(X)$, so that $\alpha$
satisfies condition $(\ref{1})$. It follows from $Y^{(p)}=A(Y,{\rm Ker}f_{p})$
that $Y^{(p)}$ is a proper subgroup in the group $Y$. Take $y_{0}\in Y$ such
that $y_{0}\notin Y^{(p)}$. Since $p\geq 3$, the numbers 2 and $p$ are
mutually prime. Therefore there exist integers $m$ and $n$ such that
$2m+pn=1$. It follows from this that $y_{0}=2my_{0}+pny_{0}$. This implies
that $2y_{0}\notin Y^{(p)}$.

Consider on the group $X$ the function

$\rho(x)=1+{\rm Re}(x,y_{0}),\ \ x\in X.$

Then $\rho(x)\geq 0$ for all $x\in X,$ and

$\int\limits_{X}\rho(x)dm_{X}(x)=1.$

Denote by $\mu$ the distribution on $X$ with the density $\rho(x)$ with
respect to $m_{X}$. It is easy to see that the characteristic function
$\hat{\mu}(y)$ is of the form

$\hat{\mu}(y)=\begin{cases}1,&\text{\ if\ }\ y=0,\\\ {1\over 2},&\text{\ if\
}\ y=\pm y_{0},\\\ 0,&\text{\ if\ }\ y\not\in\\{0,\pm y_{0}\\}.\end{cases}$
(41)

It is obvious that $\mu\not\in\Gamma(X)*I(X)$.

Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random
variables with values in the group $X$ and distribution $\mu$. Let us verify
that the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric. By
Lemma 2.2, it suffices to verify that the characteristic function
$\hat{\mu}(y)$ satisfies equation (10). Obviously, equation (10) holds true if
$v=0$. So, assume that $v\neq 0$. Then the left-hand side of equation (10)
vanishes. Otherwise, taking into account (41), we get $u+v\in\\{0,\pm
y_{0}\\}$ and $u+\alpha v\in\\{0,\pm y_{0}\\}$. This implies that
$(I-\tilde{\alpha})v=f_{p}v\in\\{0,\pm y_{0},\pm 2y_{0}\\}$, but that’s
impossible because it contradicts $y_{0},2y_{0}\notin Y^{(p)}$. Reasoning
similarly we obtain that if $v\neq 0$, then the right-hand side of equation
(10) vanishes too. Thus, in the case when the group $X$ contains elements of
finite order, the theorem is proved.

2\. Assume that $X$ is a torsion-free group. Taking into account the structure
theorem for compact Abelian torsion-free group ([17, (25.8)]) and the fact
that $X$ is connected, we get that for some $\mathfrak{n}$ the group $X$ is
topologically isomorphic to a group of the form
$(\Sigma_{\text{\boldmath$a$}})^{\mathfrak{n}}$, where
${\text{\boldmath$a$}}=(2,3,4,\dots).$ Obviously, arguing as in the proof of
necessity in Theorem 2.1 it suffices to prove the theorem for the group
$X=\Sigma_{\text{\boldmath$a$}}$, where ${\text{\boldmath$a$}}=(2,3,4,\dots)$.
Then the group $Y$ is topologically isomorphic to the group $\mathbb{Q}$. In
order not to complicate the notation we assume that $Y=\mathbb{Q}$.

Let $H$ be a subgroup of the group $Y$ of the form

$\displaystyle{H=\left\\{{m\over 5^{n}}:\ n=0,1,\dots;\ m=0,\pm 1,\pm
2,\dots\right\\}}.$

Consider on the group $H$ the function

$g(h)=\begin{cases}1,&\text{\ if\ }\ \ h\in H^{(2)},\\\ c,&\text{\ if\ }\ \
h\notin H^{(2)},\end{cases}$ (42)

where $-1<c<1,$ $c\neq 0$. It is easy to verify that $g(h)$ is a positive
definite function. Consider on the group $Y$ the function

$f(y)=\begin{cases}g(y),&\text{\ if\ }\ \ y\in H,\\\ 0,&\text{\ if\ }\ \
y\notin H.\end{cases}$ (43)

Then $f(y)$ is also a positive definite function ([18, (32.43)]). By the
Bochner theorem, there exists a distribution $\mu\in{\rm M}^{1}(X)$ such that
$\hat{\mu}(y)=f(y).$ It is obvious that $\mu\notin\Gamma(X)*I(X).$

Put $\alpha=-f_{2}^{-2}$. Then $\alpha\in{\rm Aut}(X)$, and we have
$I+\alpha=f_{3}f_{2}^{-2}$. Obviously, $I+\alpha\in{\rm Aut}(X)$, so that
$\alpha$ satisfies condition $(\ref{1})$.

Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random
variables with values in the group $X$ and distribution $\mu$. Verify that the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric. This is equivalent to the fact
that the conditional distribution of the linear form $N_{2}=4\xi_{1}-\xi_{2}$
given $N_{1}=\xi_{1}+\xi_{2}$ is symmetric. By Lemma 2.2, it suffices to
verify that the characteristic function $\hat{\mu}(y)$ satisfies the equation
(10) which takes the form

$f(u+4v)f(u-v)=f(u-4v)f(u+v),\ \ u,v\in Y.$ (44)

It follows from (42) and (43) that for $u,v\in H$ the equation (44) becomes an
equality. Note that if either $u\in H$, $v\notin H$ or $v\in H$, $u\notin H,$
then $u\pm v\notin H$, and it follows from (43) that both sides of equation
(44) are equal to zero. So, equation (44) becomes also an equality. Assume
that $u,v\notin H.$ Then the left-hand side in (44) is equal to zero. In the
opposite case (43) implies that $u+4v,u-v\in H.$ It follows from this that
$5v\in H$, and hence, $v\in H$. The obtained contradiction shows that the
left-hand side in (44) is equal to zero. Arguing similarly, we verify that if
$u,v\notin H,$ then the right-hand side in (44) is also equal to zero.
$\blacksquare$

In connection with Theorem 4.1, it is interesting to note that the following
statement is true (see [11, Theorem 3]).

Let $X$ be a locally compact Abelian group containing no elements of order
$2$. Let $\alpha$ be a topological automorphism of the group $X$ satisfying
condition $(\ref{1})$. Let $\xi_{1}$ and $\xi_{2}$ be independent random
variables with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$ with
non-vanishing characteristic functions. If the conditional distribution of the
linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then $\mu_{j}\in\Gamma(X)$, $j=1,2$.

We will complement Theorem 4.1 with two propositions. The first proposition
shows that, generally speaking, we can not replace in Theorem 4.1 the
statement:

(A) Then there exist a topological automorphism $\alpha$ of the group $X$
satisfying condition $(\ref{1})$ and independent random variables $\xi_{1}$
and $\xi_{2}$ with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$
such that the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
whereas $\mu_{j}\notin\Gamma(X)*I(X)$, $j=1,2$.

by the statement:

(B) Then for each topological automorphism $\alpha$ of the group $X$,
$\alpha\neq I$, satisfying condition $(\ref{1})$ there exist independent
random variables $\xi_{1}$ and $\xi_{2}$ with values in $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ such that the conditional distribution of the linear
form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
whereas $\mu_{j}\notin\Gamma(X)*I(X)$, $j=1,2$.

The second proposition shows that there is a compact connected Abelian group
$X$ containing no elements of order $2$ when we can replace in Theorem 4.1 (A)
by (B). We should exclude the case when $\alpha=I$, because if a compact
Abelian group $X$ contains no elements of order 2, then ${Y^{(2)}}=Y$, and by
Lemma 2.2, the symmetry of the conditional distribution of the linear form
$L_{2}=\xi_{1}+\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ implies that $\xi_{1}$
and $\xi_{2}$ have degenerate distributions.

######  Proposition 4.2

There exist a compact connected Abelian group $X$ containing no elements of
order $2$ and a topological automorphism $\alpha$ of the group $X$,
$\alpha\not=I$, satisfying condition $(\ref{1})$ such that the following
statement holds: If $\xi_{1}$ and $\xi_{2}$ are independent random variables
with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$ such that the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then $\mu_{j}\in\Gamma(X)*I(X)$,
$j=1,2$.

To prove Proposition 4.2 we need two lemmas.

######  Lemma 4.3 ([22, Lemma 6])

Let $X$ be a locally compact Abelian group, and let $\alpha$ be a topological
automorphism of the group $X$. Let $\xi_{1}$ and $\xi_{2}$ be independent
random variables with values in $X$. If the conditional distribution of the
linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is
symmetric, then the linear forms $M_{1}=(I+\alpha)\xi_{1}+2\alpha\xi_{2}$ and
$M_{2}=2\xi_{1}+(I+\alpha)\xi_{2}$ are independent.

######  Lemma 4.4 ([3], see also [7, Theorem 7.10])

Let $X$ be a locally compact Abelian group. Assume that the connected
component of zero of the group $X$ contains no elements of order $2$. Let
$\xi_{1}$ and $\xi_{2}$ be independent random variables with values in $X$ and
distributions $\mu_{1}$ and $\mu_{2}$. If the sum $\xi_{1}+\xi_{2}$ and the
difference $\xi_{1}-\xi_{2}$ are independent, then $\mu_{j}\in\Gamma(X)*I(X)$,
$j=1,2$.

Proof of Proposition 4.2 Let $X=\Sigma_{\text{\boldmath$a$}}^{2}$, where
${\text{\boldmath$a$}}=(2,3,4,\dots)$. Since the character group of the group
$X$ is topologically isomorphic to the group $\mathbb{Q}^{2}$, we have
${f_{n}\in{\rm Aut}(X)}$ for any natural $n$. This implies in particular, that
the group $X$ contains no elements of order 2. Denote by $x=(a,b)$, where
$a,b\in\Sigma_{\text{\boldmath$a$}}$, elements of the group $X$. Define the
mapping $\alpha:X\rightarrow X$ by the formula

$\alpha(a,b)=(-3a+4b,2a-3b),\ (a,b)\in X.$

Then $\alpha$ is a topological automorphism of the group $X$. Since
$I+\alpha\in{\rm Aut}(X)$, condition (2) is fulfilled. By Lemma 4.3, the
symmetry of the conditional distribution of the linear form
$L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ implies that the
linear forms $M_{1}=(I+\alpha)\xi_{1}+2\alpha\xi_{2}$ and
$M_{2}=2\xi_{1}+(I+\alpha)\xi_{2}$ are independent. Note that coefficients of
the linear forms $M_{1}$ and $M_{2}$ are topological automorphisms of the
group $X$. Consider the new independent random variables
$\eta_{1}=(I+\alpha)\xi_{1}$ and $\eta_{2}=2\alpha\xi_{2}$. The independence
of the linear forms $M_{1}$ and $M_{2}$ implies that the linear forms
$N_{1}=\eta_{1}+\eta_{2}$ and
$N_{2}=2(I+\alpha)^{-1}\eta_{1}+f_{2}^{-1}\alpha^{-1}(I+\alpha)\eta_{2}$ are
independent. Hence, the linear forms $P_{1}=\eta_{1}+\eta_{2}$ and
$P_{2}=\eta_{1}+f_{2}^{-2}\alpha^{-1}(I+\alpha)^{2}\eta_{2}$ are also
independent. It is easy to verify that
$f_{2}^{-2}\alpha^{-1}(I+\alpha)^{2}=-I$. Hence, $P_{2}=\eta_{1}-\eta_{2}$.
Denote by $\lambda_{j}$ the distribution of the random variable $\eta_{j}$.
Since $X$ is a connected Abelian group containing no elements of order $2$, by
Lemma 4.4, $\lambda_{j}\in\Gamma(X)*I(X)$, $j=1,2$. In view of
$I+\alpha,2\alpha\in{\rm Aut}(X)$, it follows from
$\lambda_{1}=(I+\alpha)\mu_{1}$ and $\lambda_{2}=2\alpha\mu_{2}$ that
$\mu_{j}\in\Gamma(X)*I(X)$, $j=1,2$. $\blacksquare$

######  Proposition 4.5

There exists a compact connected Abelian group $X$ containing no elements of
order $2$ such that for each topological automorphism $\alpha$ of the group
$X$, $\alpha\neq I$, satisfying condition $(\ref{1})$, there exist independent
random variables $\xi_{1}$ and $\xi_{2}$ with values in $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ such that the conditional distribution of the linear
form $L_{2}=\xi_{1}+\alpha\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
while $\mu_{j}\notin\Gamma(X)*I(X)$, $j=1,2$.

Proof Let $X=\Sigma_{\text{\boldmath$a$}}$, where
$\text{\boldmath$a$}=(2,2,2,\dots)$. The group $Y$ is topologically isomorphic
to a subgroup of the group $\mathbb{Q}$ of the form

$\displaystyle H_{\text{\boldmath$a$}}=\left\\{{m\over 2^{n}}:\ n=0,1,\dots;\
m=0,\pm 1,\pm 2,\dots\right\\}.$

Since $Y^{(2)}=Y$, the group $X$ contains no elements of order 2. It is easy
to see that ${\rm Aut}(X)={\\{\pm f_{2^{k}},\pm
f_{2^{k}}^{-1}:k=0,1,2,\dots\\}.}$ Obviously, there exist only two topological
automorphisms $\alpha$, $\alpha\neq I$, of the group $X$ satisfying condition
(2). They are $\alpha=-f_{2}$ and $\alpha=-f_{2}^{-1}$. Let $\alpha=-f_{2}$.
It is easy to see that the group $X$ contains an element of order 2. Then, as
has been proved in the case 1 of Theorem 4.1, there exist independent random
variables $\xi_{1}$ and $\xi_{2}$ with values in $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ such that the conditional distribution of the linear
form $L_{2}=\xi_{1}-2\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
while $\mu_{j}\notin\Gamma(X)*I(X)$, $j=1,2$.

The proof of the existence of independent random variables $\xi_{1}$ and
$\xi_{2}$ with values in $X$ and distributions $\mu_{1}$ and $\mu_{2}$ such
that the conditional distribution of the linear form
$L_{2}=\xi_{1}-f_{2}^{-1}\xi_{2}$ given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric,
while $\mu_{j}\notin\Gamma(X)*I(X)$, $j=1,2$, is reduced to the previous case.
$\blacksquare$

######  Remark 4.6

The Heyde theorem is true for some compact connected Abelian groups if we
additionally assume that the characteristic functions of random variables do
not vanish. In particular, the following statement was proved in [13].

######  Theorem

Let $X=\Sigma_{\text{\boldmath$a$}}$ be an $a$-adic solenoid containing no
elements of order $2$, and let $\alpha$ be a topological automorphism of the
group $X$ satisfying condition $(\ref{1})$. Let $\xi_{1}$ and $\xi_{2}$ be
independent random variables with values in the group $X$ and distributions
$\mu_{1}$ and $\mu_{2}$ with nonvanishing characteristic functions. If the
conditional distribution of the linear form $L_{2}=\xi_{1}+\alpha\xi_{2}$
given $L_{1}=\xi_{1}+\xi_{2}$ is symmetric, then $\mu_{j}\in\Gamma(X)$,
$j=1,2$.

Acknowledgments My special thanks go to the referee for careful reading of my
article and many useful suggestions and remarks. I appreciate it very much.

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G. M. Feldman

e-mail<EMAIL_ADDRESS>
B. Verkin Institute for Low Temperature
Physics and Engineering of the
National Academy of Sciences of Ukraine
47, Nauky Ave
Kharkiv, 61103
Ukraine


[a]Stefan Dittmaier

# Full and approximated NLO predictions for like-sign W-boson scattering at
the LHC

Christopher Schwan  Ramon Winterhalder

###### Abstract

We report on a recent calculation of next-to-leading-order (NLO) QCD and
electroweak corrections to like-sign W-boson scattering at the Large Hadron
Collider, including all partonic channels and W-boson decays in the process
${\mathrm{p}}{\mathrm{p}}\to{\mathrm{e^{+}}}\nu_{\mathrm{e}}\,\mu^{+}\nu_{\mu}\,\mathrm{j}\mathrm{j}+X$.
The calculation is implemented in the Monte Carlo integrator Bonsay and
comprises the full tower of NLO contributions of the orders
$\alpha_{\mathrm{s}}^{3}\alpha^{4}$, $\alpha_{\mathrm{s}}^{2}\alpha^{5}$,
$\alpha_{\mathrm{s}}\alpha^{6}$, and $\alpha^{7}$. Our numerical results
confirm and extend previous results, in particular the occurrence of large
purely electroweak corrections of the order of $\sim-12\%$ for integrated
cross sections, which get even larger in distributions. We construct a VBS
approximation for the NLO prediction based on partonic channels and gauge-
invariant (sub)matrix elements potentially containing the vector-boson
scattering (VBS) subprocess and on resonance expansions of the W decays. The
VBS approximation reproduces the full NLO predictions within
$\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;1.5\%$
in the most important regions of phase space. Moreover, we discuss results
from different versions of effective vector-boson approximations at leading
order, based on the collinear emission of W bosons of incoming (anti)quarks.
However, owing to the only mild collinear enhancement and the design of VBS
analysis cuts, the quality of this approximation turns out to be only
qualitative at the LHC.

##  1 Introduction

Electroweak (EW) vector-boson scattering (VBS) is among the most interesting
classes of processes first accessible at the LHC to investigate the EW gauge
structure and EW symmetry breaking. Taking into account leptonic decays of the
EW gauge bosons, the experimental signature of VBS at the LHC is given by two
mostly forward/backward-pointing jets and four leptons, with at least two of
them charged and the others being neutrinos leading to missing transverse
energy in the detector. The corresponding $2\to 6$ matrix elements do not only
involve purely EW diagrams, but also gluon exchange or even gluon-fusion
channels, leading to a large number of partonic channels and a whole tower of
next-to-leading order (NLO) corrections featuring not only pure QCD and EW
corrections but also QCD–EW mixed contributions. Precision calculations for
VBS processes have been continuously extended and refined over the last few
decades and culminated in the knowledge of the full towers of NLO corrections
for all relevant VBS processes at the LHC and the matching of fixed-order
predictions with QCD parton showers (see Ref. [1] and references therein). The
NLO corrections of all VBS processes share the feature that the EW channels,
whose relative contribution is enhanced by dedicated VBS selection cuts,
receive particularly large EW corrections of 10–15%.

In this article we summarize the salient features and results of a recent
recalculation [2] of the NLO corrections to like-sign WW scattering. Previous
results on QCD corrections [3, 4] including parton-shower matching [5] as well
as on all NLO orders including pure EW and QCD–EW contributions [6] have been
presented in the literature before. Apart from providing cross-checks to
existing results, we also present an approximation of the NLO corrections
based on gauge-invariant subcontributions featuring the VBS subprocess and on
resonance expansions for the W-boson decays, as well as a detailed discussion
of the so-called effective vector-boson approximation for leading-order (LO)
predictions based on the picture of massive vector bosons as partons of the
proton.

##  2 Like-sign W-boson scattering at NLO

Like-sign ${\mathrm{W}}{\mathrm{W}}$ production at the LHC has the signature
of two like-sign leptons, two jets, which are typically forward/backward
pointing, and missing transverse momentum carried away by two neutrinos. For
definiteness, we consider the process
${\mathrm{p}}{\mathrm{p}}\to{\mathrm{e^{+}}}\nu_{\mathrm{e}}\,\mu^{+}\nu_{\mu}\,\mathrm{j}\mathrm{j}+X$.
The LO cross section for ${\mathrm{W}}^{\pm}{\mathrm{W}}^{\pm}$ scattering
receives contributions from $qq$, $q\bar{q}$, and $\bar{q}\bar{q}$ partonic
channels involving various (anti)quark combinations with positive/negative net
electric charge, i.e. at LO there are no partonic channels involving initial-
state gluons. Denoting the EW and the strong coupling constants as $e$ and
$g_{\mathrm{s}}$, respectively, the LO matrix elements scale like
$\mathcal{O}(e^{6})$ or $\mathcal{O}(g_{\mathrm{s}}^{2}e^{4})$; the respective
contributions to the LO matrix element are denoted $\mathcal{M}_{e^{6}}$ and
$\mathcal{M}_{g_{\mathrm{s}}^{2}e^{4}}$ in the following. The structural
diagram featuring the VBS subprocess, an EW background diagram as well as a
diagram with gluon exchange are illustrated in Fig. 1.

(a) VBS, doubly-resonant

(b) non-resonant

(c) QCD contribution

Figure 1: Examples of LO Feynman diagrams for the partonic subprocess
${\mathrm{u}}{\mathrm{u}}\to{\mathrm{d}}{\mathrm{d}}\,{\mathrm{e^{+}}}\nu_{\mathrm{e}}\,\mu^{+}\nu_{\mu}$.
The shaded blob represents tree-level subdiagrams for
${\mathrm{W^{+}}}{\mathrm{W^{+}}}\to{\mathrm{W^{+}}}{\mathrm{W^{+}}}$.

The different types of VBS subdiagrams contained in $\mathcal{M}_{e^{6}}$ are
shown in Fig. 2.

(a) 4-point interaction

(b) $t$ channel

(c) $t$-channel Higgs

Figure 2: Typical VBS subdiagrams for
${\mathrm{W^{+}}}{\mathrm{W^{+}}}\to{\mathrm{W^{+}}}{\mathrm{W^{+}}}$
contained in the shaded blob of diagram Fig. 1(a).

Squaring the LO matrix element leads to three different perturbative orders,
scaling like $\alpha^{6}$, $\alpha_{\mathrm{s}}^{2}\alpha^{4}$, and
$\alpha_{\mathrm{s}}\alpha^{5}$. The latter interference contribution is
numerically strongly suppressed due to its colour structure (which demands
identical quark generations in the two quark chains) and due to the fact that
forward- or backward-enhanced $t/u$-channel propagators are not further
enhanced by squaring. At NLO, each of the LO matrix elements can receive EW
and QCD corrections, i.e. there are one-loop amplitudes of
$\mathcal{O}(e^{8})$, $\mathcal{O}(g_{\mathrm{s}}^{2}e^{6})$, and
$\mathcal{O}(g_{\mathrm{s}}^{4}e^{4})$ at NLO. Some corresponding one-loop
diagrams of these orders are shown in Fig. 3.

(a) 6-point function with Higgs

(b) gluon at single quark line

(c) double gluon exchange

Figure 3: Examples of one-loop diagrams for the partonic subprocess
${\mathrm{u}}{\mathrm{u}}\to{\mathrm{d}}{\mathrm{d}}\,{\mathrm{e^{+}}}\nu_{\mathrm{e}}\,\mu^{+}\nu_{\mu}$.

The cross sections are evaluated with the Monte Carlo program Bonsay, which is
based on multi-channel Monte Carlo integration using adaptive weight
optimization [7], similar to the approach described in Ref. [8]. Bonsay
supports the parallel computation of uncertainties induced by different scale
choices and errors in parton distribution functions. Both the tree-level and
one-loop matrix elements are provided by OpenLoops2 [9] by default and have
been cross-validated against respective results obtained with Recola [10]. The
one-loop integrals are numerically evaluated using Collier [11], which employs
the methods and results of Ref. [12] to numerically stabilize the results in
the vicinity of exceptional phase-space configurations. Particle resonances
are described in the complex-mass scheme [13, 14] to guarantee gauge
independence of amplitudes and NLO accuracy in both resonant and non-resonant
phase-space regions. The extraction and cancellation of (soft and collinear)
infrared singularities is accomplished within the dipole subtraction formalism
both for QCD [15] and EW [16] corrections.

In addition to performing the full NLO calculation we have constructed a VBS
approximation (VBSA) which can serve as a proxy for the full calculation with
a precision that is sufficient for most phenomenological analyses and which is
much less costly in terms of CPU time. At LO, the VBSA keeps the full $2\to 6$
matrix elements. At NLO, the VBSA merges two different approximative steps:
Similar to the approach used in Ref. [4], step 1 selects all partonic channels
that contain the VBS subprocess and further strips contributions of minor
importance. In detail, matrix elements are decomposed into gauge-invariant
parts characterized by different fermion-number flows, and only channels
related to VBS are kept. This step, in particular, eliminates all channels
featuring WWW production instead of VBS. Moreover, in the process of squaring
the amplitude, all interference terms from different fermion-number flows are
discarded, since they are colour suppressed and do not receive the kinematical
VBS enhancements from squared $t/u$-channel W propagators. In step 1, thus,
all one-loop amplitudes can be constructed from one prototype channel, e.g.
$\mathrm{c}\mathrm{u}\to{\mathrm{e^{+}}}\nu_{\mathrm{e}}\,\mu^{+}\nu_{\mu}\,\mathrm{d}\mathrm{s}$,
via crossing. Step 2 in the VBSA construction applies the double-pole
approximation (DPA) to the produced W bosons in the virtual corrections, i.e.
all one-loop matrix elements are expanded about the two W resonance poles.
This procedure splits the virtual corrections into factorizable and non-
factorizable parts, the former containing the corrections to the W-pair
production and the W-decay subprocesses, the latter accounting for doubly-
resonant effects from soft-photon exchange between the subprocesses. More
details on the DPA concept and the specifically employed variant can be found
in Refs. [17, 18, 14] and references therein. We finally note that care has to
be taken in the approximation of the real-emission corrections since they are
related to different underlying LO channels in different collinear limits of
phase space. For the full construction of the VBSA we refer to Ref. [2].

##  3 Cross-section predictions at NLO

In Figs. 4–6 we compare the full relative NLO corrections
$\delta=\mathrm{d}\sigma_{\text{NLO}}/\mathrm{d}\sigma_{\text{LO}}-1$ with its
approximated version $\delta_{\text{VBSA}}=\mathrm{d}\sigma_{\text{NLO,
VBSA}}/\mathrm{d}\sigma_{\text{LO}}-1$ in VBSA. The bands in the figures
illustrate the residual scale uncertainties, obtained from the envelope of the
usual seven-point scale variation with factors of $0.5$ and $2$ using the
geometric mean of the transverse momenta of the tagging jets as central scale.

Figure 4: Distributions in the transverse momenta of the leading jet and the
positron and corresponding corrections: absolute predictions (top panels),
relative NLO corrections in VBSA (middle panels), and difference between
relative full NLO corrections and corresponding VBSA (bottom panels). [Taken
from Ref. [2].]

Figure 4 shows the results for the distributions in the transverse momenta of
the leading jet and the positron. In both cases, for moderate and large
$p_{\mathrm{T}}$, the purely EW corrections of $\mathcal{O}(\alpha^{7})$ are
dominant, negative, and increasing in magnitude for increasing transverse
momenta, reaching about $\sim-20\%$ and $\sim-30\%$ at
$p_{\mathrm{T}}=600\,\mathrm{GeV}$, respectively. This behaviour is a typical
sign for the appearance of EW Sudakov logarithms at high energies. The
different size of the effects for jets and W-decay leptons can be understood
as follows: The $p_{\mathrm{T}}$ distributions of the jets do not entirely
zoom into the Sudakov regime of the WW$\to$WW subprocess, which demands large
Mandelstam variables in the $2\to 2$ subprocess and small virtualities of the
incoming (off-shell) W bosons. For large $p_{\mathrm{T}}$ values of a jet, the
virtuality of at least one of the incoming W bosons is not small, and the
$t$-channel momentum transfer in the WW$\to$WW subprocess is not forced to be
large. Therefore, the EW Sudakov double logarithms cannot fully dominate the
corrections to the $p_{\mathrm{T}}$ of the jets, and all kinds of nominally
subleading EW high-energy corrections become relevant. On the other hand, the
domain of large $p_{\mathrm{T}}$ of any of the decay leptons is dominated by
the Sudakov regime of the WW$\to$WW subprocess, because the preference of
small jet transverse momenta leads to small virtualities of the incoming W
bosons and the large transverse momentum of a decay lepton requires both a
large scattering energy (Mandelstam variable $s$) and large momentum transfer
(Mandelstam variable $t$) of the subprocess. The observed $\sim-30\%$ can be
qualitatively reproduced by just taking into account the EW Sudakov correction
factor for the WW$\to$WW subprocess,
$\delta_{\mathrm{Sud}}=-\frac{2\alpha}{\pi
s_{\mathrm{w}}^{2}}\ln^{2}(\hat{s}/M_{\mathrm{W}}^{2})$, where
$\sqrt{\hat{s}}=\mathcal{O}(p_{\mathrm{T}})$ is the WW centre-of-mass energy
and $s_{\mathrm{w}}$ the sine of the weak mixing angle. These features were
already highlighted in Ref. [6]. The next-to-largest corrections to the
$p_{\mathrm{T}}$ distributions are the mixed QCD–EW corrections of
$\mathcal{O}(\alpha_{s}\alpha^{6})$, which amount to 5–10% above the maximum
in the leading-jet distribution and are almost uniformly $\sim-5\%$ in the
$p_{\mathrm{T}}$ spectrum of the electron. The suppression of the remaining
corrections of $\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$ and
$\mathcal{O}(\alpha_{s}^{3}\alpha^{4})$ is mostly due to the fact that the
cross-section contributions widely inherit the kinematic behaviour of the LO
QCD contribution, which is small compared to the EW contribution over the
whole distribution as a consequence of the VBS cuts. The approximative quality
of the VBSA is typically at the 1% level for all orders of the NLO tower in
the regions of phase space where the relevant part of the cross section is
concentrated, with the exception of the
$\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$ contribution where the difference
$\delta_{\text{VBSA}}-\delta$ can reach the order of 1.5% in size.

Figure 5: As in Fig. 4, but for the distributions in the rapidities of the
leading jet and the muon. [Taken from Ref. [2].]

Figure 5 shows the distributions in the rapidities of the leading jet and the
muon. The hierarchy among the various NLO contributions is similar as for the
$p_{\mathrm{T}}$ distributions shown above, i.e. the purely EW corrections of
$\mathcal{O}(\alpha^{7})$ are the dominating ones, followed by the order
$\mathcal{O}(\alpha_{s}\alpha^{6})$, while the other two orders with higher
powers of $\alpha_{s}$ are widely suppressed. The corrections of
$\mathcal{O}(\alpha^{7})$ show much less variations in shape than for the
$p_{\mathrm{T}}$ distributions and are typically about $-10\%$ to $-12\%$.
This is due to the fact that the large logarithmic EW high-energy corrections
uniformly contribute to all rapidities, in contrast to the $p_{\mathrm{T}}$
distributions where they appear at high scales only. The moderate variations
in the $\mathcal{O}(\alpha^{7})$ corrections mostly result from the change in
the LO normalization induced by the variation in its composition from EW and
QCD parts; normalizing the $\mathcal{O}(\alpha^{7})$ contribution to the
$\mathcal{O}(\alpha^{6})$ LO part would produce a nearly flat relative
$\mathcal{O}(\alpha)$ correction. The overall second-largest corrections are
again the ones of $\mathcal{O}(\alpha_{s}\alpha^{6})$, which are dominated by
the QCD corrections to the EW LO channel. Their largest impact, growing even
to $\sim 10\%$, is on the leading jet at high rapidities. In the other
rapidity regions those corrections hardly exceed 5%. The pure QCD corrections
of $\mathcal{O}(\alpha_{s}^{3}\alpha^{4})$ only exceed the 1% level
significantly for central rapidities. The mixed corrections of order
$\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$ never exceed the 1% level at all.
Whenever the cross section is sizeable, the approximative quality of the VBSA
is again at the level of 1% or better for all NLO orders but
$\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$, where it is of the order of 1.5%.

Figure 6: As in Fig. 4, but for the distributions in the jet-pair (left) and
charged-lepton-pair (right). [Taken from Ref. [2].]

Finally, we show the distributions in the invariant mass of the two leading
jets (left) and in the invariant mass of the charged leptons (right) in Fig.
6. The hierarchy of the various corrections and their behaviour can be widely
explained following similar arguments as above. The trend of the dominating
genuine weak corrections of $\mathcal{O}(\alpha^{7})$ towards increasingly
negative corrections for larger scales is visible as for the $p_{\mathrm{T}}$
distributions, but the increase in size to about $-15\%$ for the largest
considered scales is much less dramatic. This is due to the fact that the
domain of large invariant masses $M_{\mathrm{j}_{1}\mathrm{j}_{2}}$ or
$M_{{\mathrm{e^{+}}}\mu^{+}}$ is not fully dominated by the Sudakov regime of
the WW$\to$WW subprocess, because the $t/u$-channel-like momentum transfer in
the subprocess is not forced to be large. Thus, the impact of the leading EW
Sudakov corrections is damped to the size of the subleading EW high-energy
corrections. In the $M_{\mathrm{j}_{1}\mathrm{j}_{2}}$ distribution, which is
mostly dominated by EW contributions at LO, the corrections of
$\mathcal{O}(\alpha_{s}\alpha^{6})$ typically have an impact at the 5% level,
while the remaining two NLO orders with higher powers of $\alpha_{s}$ hardly
reach 1%. The mixed QCD–EW and the pure QCD corrections show, however, an
interesting crossover in the $M_{{\mathrm{e^{+}}}\mu^{+}}$ distribution at
$M_{{\mathrm{e^{+}}}\mu^{+}}\sim 400\,\mathrm{GeV}$, which we attribute to the
increasing influence of the LO QCD contribution. For
$M_{{\mathrm{e^{+}}}\mu^{+}}<400\,\mathrm{GeV}$, where the EW part strongly
dominates the LO cross section, the $\mathcal{O}(\alpha_{s}\alpha^{6})$
correction is the second largest after the genuine EW correction, and the
remaining NLO orders are at the 1% level. For
$M_{{\mathrm{e^{+}}}\mu^{+}}>400\,\mathrm{GeV}$, where the LO QCD part
competes in size with the EW LO part, the corrections of
$\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$ and
$\mathcal{O}(\alpha_{s}^{3}\alpha^{4})$ dominate over
$\mathcal{O}(\alpha_{s}\alpha^{6})$ and reach $\sim-5\%$ for large
$M_{{\mathrm{e^{+}}}\mu^{+}}$. Similar to the previously considered
distributions, the approximative quality of the VBSA is at the level of 1.5%
for the $\mathcal{O}(\alpha_{s}^{2}\alpha^{5})$ corrections and of 1% for the
other NLO orders.

##  4 Effective W-boson approximation at LO

The effective vector-boson approximation (EVA), the idea of which goes back to
Ref. [19], extends the idea of partons inside hadrons to the case of weak
vector bosons, which play the role of partons in (anti)quarks, just like
(anti)quarks and gluons in hadrons. The vector-boson emission $q\to qV$ is
approximated by its asymptotic behaviour in the collinear limit, where it is
logarithmically enhanced. Previous studies in the literature have already
indicated that the approximation quality of the EVA is rather limited (see,
e.g., Refs. [20, 21] and references therein).

A comprehensive description of our construction of the LO EVA matrix elements
is given in the appendix of Ref. [2]. Figure 7 schematically illustrates the
factorization of VBS matrix elements into W radiation off (anti)quarks, VBS
core process, and subsequent W-boson decays.

(25,40)(47,50) (25,35)(47,25) enhanced$t$-channelpropagators

nearly collinear ${\mathrm{W}}$ radiation ${\mathrm{W}}{\mathrm{W}}$
scattering ${\mathrm{W}}$ decays

Figure 7: Schematic illustration of the EVA factorization of VBS matrix
elements.

We do not merely take over existing proposals from the literature, but compare
various formulations that differ in the details of handling intermediate (off-
shell) polarization vectors and external currents describing the W radiation
off the (anti)quarks and the W decays into leptons, in order to account for
spin correlations and off-shell effects as much as possible.

Figure 8 exemplarily illustrates the quality of different EVA versions for the
invariant-mass distributions of the two jets and the two charged leptons,
respectively.

Figure 8: LO distributions in the invariant masses of the two jets and the two
charged leptons, respectively, in like-sign WW scattering, based on full LO
matrix elements (LO) and different EVA versions (EVA1–4 and EVA KS, see main
text), employing typical VBS selection cuts but with an “inverted cut” on the
jets transverse momenta, $p_{\mathrm{T,jet}}<100\,\mathrm{GeV}$. The relative
deviation of the EVA predictions from LO is quantified by
$\Delta_{\mathrm{EVA}}=\mathrm{d}\sigma_{\mathrm{EVA}}/\mathrm{d}\sigma_{\mathrm{LO}}-1$.
[Taken from Ref. [2].]

More results can be found in Ref. [2]. The most important difference between
the EVA versions is that EVA1 and EVA3 restore the transversality of the
polarization vectors of initial-state W bosons in VBS amplitude, but not EVA2
and EVA4. Other, less important differences concern the transversality
conditions for the final-state W bosons of the VBS process and for the
leptonic decay currents, and an optional relative sign factor between
transverse and longitudinal polarization vectors of the incoming (off-shell) W
bosons, as described in detail in Ref. [2]. EVA KS refers to the EVA variant
of Ref. [20], where an extra weight factor was introduced for longitudinal
incoming off-shell W bosons. All EVA versions are evaluated with an on-shell
projection of the W momenta in the VBS subprocess that forces the W momenta
on-shell to guarantee gauge independence of the WW$\to$WW matrix elements and
preserves the locations of the photon poles in the $t$\- and $u$-channel
subdiagrams (b) of Fig. 2 to avoid extra enhancements in the intrinsic
uncertainty of the EVA.

Generically, we find that all EVA versions can only qualitatively describe the
full VBS process as long as typical VBS selection cuts are applied, which
exclude the very forward region of jet emission for which the EVA is actually
designed. Only if we invert the cut on the jets transverse momenta to
$p_{\mathrm{T,jet}}<100{-}150\,\mathrm{GeV}$, the EVA delivers results of some
reasonable approximative quality. In regions where the cross section is
maximal, some EVA versions are good within 10–20% for distributions defined
from the jet kinematics, but none are better than 50–100% for leptonic
observables.

##  5 Conclusions

We have reported on a recent calculation [2] of the full tower of NLO
corrections to like-sign W-boson scattering at the LHC, including all partonic
channels and W-boson decays. Our calculation, which is implemented in the
Monte Carlo integrator Bonsay, confirms the results of a previous calculation
(up to a glitch in a numerically unimportant contribution) and in particular
the occurrence of large pure EW corrections of the order of $\sim-12\%$ for
integrated cross sections.

Moreover, we have constructed a VBS approximation for the NLO prediction based
on partonic channels and gauge-invariant (sub)matrix elements featuring the
VBS subprocess and on resonance expansions of the W decays. The VBS
approximation reproduces the full NLO predictions within
$\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;1.5\%$
in the most important regions of phase space. Finally, we have discussed
results from different versions of effective vector-boson approximations at
LO, based on the collinear emission of W bosons of incoming (anti)quarks. In
line with previous findings for similarly constructed approximations, we find
that the approximative quality is only qualitative at the LHC owing to the
mild collinear enhancement of the W-boson emission and the design of VBS
analysis cuts, which excludes very forward/backward pointing jets.

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