# Orders of Oscillation Motivated by Sarnak’s Conjecture, Part II 1112010
Mathematics Subject Classification. Primary 37A35, 11K65; Secondary 37A25,
11N05 222Key words and phrases. oscillating sequence of order $d$, oscillating
sequence of order $d$ in the arithmetic sense, polynomial skew products on the
$d$-torus, MMA and MQDS flow. Chowla sequence.

Yunping JIANG

###### Abstract

I have investigated orders of oscillating sequences motivated by Sarnak’s
conjecture in [13] and proved that an oscillating sequence of order $d$ is
linearly disjoint from affine distal flows on the $d$-torus. One of the
consequences is that an oscillating sequence of order $d$ in the arithmetic
sense is linearly disjoint from affine flows with zero topological entropy on
the $d$-torus. In this paper, I will extend these results to polynomial skew
products on the $d$-torus, that is, given a polynomial skew product on the
$d$-torus, there is a positive integer $m$ such that any oscillating sequence
of order $m$ is linearly disjoint from this polynomial skew product. In
particular, when all polynomials depend only on the first variable, I have
that an oscillating sequence of order $m=d+k-1$ is linearly disjoint from all
polynomial skew products on the $d$-torus with polynomials of degree less than
or equal to $k$. One of the consequences is the linear disjointness for flows
which are automorphisms of the $d$-torus with absolute values of eigenvalues
$1$ plus a polynomial vector and oscillating sequences of order $m$ in the
arithmetic sense. Furthermore, I will prove that an oscillating sequence of
order $d$ is linearly disjoint from minimal mean attractable and minimal
quasi-discrete spectrum of order $d$ flows. Finally, I define and give some
examples of Chowla sequences from our paper [4].

##  1 Introduction

This paper uses ${\mathbb{N}}$, ${\mathbb{Z}}$, ${\mathbb{R}}$, and
${\mathbb{C}}$ to mean natural numbers, integers, real numbers, and complex
numbers. Let ${\mathbb{T}}$ be the unit circle in ${\mathbb{C}}$. Suppose
$d\geq 2$ is an integer. Let ${\mathbb{R}}^{d}$ be the $d$-Euclidean real
vector space and let ${\mathbb{Z}}^{d}$ be the integer lattice in
${\mathbb{R}}^{d}$. Let $\mathbb{T}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ be
the $d$-torus.

We use ${\bf c}=(c_{n})_{n\in{\mathbb{N}}}$ to denote a sequence of complex
numbers. An example is the Möbius sequence ${\bf
u}=(\mu(n))_{n\in\mathbb{N}}$, the one generated by the Möbius function
defined as $\mu(1)=1$, $\mu(n)=(-1)^{r}$ if $n$ is a product of $r$ distinct
prime numbers, and $\mu(n)=0$ if $p^{2}|n$ for a prime number $p$.

Suppose $X$ is a compact metric space with a metric $d(\cdot,\cdot)$ and
$f:X\to X$ is a (piece-wise) continuous map. We use $f^{n}$ to denote $n$
compositions of $f$ for $n\in\mathbb{N}$ and $f^{0}$ to denote the identity.
Then $\\{f^{n}\\}_{n=0}^{\infty}$ is a (discrete) dynamical system. Following
[14, 15], we call it or just $f$ a (discrete) flow. Let $C(X)$ be the space of
all continuous complex-valued functions on $X$ with the maximum norm
$\|\phi\|=\max_{x\in X}|\phi(x)|$ .

######  Definition 1.

We say ${\bf c}$ is linearly disjoint from (or orthogonal to) $f$ if for any
$\varphi\in C(X)$ and any $x\in X$,

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}c_{n}\varphi(f^{n}x)=0.$ (1)

Sarnak [14, 15] made a conjecture connecting number theory and ergodic theory
as follows.

######  Conjecture 1.

The Möbius sequence ${\bf u}$ is linearly disjoint from all flows $f$ with
zero topological entropy.

In the paper [13], I defined the order of an oscillating sequence ${\bf c}$ as
follows and showed why orders of oscillation on ${\bf c}$ is important for
understanding Conjecture 1.

######  Definition 2.

We call ${\bf c}=(c_{n})_{n\in{\mathbb{N}}}$ an oscillating sequence of order
$d\geq 2$ if for every real coefficient polynomial $P$ of degree $\leq d$,

$~{}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}c_{n}e^{2\pi iP(n)}=0$ (2)

with the technical condition (3),

$~{}\frac{1}{N}\sum_{n=1}^{N}|c_{n}|^{\lambda}\leq C,\quad\forall\;N\geq 1.$
(3)

I also defined a stronger version in the arithmetic sense.

######  Definition 3.

We call ${\bf c}=(c_{n})_{n\in{\mathbb{N}}}$ an oscillating sequence of order
$d\geq 2$ in the arithmetic sense if for every real coefficient polynomial $P$
of degree $\leq d$ and every pair of integers $0\leq l<k$,

$~{}\lim_{N\to\infty}\frac{1}{N}\sum_{1\leq n=mk+l\leq N}c_{n}e^{2\pi
iP(n)}=0$ (4)

with the technical condition (3).

######  Remark 1.

An oscillating sequence of order $1$ is just called an oscillating sequence
which is defined in [8].

Due to Davenport [6] and Hua [11], we know that the Möbius sequence ${\bf
u}=(\mu(n))_{n\in{\mathbb{N}}}$ is an example of an oscillating sequence of
order $d$ as well as in the arithmetic sense for all $d\geq 1$. In the view of
expanding dynamical systems, we found in [4] that for almost all $\beta>1$,
sequences $(e^{2\pi i\beta^{n}})_{n\in{\mathbb{N}}}$ are oscillating sequences
of order $d$ as well as in the arithmetic sense for all $d\geq 1$. We actually
proved a more general result (refer to Theorem 4 in section 4). It is still an
interesting problem to find a concrete number $\beta>1$ such that $(e^{2\pi
i\beta^{n}})_{n\in{\mathbb{N}}}$ is an example of an oscillating sequence of
order $d$ as well as in the arithmetic sense for all $d\geq 1$.

We are interested in the following conjecture.

######  Conjecture 2.

An oscillating sequence ${\bf c}$ of order $d$ for all $d\geq 2$ in the
arithmetic sense is linearly disjoint from flows $f$ on a torus with zero
topological entropy.

I have studied Conjecture 2 for all affine flows with zero topological entropy
on a torus in [13] as follows.

For ${\bf x}\in{\mathbb{T}}^{d}$, let ${\bf x}^{t}=(x_{1},\cdots,x_{d})$ mean
the transpose of ${\bf x}$. Let $GL(d,{\mathbb{Z}})$ be the space of all
$d\times d-$matrices $A$ of integer entries with determinants $\det(A)=\pm 1$.
For any $A\in GL(d,{\mathbb{Z}})$, the map $A{\bf x}$ is an automorphism of
${\mathbb{T}}^{d}$. For ${\bf a}\in{\mathbb{T}}^{d}$ with ${\bf
a}^{t}=(a_{1},\cdots,a_{d})$, we have an affine flow

$~{}T_{A,a}({\bf x})=A{\bf x}+{\bf a}:{\mathbb{T}}^{d}\to{\mathbb{T}}^{d}.$
(5)

A result of Sinai says that the flow $T_{A,a}$ has zero topological entropy if
and only if the absolute values of all eigenvalues of $A$ are $1$.
Furthermore, a result of Kronecker implies that all eigenvalues of $A$ are
roots of unity. Therefore, for some positive integer $m$, all eigenvalues of
$A^{m}$ are $1$. Without loss of generality, we only need to consider $A\in
GL(d,{\mathbb{Z}})$ such that its all eigenvalues are $1$. We have that

######  Lemma 1.

Suppose $A\in GL(d,{\mathbb{Z}})$ such that its all eigenvalues are $1$. Then
we have a matrix $P\in GL(d,{\mathbb{Z}})$ whose determinant is $1$ such that
$L=P^{-1}AP\in GL(d,{\mathbb{Z}})$ is a lower triangular matrix.

Thus without loss of generality, we only need to consider a lower triangular
matrix $A\in GL(d,{\mathbb{Z}})$ whose entries on the main diagonal are $1$.
Under this assumption, the map in (5) is an affine skew product,

$~{}T_{A,a}({\bf x})=\left(\begin{array}[]{c}x_{1}+a_{1}\\\
x_{2}+b_{21}x_{1}+a_{2}\\\ \vdots\\\
x_{d}+b_{d(d-1)}x_{d-1}\cdots+b_{d1}x_{1}+a_{d}\end{array}\right)$ (6)

where $b_{ij}$, $2\leq i\leq d$, $1\leq j\leq i-1$, are constants. In [13], I
proved the following theorem.

######  Theorem A.

An oscillating sequence ${\bf c}$ of order $d\geq 2$ is linearly disjoint from
affine skew products $T_{A,a}({\bf x})$ in (6) on the $d$-torus.

One of the consequences of Theorem A is that

######  Corollary A.

An oscillating sequence ${\bf c}$ of order $d\geq 2$ in the arithmetic sense
is linearly disjoint from affine flows $T_{A,a}$ on the $d$-torus with zero
topological entropy.

One purpose of this paper is to extend these results to some nonlinear flows
on a torus, which we will present in section 2. Another goal of this paper is
to generalize a result of Abdalaoui in [1] about quasi-discrete spectrum
flows, which we will present in section 3. Finally, I will define and
construct Chowla sequences, which will be presented in section 5.

Acknowledgment:  This material is based upon work supported by the National
Science Foundation from 2017 to 2019. It is also partially supported by a
collaboration grant from the Simons Foundation (grant number 523341) and PSC-
CUNY awards.

##  2 Polynomial skew products.

We call a map $f$ on the $d$-torus a polynomial skew products of degree $k$ if

$~{}f({\bf x})=\left(\begin{array}[]{c}x_{1}+a\\\ x_{2}+h_{2}(x_{1})\\\
x_{3}+h_{3}(x_{1},x_{2})\\\ \vdots\\\
x_{d}+h_{d}(x_{1},x_{2},\cdots,x_{d-1})\end{array}\right)$ (7)

where $a$ is a constant and $h_{2}(x_{1})$ is a polynomial of $x_{1}$ of
degree $\leq k$, $h_{3}(x_{1},x_{2})$ is a polynomial of $x_{1}$ and $x_{2}$
of degree $\leq k$, $\cdots$, $h_{d}(x_{1},\cdots,x_{d-1})$ is a polynomial of
$x_{1},\cdots,x_{d-1}$ of degree $\leq k$. We call $f$ a simple polynomial
skew product if it is in the form

$~{}f({\bf x})=\left(\begin{array}[]{c}x_{1}+a\\\ x_{2}+h_{2}(x_{1})\\\
x_{3}+b_{32}x_{2}+h_{3}(x_{1})\\\ \vdots\\\
x_{d}+b_{d(d-1)}x_{d-1}\cdots+b_{d2}x_{2}+h_{d}(x_{1})\end{array}\right)$ (8)

where $a$ and $b_{ij}$, $3\leq i\leq d$, $2\leq j\leq i-1$, are constants and
$h_{i}(x_{1})$, $2\leq i\leq d$, are polynomials of $x_{1}$ of degree $\leq
k$. One of the results in this paper is the following theorem.

######  Theorem 1.

An oscillating sequence ${\bf c}$ of order $m=d+k-1$ is linearly disjoint from
all simple polynomial skew products $f({\bf x})$ of degree $k$ in (8) on the
$d$-torus for any $d\geq 2$ and $k\geq 2$.

######  Corollary 1.

Suppose $A\in GL(d,{\mathbb{Z}})$ and all eigenvalues of $A$ have the absolute
values $1$. Let $q\geq 1$ be the smallest integer such that all eigenvalues of
$A^{q}$ are $1$. Suppose

$f^{q}({\bf x})=A^{q}{\bf x}+{\bf h}(x_{1})$

where ${\bf
h}^{t}(x_{1})=(a,h_{2}(x_{1}),\cdots,h_{d}(x_{1}))\in{\mathbb{T}}^{d}$ and $a$
is a constant and $h_{i}(x_{1})$, $2\leq i\leq d$, are polynomials of $x_{1}$
of degree $\leq k$ for $k\geq 2$. Then any oscillating sequence ${\bf c}$ of
order $m=d+k-1$ in the arithmetic sense is linearly disjoint from the flow
$f$.

######  Remark 2.

For $k=1$, they are Theorem A and Corollary A.

###### Proof of Theorem 1..

Let ${\bf k}\in{\mathbb{Z}}^{d}$ with ${\bf k}^{t}=(k_{1},\cdots,k_{d})$,
define

$e({\bf k}\cdot{\bf x})=e^{2\pi i(k_{1}x_{1}+\cdots+k_{d}x_{d})},\quad{\bf
x}\in{\mathbb{T}}^{d}.$

From the Stone-Weierstrass theorem, the set $S=\Big{\\{}e({\bf k}\cdot{\bf
x})\Big{\\}}_{{\bf k}\in{\mathbb{Z}}^{d}}$ forms a dense subset in
$C({\mathbb{T}}^{d})$. A linear combination $p$ of elements in $S$ is called a
trigonometric polynomial. We can write $p$ as

$p({\bf x})=\sum_{m_{1}\leq k_{1}\leq s_{1}}\cdots\sum_{m_{d}\leq k_{d}\leq
s_{d}}a_{\bf k}e^{2\pi i(k_{1}x_{1}+\cdots+k_{d}x_{d})}.$

For any $\phi\in C({\mathbb{T}}^{d})$, we have a sequence of trigonometric
polynomials

$~{}p_{q}({\bf x})=\sum_{m_{1q}\leq k_{1}\leq s_{1q}}\cdots\sum_{m_{dq}\leq
k_{d}\leq s_{dq}}a_{{\bf k},q}e^{2\pi i(k_{1}x_{1}+\cdots+k_{d}x_{d})}.$ (9)

such that $\|\phi-p_{q}\|\to 0$ as $q\to\infty$. The sequence
$\\{p_{q}\\}_{q\in{\mathbb{N}}}$ is called the trigonometric approximation of
$\phi$. Consider

$S_{N}\phi({\bf x})=\frac{1}{N}\sum_{n=1}^{N}c_{n}\phi(f^{n}{\bf x}).$

For any $\epsilon>0$, we have an integer $r>0$ such that

$\|\phi-p_{r}\|<\frac{\epsilon}{2C^{\frac{1}{\lambda}}}.$

Then

$S_{N}\phi({\bf x})=\Big{(}\frac{1}{N}\sum_{n=1}^{N}c_{n}\big{(}\phi(f^{n}{\bf
x})-p_{r}(f^{n}{\bf
x})\big{)}\Big{)}+\Big{(}\frac{1}{N}\sum_{n=1}^{N}c_{n}p_{r}(f^{n}{\bf
x})\Big{)}=I+II.$

For the estimation of $I$, we apply the Hölder inequality. Let $\lambda$ and
$C$ be the numbers in (3) and $\lambda^{\prime}>1$ be the dual number of
$\lambda$, that is,

$\frac{1}{\lambda}+\frac{1}{\lambda^{\prime}}=1.$

Then we have that

$|I|\leq\Big{(}\frac{1}{N}\sum_{l=1}^{N}|c_{n}|^{\lambda}\Big{)}^{\frac{1}{\lambda}}\Big{(}\frac{1}{N}\sum_{n=1}^{N}|\phi(f^{n}{\bf
x})-q_{r}(f^{n}{\bf
x})|^{\lambda^{\prime}}\Big{)}^{\frac{1}{\lambda^{\prime}}}\leq
C^{\frac{1}{\lambda}}\cdot\frac{\epsilon}{2C^{\frac{1}{\lambda}}}=\frac{\epsilon}{2}.$

For the estimation of $II$, let ${\bf x}_{n}=f^{n}{\bf x}$ for $n=0,1,\cdots$.
Denote

${\bf x}_{n}^{t}=(x_{1}^{n},\cdots,x_{d}^{n}).$

We have that

$x_{1}^{n}=x_{1}+na,$
$x_{2}^{n}=x_{2}+\sum_{l=0}^{n-1}h_{2}(x_{1}+la)=x_{2}+\sum_{l=0}^{n-1}\sum_{i=0}^{k}r_{i}(x+la)^{i}$
$=x_{2}+\sum_{i=0}^{k}r_{i}\sum_{l=0}^{n-1}(x+la)^{i}=x_{2}+\sum_{i=0}^{k}r_{i}\sum_{l=0}^{n-1}\sum_{j=0}^{i}\binom{i}{j}x^{i-j}(la)^{j}$
$=x_{2}+\sum_{i=0}^{k}r_{i}\sum_{j=0}^{i}\sum_{l=0}^{n-1}\binom{i}{j}x^{i-j}(la)^{j}=y+\sum_{i=0}^{k}\Big{(}r_{i}\sum_{j=0}^{i}\binom{i}{j}x^{i-j}a^{j}\Big{)}\Big{(}\sum_{l=0}^{n-1}l^{j}\Big{)}.$

Since $\sum_{l=0}^{n-1}l^{j}$ is a polynomial of $n$ of degree $j+1$ for
$1\leq j\leq k$, $x_{2}^{n}$ is a polynomial of $n$ of degree $\leq
k+1=2+k-1$. Next, we calculate

$x_{3}^{n}=x_{3}+(b_{32}x_{2}+b_{32}b_{31}x_{1})n+b_{31}a\sum_{j=1}^{n-1}j+b_{32}\sum_{j=1}^{n-2}\sum_{l=0}^{j}h_{2}(x_{1}+la)+\sum_{l=0}^{n-1}h_{3}(x_{1}+la).$

So we see that $x_{3}^{n}$ is a polynomial of $n$ of degree at most
$k+2=3+k-1$. In general,

$x_{i}^{n}=x_{i}^{n-1}+b_{i(i-1)}x_{i-1}^{n-1}+\cdots+b_{i2}x_{2}^{n-1}+\sum_{l=0}^{n-1}h_{i}(x_{1}+la),\;\;4\leq
i\leq d.$

Inductively, one can see that the term $b_{i2}x_{2}^{n-1}$ in $x_{i}^{n}$
contains

$\sum_{j_{1}=0}^{n-2}\sum_{j_{2}=0}^{j_{1}}\cdots\sum_{j_{i-2}=0}^{j_{i-3}}\sum_{l=0}^{j_{i-2}}h_{2}(x_{1}+la).$

So $x^{n}_{i}$ is a polynomial of degree $i+k-1$. Thus we have that

${\bf x}_{n}^{t}=(P_{1}(n),\cdots,P_{d}(n)),$

where $P_{1}(n)$ is polynomial of degree $1$ and $P_{i}(n)$ is a polynomial of
degree at most $i+k-1$ for every $2\leq i\leq d$. Now using the formula (9),
we have that

$p_{r}(f^{n}{\bf x})=\sum_{m_{1r}\leq k_{1}\leq s_{1r}}\cdots\sum_{m_{dr}\leq
k_{d}\leq s_{dr}}a_{{\bf k},r}e^{2\pi iP_{{\bf k},r}(n)},$

where

$P_{{\bf k},r}(n)=c_{{\bf k},r,d}n^{d+k-1}+\cdots+c_{{\bf k},r,1}n+c_{{\bf
k},r,0}$

is a real coefficient polynomial of degree at most $d+k-1$.

After the above calculations, we get

$|II|=\Big{|}\frac{1}{N}\sum_{n=1}^{N}c_{n}\sum_{m_{1r}\leq k_{1}\leq
s_{1r}}\cdots\sum_{m_{dr}\leq k_{d}\leq s_{dr}}a_{{\bf k},r}e^{2\pi iP_{{\bf
k},r}(n)}\Big{|}$ $=\Big{|}\sum_{m_{1r}\leq k_{1}\leq
s_{1r}}\cdots\sum_{m_{dr}\leq k_{d}\leq s_{dr}}a_{{\bf
k},r}\frac{1}{N}\sum_{n=1}^{N}c_{n}e^{2\pi iP_{{\bf k},r}(n)}\Big{|}.$

Let

$L=\max\\{|m_{1r}|,\cdots,|m_{dr}|,|s_{1r}|,\cdots,|s_{dr}|,|a_{{\bf
k},r}|\;|\;m_{jr}\leq k_{j}\leq s_{jr},1\leq j\leq d\\}.$

Since ${\bf c}$ is an oscillating sequence of order $d+k-1$, we can find an
integer $M>r$ such that for $N>M$,

$\Big{|}\frac{1}{N}\sum_{n=1}^{N}c_{n}e^{2\pi iP_{{\bf
k},r}(n)}\Big{|}<\frac{\epsilon}{2L^{d}},\quad\forall m_{1r}\leq k_{1}\leq
s_{1r},\cdots,m_{dr}\leq k_{n}\leq s_{dr}.$

This implies that $|II|<\epsilon/2.$ Therefore, we get that for all $N>M$,

$|S_{N}\phi({\bf x})|<\epsilon.$

This says that $\lim_{N\to\infty}S_{N}\phi({\bf x})=0$. We completed the
proof. ∎

Theorem 1 can be generalized to the following theorem.

######  Theorem 2.

Suppose $f$ is a polynomial skew products of degree $k$ on the $d$-torus in
the form (7). Then there is a positive integer $m=m(d,k,h_{1},\cdots,h_{d})$
such that any oscillating sequence ${\bf c}$ of order $m$ is linearly disjoint
from $f$.

###### Proof.

The most of the proof is similar to the proof of Theorem 1. Here we just
outline some different steps. Let ${\bf x}_{n}=f^{n}{\bf x}$ for
$n=0,1,\cdots$. Denote ${\bf x}_{n}^{t}=(x_{1}^{n},\cdots,x_{d}^{n})$. Then

$x_{1}^{n}=x_{1}+na$

is a linear map of $n$,

$x_{2}^{n}=x_{2}+\sum_{l=0}^{n-1}h_{2}(x_{1}+la),$

is a polynomial of $n$ of degree $k_{2}\leq k+1$,

$x_{3}^{n}=x_{3}+\sum_{j=1}^{n-2}\sum_{l=0}^{j}h_{2}(x_{1}+la)+\sum_{l=0}^{n-1}h_{3}(x_{1}+la,x_{2}+\sum_{s=0}^{r-1}h_{2}(x_{1}+sa)).$

is a polynomial of $n$ of degree $k_{3}\leq k^{2}+1$, which depends on the
terms of $x_{1}$ and $x_{2}$ appeared in $h_{3}$. Inductively, we have that

$x_{i}^{n}=x_{i}^{n-1}+\sum_{l=0}^{n-1}h_{i}(x^{n-1}_{1},x_{2}^{n-1},\cdots,x_{i-1}^{n-1})$

is a polynomial of $n$ of degree $k_{i}\leq k^{i-1}+1$ for $4\leq i\leq d$.
Thus we have that

${\bf x}_{n}^{t}=(P_{1}(n),\cdots,P_{d}(n)),$

where $P_{1}(n)$ is polynomial of degree $1$ and $P_{i}(n)$ is a polynomial of
degree at most $k_{i}$ for every $2\leq i\leq d$ and
$m=\max\\{1,k_{2},\cdots,k_{d}\\}$. ∎

######  Corollary 2.

Suppose $A\in GL(d,{\mathbb{Z}})$ and all eigenvalues of $A$ have the absolute
values $1$. Let $q\geq 1$ be the smallest integer such that all eigenvalues of
$A^{q}$ are $1$. Suppose

$f^{q}({\bf x})=A^{q}{\bf x}+{\bf h}$

where ${\bf
h}^{t}=(a,h_{2}(x_{1}),h_{3}(x_{1},x_{2}),\cdots,h_{d}(x_{1},\cdots,x_{d-1}))\in{\mathbb{T}}^{d}$
and $h_{i}(x_{1},\cdots,x_{i-1})$, $2\leq i\leq d$, are polynomials of
$x_{1},\cdots,x_{i-1}$ of degree $\leq k$. Then there is a positive integer
$m=m(d,k,h_{1},\cdots,h_{d})$ such that any oscillating sequence ${\bf c}$ of
order $m$ in the arithmetic sense is linearly disjoint from the flow $f$.

##  3 Minimal Mean Attractable and Minimal Quasi-Discrete Spectrum Flows.

In [8], we extended the concept of the mean-Lyapunov-stability (abbreviated
MLS) of Fomin [7] to the concept of minimal MLS (abbreviated MMLS). We also
define a new concept called minimal attractability (abbreviated MMA) in [8].
We studied the linear disjointness of an oscillating sequence (of order $1$)
from MMA and MMLS flows using these two concepts. In this paper, we will
extend the concept of the quasi-discrete spectrum (abbreviated QDS) of
Abramov, Hahn, and Parry [3, 10] to the concept of the minimal QDS
(abbreviated MQDS) and study the linear disjointness of an oscillating
sequence of order $d$ from MMA and MQDS(d) flows.

Suppose $K$ is a subset of $X$. We say $K$ is minimal (respective to $f$) if
$f:K\to K$ and for any $x\in K$, the closure of the forward orbit of $x$ is
$K$, that is,

$\overline{\\{f^{n}x\\}_{n=0}^{\infty}}=K.$

######  Definition 4 (MMA).

Suppose $K\subseteq X$ is minimal. We say $x\in X$ is mean attracted to $K$ if
for any $\epsilon>0$, there is a point $z=z_{\epsilon,x}\in K$ such that

$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}d(f^{n}x,f^{n}z)<\epsilon.$ (10)

The basin of attraction of $K$, denoted as $\hbox{\rm Basin}(K)$, is defined
to be the set of all points $x$ which are mean attracted to $K$. We call $f$
minimally mean attractable (abbreviated MMA) if

$X=\bigcup_{K}\mbox{\rm Basin}(K)$ (11)

where $K$ varies among all minimal subsets of $X$.

######  Remark 3.

It is clear that $K\subseteq\hbox{\rm Basin}(K)$. The union in the above
definition could be uncountable.

We use [9] as a general reference in the reviewing of a QDS flow. Suppose $G$
is an Abelian group in multiplication and $\Lambda:G\to G$ is homomorphism.
Consider another homomorphism $\Phi:G\to G$ defined as

$\Phi g:=g\cdot\Lambda g,\quad g\in G.$

Then we have that

$\Phi^{n}g=\prod_{j=0}^{n}(\Lambda^{j}g)^{\binom{n}{j}},\quad g\in G.$

Let $G_{n}=\ker(\Lambda^{n})$ be a subgroup of $G$. Then we have a filter

$\\{1\\}=G_{0}\subseteq G_{1}\subseteq G_{n}\subseteq G_{n+1}\subseteq\cdots$

and $\Lambda:G_{n}\to G_{n-1}$ for all $n\geq 1$. Note that $\Phi:G_{n}\to
G_{n}$ is an automorphism for every $n\geq 1$. The homomorphism $\Lambda:G\to
G$ is called nilpotent if $G=G_{n}$ for some $n\geq 0$. It is called quasi-
nilpotent if $G=\cup_{n=0}^{\infty}G_{n}$.

A triple $(G,\Lambda,\iota)$ is called a signature if $G$ is an Abelian group,
$\Lambda:G\to G$ is a quasi-nilpotent homomorphism and

$\iota:G_{1}\to{\mathbb{T}}^{2}$

is an injective homomorphism (i.e., monomorphism), where
$G_{1}=\ker(\Lambda)$. The order of the signature $(G,\Lambda,\iota)$ is

$ord(G,\Lambda,\iota):=\inf\\{n\in{\mathbb{N}}\;|\;G=G_{n}\\}\in{\mathbb{N}}\cup\\{\infty\\}.$

Note that the order is infinite if $G\not=G_{n}$ for all $n\in{\mathbb{N}}$,
that is, $G$ is not nilpotent.

Now let us return to $f:X\to X$. Suppose $K\subseteq X$ is minimal. Consider
the Koopman operator $\Phi:C(K)\to C(K)$ defined as

$\Phi\phi=\phi\circ f,\quad\phi\in C(K).$

Let

$G=\\{g\in C(K)\;|\;|g|=1\\}.$

Then it is an Abelian group in multiplication. Consider the homomorphism
$\Lambda:G\to G$ defined as

$\Lambda g=\Phi g\cdot\overline{g},\quad g\in G.$

Then

$\Phi g=g\cdot\Lambda g,\quad g\in G.$

One can see that $G_{1}=Fix(\Phi)\cap G$ where $Fix(\Phi)$ is the set of all
fixed points of $\Phi$. Since $K$ is minimal, we have that
$G_{1}={\mathbb{T}}$ and

$Fix(\Phi)=span_{{\mathbb{C}}}(G_{1})={\mathbb{C}}.$

Let us consider $G_{2}=\ker(\Lambda^{2})$. For any $g\in G_{2}$, we have
$\Lambda^{2}g=1$. This implies that

$\Phi^{2}g=g\cdot(\Lambda g)^{2}\cdot\Lambda^{2}g=\Lambda g\cdot\Phi g$

Since $\Lambda g\in G_{1}$, $\lambda=\Lambda g$ with $|\lambda|=1$ is a
unimodular eigenvalue of $\Phi$ with an unimodular eigenvector
$g^{\prime}=\Phi g\in G_{2}$. Conversely, suppose $\lambda$ is a unimodular
eigenvalue of $\Phi$ with a unimodular eigenvector $g$, that is, $\Phi
g=\lambda g$ and $|\lambda|=1$. Then $|g|=1\in G_{1}$ and $\Lambda g=\Phi
g\cdot\overline{g}=\lambda g\cdot\overline{g}=\lambda\in G_{1}$. This says
that $g\in G_{2}$. Thus $G_{2}$ is the set of all unimodular eigenvectors of
$\Phi$. In general, elements in $G_{n}$ are called unimodular quasi-
eigenvectors of order $n-1$. We say that $f:K\to K$ is a quasi-discrete
spectrum (abbreviated QDS) flow if the linear hull of all unimodular quasi-
eigenvectors is dense in $C(K)$, that is, $G=\cup_{n=0}^{\infty}G_{n}$ and

$\overline{span_{{\mathbb{C}}}(G)}=C(K).$

We say that $f:K\to K$ is a quasi-discrete spectrum of order $d$ (abbreviated
QDS(d)) if it is a QDS flow and $ord(G,\Lambda,\iota)=d+1$.

######  Definition 5 (MMA and MQDS).

We call $f:X\to X$ a MMA and MQDS (d) flow if $f$ is MMA and for every minimal
set $K\subseteq X$, $f:K\to K$ is QDS (k) for some $k\leq d$.

Now we can state another result in this paper.

######  Theorem 3.

An oscillating sequence ${\bf c}$ of order $d\geq 1$ is linearly disjoint from
MMA and MQDS (d) flows.

###### Proof.

Let $\lambda>1$ and $C$ be the numbers in (3). Let $\lambda^{\prime}>1$ be the
dual number of $\lambda$, that is,

$\frac{1}{\lambda}+\frac{1}{\lambda^{\prime}}=1.$

For any $x\in X$ and any $\phi\in C(X)$, we need to prove that (1) holds.
Since $f$ is MMA, $x\in Basin(K)$ for some minimal subset $K\subseteq X$.
Assume that we already know that (1) holds for $z\in K$. For an arbitrarily
small number $\epsilon>0$, the uniform continuity of $\phi$ implies that there
is a $\delta>0$ such that

$|\phi(u)-\phi(v)|<\frac{\epsilon}{2^{\frac{1+\lambda^{\prime}}{\lambda^{\prime}}}C^{\frac{\lambda^{\prime}}{\lambda}}}$

whenever $d(u,v)<\delta$. Since $f$ is MMA, there exists $z=z_{\delta,x}\in K$
such that

$\limsup_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}d(f^{n}x,f^{n}z)<\delta^{2}.$

This implies that

$\overline{D}(E)=\limsup_{n\to\infty}\frac{\\#(E\cap[1,N])}{N}\leq\delta$

where $E=\\{n\geq 1\;|\;d(f^{n}x,f^{n}z)\geq\delta\\}$, because

$\delta\sharp(E\cap[1,N])\leq\sum_{n=1}^{N}d(f^{n}x,f^{n}z).$

Consider

$S_{N}\phi(x)=\frac{1}{N}\sum_{n=1}^{N}c_{n}\phi(f^{n}x).$

Write

$S_{N}\phi(x)=\Big{(}S_{N}\phi(x)-S_{N}\phi(z)\Big{)}+S_{N}\phi(z)=I+II$

Since we assume that (1) holds for $z\in K$, we have a $N_{0}>0$ such that
$|II|=|S_{N}\phi(z)|<\epsilon/2$ for all $N>N_{0}$. From the Hölder
inequality,

$|I|=|S_{N}\phi(x)-S_{N}\phi(z)|\leq\Big{(}\frac{1}{N}\sum_{n=1}^{N}|c_{n}|^{\lambda}\Big{)}^{\frac{1}{\lambda}}\Big{(}\frac{1}{N}\sum_{n=1}^{N-1}|\phi(f^{n}x)-\phi(f^{n}y)|^{\lambda^{\prime}}\Big{)}^{1/\lambda^{\prime}}$
$\leq
C^{\frac{1}{\lambda}}\Big{(}\frac{1}{N}\sum_{n=1}^{N-1}|\phi(f^{n}x)-\phi(f^{n}y)|^{\lambda^{\prime}}\Big{)}^{1/\lambda^{\prime}}.$

This implies that

$|I|\leq C^{\frac{1}{\lambda}}\Big{(}\frac{1}{N}\sum_{n\in[1,N]\setminus
E}|\phi(f^{n}x)-\phi(f^{n}y)|^{\lambda^{\prime}}+\frac{1}{N}\sum_{n\in
E}|\phi(f^{n}x)-\phi(f^{n}y)|^{\lambda^{\prime}}\Big{)}^{1/\lambda^{\prime}}$
$\leq
C^{\frac{1}{\lambda}}\Big{(}\frac{\epsilon^{\lambda^{\prime}}}{2^{1+\lambda^{\prime}}C^{\frac{\lambda^{\prime}}{\lambda}}}+(2\|\phi\|_{\infty})^{\lambda^{\prime}}\delta\Big{)}^{\frac{1}{\lambda^{\prime}}}.$

Here we can take $\delta$ small enough such that

$(2\|\phi\|_{\infty})^{\lambda^{\prime}}\delta<\frac{\epsilon^{\lambda^{\prime}}}{2^{1+\lambda^{\prime}}C^{\frac{\lambda^{\prime}}{\lambda}}}.$

So we have that

$|S_{N}\phi(x)|\leq|I|+|II|\leq\epsilon$

for all $N>N_{0}$. We proved (1) under the assumption.

Now we prove the assumption that for any $z\in K$, (1) holds. Since $f:K\to K$
is a QDS (d) flow, we have that

$C(K)=\overline{span_{{\mathbb{C}}}(G_{d+1})}.$

Similar to the proof of Theorem 1, we need only to prove (1) for any $g\in
G_{d+1}$ and any $z\in K$. Since $\Lambda^{d+1}g=1$, For any $n\geq d+1$, we
have

$\Phi^{n}g(z)=g\circ
f^{n}(z)=\prod_{j=0}^{n}(\Lambda^{j}g(z))^{\binom{n}{j}}=\prod_{j=0}^{d}(\Lambda^{j}g(z))^{\binom{n}{j}}.$

Since $\Lambda^{j}g\in G_{d+1-j}$ for $0\leq j\leq d$, we have
$\Lambda^{j}g(z)=e^{2\pi i\theta_{j}}$ for some real number
$\theta_{j}\in[0,2\pi]$. This implies that

$\Phi^{n}g(z)=e^{2\pi i\sum_{j=0}^{d}\theta_{j}\binom{n}{j}}=e^{2\pi iP(n)}$

where

$P(n)=\sum_{j=0}^{d}a_{j}n^{j}$

is a real-coefficient polynomial of $n$ of degree $\leq d$. Now

$S_{N}g(z)=\frac{1}{N}\sum_{n=1}^{N}c_{n}g\circ
f^{n}=\frac{1}{N}\sum_{n=1}^{d}c_{n}\Phi^{n}g(z)+\frac{1}{N}\sum_{n=d+1}^{N}c_{n}\Phi^{n}g(z)$
$=\frac{1}{N}\sum_{n=1}^{d}c_{n}\Phi^{n}g(z)+\frac{1}{N}\sum_{n=d+1}^{N}c_{n}e^{2\pi
iP(n)}.$

Since ${\bf c}$ is an oscillating sequence of order $d$,

$\lim_{n\to\infty}\frac{1}{N}\sum_{n=d+1}^{N}c_{n}e^{2\pi P(n)}=0.$

Therefore, we have that

$\lim_{n\to\infty}S_{N}g(z)=0.$

We completed the proof of the assumption and thus the proof of the theorem. ∎

######  Remark 4.

From Theorem 1 and Theorem 2 and Theorem 3, we can see that in general the
flow in the form (8) or (7) will not be a MQDS(d) flow if $k\geq 2$.

##  4 Sarnak’s conjecture

One of the consequences of this paper is the confirmation of Conjecture 1 for
all polynomial skew products (for $d=2$, it is a consequence of the result in
[19]. Still, we have a completely different approach, and hopefully, this
proof will lead to a new proof of the result of Wang in [19] and Conjecture 2
for $d=2$. For $d>2$, it is a new result) and for all MMA and MQDS (d) flows
in this paper.

##  5 Chowla Sequences.

It has been a question that if it is enough to study Sarnak’s conjecture just
through the study of orders of oscillation sequences? It turns out it is not
the case. The paper [1] contains an example of an oscillating sequence of
order $d$ for all $d\geq 2$, which is not linear disjoint from a flow with
zero topological entropy. So we would also like to study Chowla sequences
defined as follows.

######  Definition 6 (Chowla Sequence).

We call a sequence ${\bf c}$ of complex numbers a Chowla sequence if

$~{}\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}c_{n+l_{1}}^{k_{1}}c_{n+l_{2}}^{k_{2}}\cdots
c_{n+l_{r}}^{k_{r}}=0$ (12)

for all $r\geq 1$, all choices of integers $0\leq l_{1}<l_{2}<\cdots<l_{r}$
and $k_{i}\in{\mathbb{N}}$, $i=1,2,\cdots,r$, such that not all

$c_{n+l_{i}}^{k_{i}}=|c_{n+l_{i}}|,\quad 1\leq i\leq r$

with the control condition (3).

Chowla’s conjecture [5] states that the Möbius sequence ${\bf u}$ is a Chowla
sequence. Due to the recent work of Sanark [14, 15], Veech [18] and Tao [16,
17] (see also [2]), Chowla’s conjecture is equivalent to Sarnak’s conjecture.
In this section, we give some examples of Chowla sequences using our work in
[4].

Take a non empty interval $I$ in the real line ${\mathbb{R}}$, which can be
closed, open or semi-open. Let $\mathcal{C}^{k}_{+}(I)$ be the space of all
positive real valued $k$-times continuously differentiable functions on an
interval $I$, whose $i$-th derivative is non-negative for $i\leq k$. Then it
is closed under addition and multiplication, that is, if
$f,g\in\mathcal{C}^{k}_{+}(I)$, then $f+g,fg\in\mathcal{C}^{k}_{+}(I)$. In
what follows, we often use this closure property of $\mathcal{C}^{k}_{+}(I)$.

Let $g$ be a function in $C^{2}_{+}((0,\infty))$ and $\alpha\not=0$ and
$\beta>1$ be real numbers. Consider the sequence of complex number ${\bf
c}=(c_{n})_{n\in{\mathbb{N}}}$ where

$c_{n}=\exp\Big{(}2\pi i(\alpha\beta^{n}g(\beta))\Big{)}=e^{2\pi
i\big{(}\alpha\beta^{n}g(\beta)\big{)}}.$

Then we have that

$c_{n+l_{1}}^{k_{1}}c_{n+l_{2}}^{k_{2}}\cdots
c_{n+l_{r}}^{k_{r}}=\exp\Big{(}2\pi
i\big{(}\sum_{i=1}^{r}k_{i}\big{(}\alpha\beta^{n+l_{i}}g(\beta)\big{)}\big{)}\Big{)}$

This says that in this case, (12) relates with the uniformly distributed
modulo $1$ (abbreviated u. d. mod $1$) of the sequence

$\Big{(}\sum_{i=1}^{r}k_{i}\big{(}\alpha\beta^{n+l_{i}}g(\beta)\big{)}\Big{)}_{n\in{\mathbb{N}}}$

on the unit interval $[0,1]$ as follows.

For a real number $x$, let $\\{x\\}=x\mod{1}$ be the fractional part of $x$.

######  Definition 7 (Uniform Distribution).

We say a sequence ${\bf x}=(x_{n})_{n\in{\mathbb{N}}}$ of real numbers is u.
d. mod $1$ if for any $0\leq a<b\leq 1$, we have

$\lim_{N\to\infty}\frac{\\#(\\{n\in[1,N]\;|\;\\{x_{n}\\}\in[a,b]\\})}{N}=b-a.$

######  Theorem B (The Weyl Criterion).

The sequence ${\bf x}=(x_{n})_{n\in{\mathbb{N}}}$ is u. d. mod $1$ if and only
if

$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi ihx_{n}}=0\quad\hbox{for
all integers $h\not=0$}.$

The following theorem is essentially [4, Theorem 1] (that is, take $l=0$ in
the proof in [4]). For the reader’s convenience, I will also provide detailed
proof in this paper.

######  Theorem 4.

Let us take $g\in\mathcal{C}^{2}_{+}((1,\infty))$. Then, for a fixed real
number $\alpha\not=0$ and almost all real numbers $\beta>1$ (alternatively,
for a fixed real number $\beta>1$ and almost all real numbers $\alpha$) and
for all $r\geq 1$, all choices of integers $0\leq l_{1}<l_{2}<\cdots<l_{r}$
and $k_{i}\in{\mathbb{N}}$, $i=1,2,\cdots,r$, sequences

$~{}\Big{(}\sum_{i=1}^{r}k_{i}\big{(}\alpha\beta^{n+l_{i}}g(\beta)\big{)}\Big{)}_{n\in{\mathbb{N}}}$
(13)

are u. d. mod $1$.

The following corollary is a consequence of this theorem because of the Weyl
criterion.

######  Corollary 3.

Let us take $g\in\mathcal{C}^{2}_{+}((1,\infty))$. Then, for a fixed real
number $\alpha\not=0$ and almost all real numbers $\beta>1$ (alternatively,
for a fixed real number $\beta>1$ and almost all real numbers $\alpha$),
sequences

${\bf c}=\Big{(}\exp(2\pi
i\big{(}\alpha\beta^{n}g(\beta)\big{)}\Big{)}_{n\in{\mathbb{N}}}$

are Chowla sequences.

######  Remark 5.

Again it is an interesting problem to find a concrete number of $\beta>1$ such
that

${\bf c}=\big{(}e^{2\pi i\beta^{n}}\big{)}_{n\in{\mathbb{N}}}$

is a Chowla sequence.

######  Theorem C (Koksma’s Theorem).

Let $(y_{n}(x))_{n\in{\mathbb{N}}}$ be a sequence of real valued $C^{1}$
functions defined on an interval $[a,b]$. Suppose
$y_{m}^{\prime}(x)-y_{n}^{\prime}(x)$ is monotone on $[a,b]$ for any two
integers $m\not=n$ and suppose

$\inf_{m\not=n}\min_{x\in[a,b]}|y_{m}^{\prime}(x)-y_{n}^{\prime}(x)|>0.$

Then for almost all $x\in[a,b]$, the sequence ${\bf
y}=(y_{n}(x))_{n\in{\mathbb{N}}}$ is u. d. mod $1$.

The reader who is interested in Theorem B and Theorem C can find proofs in
[12].

###### Proof of Theorem 4.

Given a choice of $0\leq l_{1}<l_{2}<\cdots<l_{r}$ and $k_{i}\in{\mathbb{N}}$,
$i=1,2,\cdots,r$, consider the sequence

$\sum_{i=1}^{r}k_{i}\beta^{n+l_{i}}g(\beta)=\beta^{n}g(\beta)\sum_{i=1}^{r}k_{i}\beta^{l_{i}}=G(\beta)\beta^{n}$

where $G(\beta)=g(\beta)\sum_{i=1}^{r}k_{i}\beta^{l_{i}}\in
C^{2}_{+}((0,\infty))$ since both $g(\beta)$ and
$\sum_{i=1}^{r}k_{i}\beta^{l_{i}}$ are in $C^{2}_{+}((0,\infty))$. Define a
function

$y_{n}(x)=G(x)x^{n},\quad x\in(1,\infty).$

For $n>m$, we have

$y^{\prime}_{n}(x)-y^{\prime}_{m}(x)=G(x)(nx^{n-1}-mx^{m-1})+G^{\prime}(x)(x^{n}-x^{m})$

Since $G^{\prime},G^{\prime\prime}\geq 0$ and since $nx^{n-m}-m\geq n-m\geq 1$
for $x>1$, we see that every term in the expression of
$y^{\prime}_{n}(x)-y^{\prime}_{m}(x)$ are in $\mathcal{C}^{1}_{+}([a,\eta])$
for any $1<a<\eta<\infty$. By the closure property of
$\mathcal{C}^{1}_{+}([a,\eta])$, we have that

$~{}y^{\prime}_{n}(x)-y^{\prime}_{m}(x)\in\mathcal{C}^{1}_{+}([a,\eta])\quad\forall
n>m.$ (14)

In particular, this imply that $y^{\prime}_{n}-y^{\prime}_{m}$ is increasing
for $n>m$. Furthermore, we see that there is a constant $L>0$ such that

$~{}|y_{n}^{\prime}(x)-y^{\prime}_{m}(x)|\geq L\quad\forall
n>m\in{\mathbb{N}},\;\forall a\leq x\leq\eta.$ (15)

(14) and (15) say that the sequence of real valued $C^{1}$ functions
$(y_{n}(x))_{n\in{\mathbb{N}}}$ satisfies all hypothesizes of Theorem C.

Theorem C implies that for almost all $x$ in
$[(2^{k}+1)/2^{k},(2^{k-1}+1)/2^{k-1}]$ or $[k,k+1]$ for $k\geq 2$, the
sequence $(\alpha y_{n}(x))_{n\in{\mathbb{N}}}$ is u. d. mod $1$. Further,
this implies that for almost all

$x\in(1,\infty)=\cup_{k=2}^{\infty}\Big{[}\frac{2^{k}+1}{2^{k}},\frac{2^{k-1}+1}{2^{k-1}}\Big{]}\cup\cup_{k=2}^{\infty}[k,k+1]$

the sequence $(\alpha y_{n}(x))_{n\in{\mathbb{N}}}$ is u. d. mod $1$.

Let

$A_{r,(l_{1},\cdots,l_{r}),(k_{1},\cdots,k_{r})}=\\{\beta>1\ |\
\alpha\beta^{n}g(\beta)\sum_{i=1}^{r}k_{i}\beta^{l_{i}}\text{ is not
u.\,d.\,mod\;$1$ }\\}.$

Then the one dimensional Lebesgue measure of
$A_{r,(l_{1},\cdots,l_{r}),(k_{1},\cdots,k_{r})}$ is zero. Since the set

$U=\\{(r,(l_{1},\dots,l_{r}),(k_{1},\cdots k_{r}))\ |\
r,k_{1},\cdots,k_{r},l_{2}<\cdots<l_{r}\in{\mathbb{N}},l_{1}\in{\mathbb{N}}\cup\\{0\\},l_{1}<l_{2}\\}$

is countable, the one dimensional Lebesgue measure of

$\bigcup_{(r,(l_{1},\dots,l_{r}),(k_{1},\cdots k_{r}))\in
U}A_{r,(l_{1},\cdots,l_{r}),(k_{1},\cdots,k_{r})}$

is zero too.

For a fixed real number $\alpha\neq 0$ in the theorem, take a real number

$\beta\in(1,\infty)\setminus\bigcup_{(r,(l_{1},\dots,l_{r}),(k_{1},\cdots
k_{r}))\in U}A_{r,(l_{1},\cdots,l_{r}),(k_{1},\cdots,k_{r})}.$

This says that the sequence

$\Big{(}\sum_{i=1}^{r}\alpha\beta^{n+l_{i}}g(\beta)\Big{)}_{n\in{\mathbb{N}}}$

is u. d. mod $1$ for all

$(r,(l_{1},\dots,l_{r}),(k_{1},\cdots k_{r}))\in U.$

We proved the theorem. ∎

## References

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http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.

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Yunping Jiang: Department of Mathematics, Queens College of the City
University of New York, Flushing, NY 11367-1597 and Department of Mathematics
Graduate School of the City University of New York 365 Fifth Avenue, New York,
NY 10016

<EMAIL_ADDRESS>


Deriving experimental constraints on the scalar form factor in the second-
class $\tau\to\eta\pi\nu$ mode

B. Moussallam1

1 Laboratoire Irène Joliot-Curie, Université Paris-Saclay, 91406 Orsay, France

16th International Workshop on Tau Lepton Physics (TAU2021),
September 27 – October 1, 2021 10.21468/SciPostPhysProc.?

## Abstract

The rare second-class decay mode of the $\tau$ into $\eta\pi\nu$ could be
observed for the first time at Belle II. It is important to try to derive a
reliable evaluation of the branching fraction and of the energy distribution
of this mode within the standard-model. Many predictions exist already in the
literature which can differ by one to two orders of magnitude. Here, an
approach based on a systematic use of the property of analyticity of form
factors and scattering amplitudes in QCD is discussed. In particular, we will
show that the scalar form factor in the $\tau$ decay can be related to photon-
photon scattering and radiative $\phi$ decay amplitudes for which precise
experimental measurements have been performed by the Belle and KLOE
collaborations.

##  1 Introduction

The decay $\tau\to\eta\pi\nu$ is induced by second-class type currents [1]. In
the Standard Model (SM) the amplitude must thus be proportional to the small
isospin breaking parameters: $(m_{d}-m_{u})/m_{s}$, $e^{2}$ and it has a
sensitivity to new scalar or tensor interactions (e.g. [2, 3, 4]). As Tony
Pich has reminded us in the introductory talk, this mode was once claimed to
have been observed [5] with an unexpectedly large branching fraction
$BF=(5.1\pm 1.5)\%$ but this was not confirmed. The best upper bounds now
available are $BF<7.3\times 10^{-5}$ (Belle [6]) and $BF<9.9\times 10^{-5}$
(Babar [7]). At the new Belle II facility, it has been estimated [8] that the
$\tau\to\eta\pi\nu$ decay could be observed with a significance of $2.6\sigma$
for a branching fraction of $4.4\times 10^{-5}$. A second-class scalar current
would lead to a simple modification of the scalar form factor (defined below
(2)) by a term linear in energy

$f_{0}^{\eta\pi}(s)=\left.f_{0}^{\eta\pi}(s)\right|_{SM}\left(1-\epsilon_{S}\frac{s}{m_{\tau}(m_{d}-m_{u})}\right)$
(1)

proportional to the strength $\epsilon_{S}$ (with the notation of [9]) of the
new interaction. A correct estimate of the scalar form factor in the SM is
necessary in order to derive reliable constraints on $\epsilon_{S}$ from
experimental results.

A number of evaluations of the $\eta\pi$ vector and scalar form factors have
already been performed (e.g. [10, 11, 12, 3, 13, 14, 15, 16] for a
representative list). The predictions for the branching fraction associated
with the vector form factor lie in a range $BF_{V}\simeq[0.1-0.8]\cdot
10^{-5}$ while those associated with the scalar form factor seem more
uncertain: $BF_{S}\simeq[0.3-5.3]\cdot 10^{-5}$.

We present here an evaluation of $f_{0}^{\eta\pi}$ which exploits the
properties of analyticity and unitarity of form factors in QCD (e.g. [17]). In
this framework, generating the energy dependence of the form factors can be
viewed as a final-state interaction (FSI) problem. This allows us to derive
relations between the scalar form factor and the photon-photon production
amplitude $\gamma\gamma\to\eta\pi$ and also with the $\phi$ radiative decay
amplitude $\phi\to\gamma\eta\pi$, for which FSI theory can also be applied and
which are experimentally known.

##  2 Unitarity and Omnès representations of form factors

The $\tau$ decay amplitude into two pseudoscalar mesons involves the matrix
element of the charged vector current ${\langle
P_{1}(p_{1})P_{2}(p_{2})|j_{\mu}^{uq}(0)|0\rangle}$ (with
$j_{\mu}^{uq}=\bar{u}\gamma_{\mu}{q}$, $q=d,s$), which can be expressed in
terms of two independent form factors

${\langle
P_{1}(p_{1})P_{2}(p_{2})|\bar{u}\gamma^{\mu}{q}|0\rangle}=C_{12}\Big{\\{}f_{+}^{P_{1}P_{2}}(s)\Big{[}p_{1}-p_{2}-\frac{\Delta_{12}}{s}(p_{1}+p_{2})\Big{]}^{\mu}+f_{0}^{P_{1}P_{2}}(s)\,\frac{\Delta_{12}}{s}[p_{1}+p_{2}]^{\mu}\Big{\\}}$
(2)

where $s=(p_{1}+p_{2})^{2}$, $\Delta_{12}=m_{1}^{2}-m_{2}^{2}$ and $C_{12}$ is
a numerical factor (below, we will need $C_{\eta\pi^{+}}=-\sqrt{2}$,
$C_{\bar{K}^{0}{K^{+}}}=-1$). Using the Ward identity for the vector current,
the scalar form factors $f_{0}^{P_{1}P_{2}}$ can also be expressed in terms of
the matrix elements,

$C_{12}\Delta_{12}f_{0}^{P_{1}P_{2}}(s)=-{\langle
P_{1}(p_{1})P_{2}(p_{2})|(m_{d}-m_{u})\bar{u}d-eA_{\mu}{j}^{\mu,ud}|0\rangle}\
.$ (3)

For the $\pi\pi$ or $\pi{K}$ vector form factors, simple approximate
evaluations are known to hold e.g.

$f_{+}^{\pi K}(s)\simeq f_{+}^{\pi
K}(0)\Omega_{1}(s),\quad\Omega_{1}(s)=\exp\Big{(}\frac{s}{\pi}\int_{(m_{\pi}+m_{K})^{2}}^{\infty}ds^{\prime}\,\frac{\delta^{1/2}_{1}(s^{\prime})}{s^{\prime}(s^{\prime}-s)}\Big{)}$
(4)

where $\Omega_{1}$ is the Omnès function [18] and $\delta^{1/2}_{1}$ is the
$J=0$ $I=1/2$ $\pi{K}$ scattering phase shift. This is justified because
$\pi{K}$ scattering is effectively elastic in a energy region $s<\Lambda^{2}$,
with $\Lambda\simeq 1$ GeV, which includes the leading low energy resonance
$K^{*}(892)$. Multiplying $f_{+}^{\pi K}$ by the inverse of the Omnès function
removes the cut below $\Lambda$, such that in the region $s<\Lambda^{2}$ one
can write a low energy expansion: $\Omega^{-1}(s)f_{+}^{\pi K}(s)=f_{+}^{\pi
K}(0)(1+O(s/\Lambda^{2}))$. The situation is different for the $\pi\eta$
vector form factor because contributions to unitarity arise form the states
${|\gamma\pi^{+}\rangle}$, ${|\pi^{0}\pi^{+}\rangle}$ which are lighter than
the elastic threshold ${|\eta\pi^{+}\rangle}$. The contribution from the
${|\pi^{0}\pi^{+}\rangle}$ should be the dominant one below 1 GeV as it is
enhanced by the $\rho(770)$ resonance. The corresponding discontinuity reads,

${\rm
disc\,}[f_{+}^{\pi\eta}(s)]_{\pi\pi}=2i\theta(s-4{m^{2}_{\pi}})\frac{4{m^{2}_{\pi}}-s}{\sqrt{\lambda_{\eta\pi}(s)}}F_{V}^{\pi}(s)\times(t_{J=1}^{\pi^{0}\pi^{+}\to\eta\pi^{+}}(s))^{*}$
(5)

where $\lambda_{12}(s)=(s-(m_{1}+m_{2})^{2})(s-(m_{1}-m_{2})^{2})$. It
involves the pion form factor (which is well measured) and the $J=1$
projection of the $\pi\pi\to\eta\pi$ amplitude. This amplitude was evaluated
in [15] using Khuri-Treiman equations with experimental inputs on $\eta\to
3\pi$ decays.

Let us now consider the scalar form factor $f_{0}^{\pi\eta}$. In that case,
the ${|\pi^{0}\pi^{+}\rangle}$ contribution to unitarity can be neglected
because it is quadratic in isospin breaking. The important low energy scalar
resonance is the $a_{0}(980)$ which is known to couple strongly to the two
channels ${|\eta\pi^{+}\rangle}$ and ${|\bar{K}^{0}{K^{+}}\rangle}$. The
unitarity relation for the scalar form factor, including these two channels,
reads (see ref. [15] for details on the derivation)

$\begin{array}[]{ll}{\rm
Im\,}[f_{0}^{\eta\pi^{+}}(s)]&=\sigma_{\eta\pi^{+}}(s)\,f_{0}^{\eta\pi^{+}}(s)\times(t^{\eta\pi^{+}\to\eta\pi^{+}}_{J=0}(s))^{*}\\\\[8.5359pt]
&+\sigma_{\bar{K}^{0}{K^{+}}}(s)\,\dfrac{\Delta_{\bar{K}^{0}{K^{+}}}}{\sqrt{2}{\Delta_{\eta\pi}}}f_{0}^{\bar{K}^{0}{K^{+}}}(s)\times(t^{\eta\pi^{+}\to\bar{K}^{0}{K^{+}}}_{J=0}(s))^{*}\end{array}$
(6)

where $\sigma_{12}(s)=\theta(s-(m_{1}+m_{2})^{2})\sqrt{\lambda_{12}(s)}/{s}$
and a similar relation can be written for ${\rm
Im\,}[f_{0}^{\bar{K}^{0}{K^{+}}}]$. This suggests a minimal representation of
the scalar form factors analogous to eq. (4) involving an Omnès matrix instead
of a function,

$\begin{pmatrix}f_{0}^{\eta\pi^{+}}(s)\\\\[5.69046pt]
\epsilon_{KK}f_{0}^{\bar{K}^{0}{K^{+}}}(s)\end{pmatrix}\simeq\begin{pmatrix}\Omega_{11}(s)&\Omega_{12}(s)\\\\[2.84544pt]
\Omega_{21}(s)&\Omega_{22}(s)\end{pmatrix}\begin{pmatrix}f_{0}^{\eta\pi^{+}}(0)\\\\[2.84544pt]
\epsilon_{KK}f_{0}^{\bar{K}^{0}{K^{+}}}(0)\end{pmatrix}$ (7)

with111The minus sign arises because the $\Omega$ matrix is defined with
respect to isospin eigenstates and one has
${|\eta\pi^{+}\rangle}=-{|1,1\rangle}$.
$\epsilon_{KK}=-\Delta_{\bar{K}^{0}{K^{+}}}/\sqrt{2}\Delta_{\eta\pi}$. The
coupled-channel Omnès-Muskhelishvili problem cannot be solved in closed form
in terms of phase-shifts and inelasticities as in the one-channel case, but
the equations can be solved numerically in terms of the $T$-matrix elements
[19]. Furthermore, no direct experimental determinations of the phase-shifts
and inelasticities has been performed for $\eta\pi$ scattering.

##  3 The $\Omega$ matrix in $\gamma\gamma$ scattering and $\phi(1020)$
radiative decay

Cross-sections for $\gamma\gamma\to\eta\pi^{0}$ have first been measured by
the Crystal Ball collaboration at SLAC [20] and at DESY [21]. Recently, high
statistics measurements have been performed by the Belle collaboration [22].
The $\gamma\gamma\to\eta\pi^{0}$ process is described by helicity amplitudes
$L_{\lambda\lambda^{\prime}}$ which are functions of the Mandelstam variables
$s$ (the energy squared of the $\eta\pi^{0}$ pair) and $t$, $u$. In the energy
region $\sqrt{s}\mathrel{\raise 2.58334pt\hbox{$<$}\kern-7.7778pt\lower
2.79857pt\hbox{$\sim$}}1.4$ GeV the contributions of the partial waves with
$J>2$ to the cross-sections are negligibly small, while in the region
$\sqrt{s}\mathrel{\raise 2.58334pt\hbox{$<$}\kern-7.7778pt\lower
2.79857pt\hbox{$\sim$}}1.1$ GeV, the $S$-wave $l_{0++}(s)$ dominates.
Theoretical descriptions of this amplitude based on general FSI methods have
been discussed recently [23, 24]. A closely related, but somewhat less general
approach, was used earlier in ref. [25]. $l_{0++}(s)$ is an analytic function
of $s$ except for two cuts.

The right-hand cut $s>(m_{\eta}+m_{\pi})^{2}$, as in the case of the form
factor, is associated with unitarity. We will consider an energy region in
which two-channel unitarity is a reasonable approximation such that one can
write

$\begin{array}[]{l}{\rm
Im\,}[l_{0++}(s)]=\sigma_{\eta\pi^{0}}(s)\,l_{0++}(s)\,T_{11}^{*}(s)+\sigma_{K\bar{K}}(s)\,k^{1}_{0++}(s)\,T_{12}^{*}(s)\\\
{\rm
Im\,}[k^{1}_{0++}(s)]=\sigma_{\eta\pi^{0}}(s)\,l_{0++}(s)\,T_{12}^{*}(s)+\sigma_{K\bar{K}}(s)\,k^{1}_{0++}(s)\,T_{22}^{*}(s)\end{array}$
(8)

where $k^{1}_{0++}(s)$ is the $J=0$ $\gamma\gamma\to(K\bar{K})^{I=1}$
amplitude and the $T$-matrix elements associated with the states
${|\eta\pi^{0}\rangle}$, ${|K\bar{K}\rangle}$ are denoted simply as $T_{ij}$.
In addition to the right-hand cut, the amplitudes $l_{0++}(s)$,
$k^{1}_{0++}(s)$ have a left-hand cut on the negative real axis $-\infty<s\leq
0$. There are no other singularities. The amplitudes must also satisfy Low’s
soft photon theorem [26] which states that

$\lim_{s\to 0}l_{0++}(s)=0,\quad\lim_{s\to 0}\bar{k}^{1}_{0++}(s)=0$ (9)

where $\bar{k}^{1}_{0++}\equiv k^{1}_{0++}(s)-k^{1,Born}_{0++}(s)$ and
$k^{1,Born}_{0++}$ is the $I=1$ projection of the amplitude part induced by
the Kaon pole in the $t$ and $u$ channels, which coincides with the QED Born
amplitude

$k^{1,Born}_{0++}(s)=-\frac{2\sqrt{2}{m}^{2}_{K^{+}}}{s\sigma_{K^{+}}(s)}\log\frac{1+\sigma_{K^{+}}(s)}{1-\sigma_{K^{+}}(s)}\
.$ (10)

and $\sigma_{K^{+}}(s)=\sqrt{1-4{m^{2}_{K^{+}}}/s}$. Knowing the location of
the singularities allows one to write Cauchy dispersive integral
representations 222It is also necessary to know the asymptotic behaviour. It
can be argued [24] that $|l_{0++}(s)|$, $|k^{1}_{0++}(s)|$ are bounded by
$\sqrt{s}$ when $s\to\infty$. for $l_{0++}$ and $k^{1}_{0++}$. Combining these
with the unitarity relations (8) leads to a set of Muskhelishvili-type
singular equations for the amplitudes. The solutions, taking into account the
soft photon constraints, can be expressed in the following form in terms of
the Omnès matrix [24]

$\begin{pmatrix}l_{0++}(s)\\\\[4.26773pt] k^{1}_{0++}(s)\\\
\end{pmatrix}=\begin{pmatrix}0\\\\[4.26773pt] k^{1,Born}_{0++}(s)\\\
\end{pmatrix}+s\begin{pmatrix}{\Omega_{11}(s)}&{\Omega_{12}(s)}\\\\[4.26773pt]
{\Omega_{21}(s)}&{\Omega_{22}(s)}\\\
\end{pmatrix}\begin{pmatrix}{b_{l}}+L_{1}^{V}(s)+R_{1}^{Born}(s)\\\\[4.26773pt]
{b_{k}}+L_{2}^{V}(s)+R_{2}^{Born}(s)\\\ \end{pmatrix}\ .$ (11)

In these equations the functions $R_{i}^{Born}$ are integrals involving the
Born amplitude,

$R_{i}^{Born}(s)=-\frac{s-s_{A}}{\pi}\int_{4{m^{2}_{K^{+}}}}^{\infty}\frac{ds^{\prime}}{s^{\prime}(s^{\prime}-s_{A})(s^{\prime}-s)}{\rm
Im\,}[(\Omega^{-1})_{i2}(s^{\prime})]k^{1,Born}_{0++}(s^{\prime})$ (12)

where $s_{A}$ can conveniently be taken to be equal to the Adler zero of the
amplitude $l_{0++}$ ($s_{A}\simeq m^{2}_{\eta}$) and the functions
$L_{i}^{V}(s)$ are integrals over the left-hand cut involving ${\rm
Im\,}[l_{0++}(s^{\prime})]$, ${\rm Im\,}[\bar{k}^{1}_{0++}(s^{\prime})]$.
These imaginary parts are approximated by the contributions induced by the
light vector meson poles in the $t$ (or $u$) channels, i.e.

${\rm Im\,}[l_{0++}(s^{\prime})]\simeq\sum_{V=\rho,\omega,\phi}{\rm
Im\,}[l^{V}_{0++}(s^{\prime})],\quad{\rm
Im\,}[\bar{k}^{1}_{0++}(s^{\prime})]\simeq{\rm
Im\,}[\bar{k}^{1,K^{*}}_{0++}(s^{\prime})],\quad(s^{\prime}\leq 0)\ .$ (13)

As before, the contributions from the higher energy regions of the cuts, where
the approximations made are no longer accurate, are absorbed into a set of
subtraction parameters. Eq. (11) involves two such parameters $b_{l}$,
$b_{k}$.

The representation (11) for the $S$-wave $\gamma\gamma$ amplitudes,
supplemented with a simple Breit-Wigner model for the $D$-waves, was compared
to the experimental data on photon-photon scattering to $\eta\pi$ cross-
sections in [23, 24]. In ref. [24] experimental data on $\gamma\gamma\to
K_{S}K_{S},{K^{+}}{K^{-}}$ was considered as well, which allows to constrain
not only the constants $b_{l}$, $b_{k}$ but also the $\Omega$ matrix. For that
purpose, a previously proposed six parameters model for the underlying
$T$-matrix [27] was used, from which $\Omega$ is computed by solving
numerically the related Muskhelishvili-Omnès equations. Fig. (1) shows an
illustrative comparison of cross sections computed from this fitted
theoretical model with the experimental ones.

Let us now consider the case where one photon has a non-vanishing virtuality
${q^{2}}$, and label the $S$-wave amplitudes as $l_{0++}(s,{q^{2}})$,
$k^{1}_{0++}(s,{q^{2}})$. These amplitudes are analytic functions of the
energy $s$ with two cuts (there are no anomalous thresholds), such that one
can write a dispersive representation in terms of the Omnès matrix, exactly as
before,

$\begin{pmatrix}l_{0++}(s,{q^{2}})\\\
k_{0++}(s,{q^{2}})\end{pmatrix}\\!=\\!\begin{pmatrix}0\\\
\tilde{k}^{1,Born}(s,{q^{2}})\end{pmatrix}\\!\\!+(s-{q^{2}})\begin{pmatrix}\\!\\!{\Omega_{11}}(s)&{\Omega_{12}}(s)\\\
{\Omega_{21}}(s)&{\Omega_{22}}(s)\end{pmatrix}\\!\\!\begin{pmatrix}{a_{1}}({q^{2}})+I_{1}^{LC}(s,{q^{2}})+I_{1}^{RC}(s,{q^{2}})\\\
{a_{2}}({q^{2}})+I_{2}^{LC}(s,{q^{2}})+I_{2}^{RC}(s,{q^{2}})\end{pmatrix}\ $
(14)

satisfying the soft photon theorem which now corresponds to $s={q^{2}}$. We
will be interested in the case of timelike virtualities and
${q^{2}}>(m_{\eta}+m_{\pi})^{2}$ which corresponds experimentally to the
$e^{+}e^{-}\to\gamma^{*}\to\gamma\eta\pi$ processes (and also, up to an
isospin rotation, to the vector current part of the radiative $\tau$ decays
$\tau^{+}\to\gamma\eta\pi^{+}\nu,\gamma\bar{K}^{0}K^{+}\nu$). Some care must
be taken in order to properly utilize eq. (14) because the pole singularity of
the Born amplitude at $s={q^{2}}$ occurs within the range of integration in
the integrals $I^{RC}_{i}$. It is convenient to separate this singular part
(labelled $\tilde{k}^{1,Born}$ in (14) )and perform the related integrations
analytically. Furthermore, the left-hand cut now has complex components as
well as a real component which partly overlaps with the unitarity cut (which
leads to a violation of the Fermi-Watson phase theorem). Details on how to
properly compute the left-cut integrals $I_{i}^{LC}$ can be found in [28]). In
practice, eqs. (14) where implemented with ${q^{2}}\simeq m^{2}_{\phi}$
enabling to determine the $\phi(1020)\to\gamma\eta\pi$ decay amplitude. As in
the ${q^{2}}=0$ case, the discontinuities across the left-hand cut components
are modelled based on the contributions from the light vector mesons $\rho$,
$\omega$, $\phi$ and $K^{*}$. Fig. (1) shows that a rather good description of
the accurate experimental data [29, 30] can be obtained with a two parameter
fit, using the $\Omega$ matrix as determined previously from the
$\gamma\gamma$ scattering fit.

Figure 1: Comparison of experimental data with Omnès-based $S$-wave amplitude
models, left: $\gamma\gamma\to\pi\eta$ cross-sections, middle:
$\gamma\gamma\to K_{S}K_{S}$ cross-sections, right: $\phi\to\gamma\eta\pi$
energy distribution.

##  4 Chiral inputs and results

In the minimal modelling, once the $\Omega$ matrix is known the scalar form
factors are given in terms of their values at $s=0$. Since
$f_{0}^{\bar{K}^{0}{K^{+}}}(0)$ appears multiplied by an isospin breaking
factor, we can simply set $f_{0}^{\bar{K}^{0}{K^{+}}}(0)=1$, because the
chiral corrections are $O(m_{d}-m_{u},e^{2})$. The chiral expansion of
$f_{0}^{\eta\pi^{+}}(0)$ takes the following form [12]

$f_{0}^{\eta\pi^{+}}(0)=\epsilon\times(1+O(p^{2}))+\frac{O(e^{2}p^{2})}{m^{2}_{\eta}-{m^{2}_{\pi}}},\quad\epsilon=\frac{\sqrt{3}(m_{d}-m_{u})}{4(m_{s}-\hat{m})}$
(15)

with $\hat{m}=(m_{d}+m_{u})/2$. The value of $\epsilon$ from the FLAG revue
[31] is $\epsilon=1.137(45)\cdot 10^{-2}$. The $O(p^{2})$ correction involves
the chiral couplings $L_{7}$, $L_{8}$ which can be computed using the chiral
expansions of the pseudo-scalar meson masses at NLO and the value the quark
mass ratio $m_{s}/\hat{m}$ [32], which is known from lattice QCD simulations:
$m_{s}/\hat{m}=27.42(12)$ (see [31]). The couplings which appear in the
$O(e^{2}p^{2})$ part in (15) are not precisely known, they can be estimated
approximately using resonance saturation modelling [33, 34]. Finally, from
this NLO chiral expansion, one obtains

$\left.f_{0}^{\eta\pi^{+}}(0)\right|_{\chi{PT}}=(1.40\pm 0.14)\cdot 10^{-2}\
,$ (16)

assuming a 50% error on the chiral corrections. An alternative estimate can be
performed based on a relation, which is exact at NLO, with the $K^{+}_{l3}$
and $K^{0}_{l3}$ form factors ratio $\Delta_{SU(2)}$
$\equiv{f_{0}^{{K^{+}}\pi^{0}}(0)}/{f_{0}^{{K^{0}}\pi^{-}}(0)}-1$ [12]. The
experimental value $\Delta_{SU(2)}$ $=(2.73\pm 0.41)\cdot 10^{-2}$ [35] gives
$f_{0}^{\eta\pi^{+}}(0)|_{\chi{PT}+K_{l3}}$ $=(1.51\pm 0.24)\cdot 10^{-2}$
which is compatible with eq. (16). Using the values at $s=0$ from $\chi$pt the
result for $f_{0}^{\eta\pi}(s)$ from the minimal Omnès representation is
illustrated in Fig (2) (solid black line on the left plot). Obviously, this
minimal model cannot be very precise. In fact, computing the derivatives of
the form factors at $s=0$ one finds substantial deviations from the $O(p^{4})$
$\chi$pt values:
$\dot{f}_{0}^{\eta\pi}(0)|_{min}/\dot{f}_{0}^{\eta\pi}(0)|_{\chi{pt}}\sim
0.5$,
$\dot{f}_{0}^{K\bar{K}}(0)|_{min}/\dot{f}_{0}^{K\bar{K}}(0)|_{\chi{pt}}\sim
1.6$. The values of the derivatives can be corrected by replacing
$f_{0}^{12}(0)$ in eq. (7) by a linear polynomial
$f_{0}^{12}(0)(1+\lambda_{12}s)$. Physically, it is plausible that two-channel
unitarity is insufficient in the region of the $a_{0}(1450)$ resonance.
Assuming an effective third channel, this suggests the following simple
improvement over the minimal model

$f_{0}^{\eta\pi}(s)=\left.f_{0}^{\eta\pi}(s)\right|_{min}+\Omega_{13}(s)\lambda_{13},\quad\Omega_{13}(s)\simeq\frac{s}{m^{2}_{a_{0}^{\prime}}-s-im_{a_{0}^{\prime}}\Gamma_{a_{0}^{\prime}}}$
(17)

in which the parameter $\lambda_{13}$ is adjusted such that the $\chi$pt
derivative at $s=0$ is reproduced. The effects of these various ways to modify
the derivatives at $s=0$ are illustrated in Fig. (2).

Figure 2: Left plot: absolute value of the scalar form factor
$f_{0}^{\eta\pi}$ from the minimal two-channel Omnès model (black solid line)
and from non-minimal models a) imposing the value of the derivative at $s=0$
via a linear polynomial, b) imposing this value via a third channel. The right
plot shows the energy distribution of the $\tau\to\eta\pi\nu$ branching
fractions. The black solid line corresponds to the scalar form factor and blue
error band is generated by varying both the input values at $s=0$ and those of
the derivatives. The red lines correspond to the vector form factor (see
text).

The energy dependence of the $\tau\to\eta\pi\nu$ branching fraction associated
to the scalar form factor is shown in Fig. (2) (right). Also shown for
comparison is the branching fraction associated with the vector form factor.
The dotted red line is the result from ref. [15] in which the amplitude
$t_{J=1}^{\pi^{0}\pi^{+}\to\eta\pi^{+}}$ is determined using $\eta\to 3\pi$
inputs from ref. [36]. The solid red line is an update which uses the more
recent results from [37]. One sees a substantial difference between these two
evaluations which gives a measure of the error. Finally, integrating over the
$\pi\eta$ energy we find the following results for the branching fractions

$BF_{S}=(1.8\pm 0.7)\cdot 10^{-6},\quad BF_{V}=(2.6\pm 1.30)\cdot 10^{-6}\ .$
(18)

##  5 Conclusions

We have proposed a determination of the scalar form factor component of the
$\tau\to\eta\pi\nu$ amplitude based on a two-channel Omnès matrix constrained
by photon-photon scattering data and shown to be compatible with $\phi$
radiative decay data. The result for the branching fraction $BF_{S}$ (eq.
(18)) should be more precise than the one given previously in [15] 333In that
reference and idea proposed in ref. [38] was followed, which consists in
making estimates of the behavior of the phase of the form factor in the
inelastic region using the QCD asymptotic constraint.. We have also presented
an update on the branching fraction generated by the vector form factor. If
our results are correct, then the forthcoming search for this rare $\tau$
decay mode at Belle II might not be able to clearly see it. However, these new
measurements should be able to rule out some of the theoretical models.

## Acknowledgements

This work is supported by the European Union’s Horizon2020 research and
innovation programme (HADRON-2020) under the Grant Agreement n∘ 824093.

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